New results and methods in the study of some models described by partial differential equations Cosma (Barbu) Elena Luminit ¸a Faculty of Mathematics… [619781]

Licență

New results and methods in the study of some models described by partial differential equations Cosma (Barbu) Elena Luminit ¸a Faculty of Mathematics… [619781]

Byadmin ianuarie 1, 2024

Habilitation Thesis
New results and methods in the study of some models
described by partial differential equations
Cosma (Barbu) Elena Luminit ¸a
Faculty of Mathematics and Informatics, Ovidius University of Constant ¸a
2018

Abstract
The purpose of this habilitation thesis is to present the author’s main scientiﬁc results, as well as
future research prospects. The author’s work in the ﬁeld of singularly perturbed problems beganwhen she started writing her doctoral thesis, Asymptotic analysis of some boundary value problems
with small parameter , which was defended in 1998, at the Simion Stoilow Institute of Mathematics
of the Romanian Academy, and it is still ongoing. During the last few years, maximum principlesfor P(ayne)–functions and their applications regarding partial differential equations were also addedto the author’s ﬁeld of research. The results highlighted in this thesis belong to these two ﬁelds ofresearch and they were published after 1998, the year when the Ph.D thesis was defended.
The theory of singular perturbations is a very wide domain in mathematics, with numerous
subdivisions, whose study involved signiﬁcants efforts. A. N. Tikhonov ([95], [96], [97]) markedthe beginning of the systematic study of the singular perturbation theory. In 1957, M. I. Vishik andL. A. Lyusternik [102] studied a linear partial differential equations with singular perturbations,introducing the famous method which is today called the Vishik-Lyusternik method . Afterwards,
an entire literature has been devoted to this subject: A. B. Vasilieva-V . F. Butuzov-L. V . Kalachev[100], W. Eckhaus [47], J. L. Lions [69], R. E. O’Malley [78], [79] and the list may go on.
In order to underline the importance of the method that was previously mentioned, we are going
to brieﬂy discuss one of its many applications. It is well known that a lot of real life phenomenacan be described by boundary value problems associated with various types of partial differentialequations or systems. Most of the time, the mathematical model is simpliﬁed, so that it can be more
easily investigated. For instance, if certain terms are negligible compared to others, they will beignored. The initial model, which contains these small parameters, is called the perturbed model ,
whereas the simpliﬁed model is called the reduced model . We are interested to know if the latter
can be used to describe faithfully enough the respective phenomenon, which means that its solutionmust be close enough to the solution of the corresponding perturbed model.Therefore, there are two
types of perturbations: regular and singular. Regular perturbations determine small variations of
the solution, while singular perturbations cause signiﬁcant variations of the solution, at least in
one portion of its deﬁnition domain.
The ﬁrst chapter of the present thesis includes asymptotic analysis of some singularly bound-
ary value problems and it is based on the following papers: [10], [14], [16], [17], [23]. In the ﬁrstsection we present the Vishik-Lyusternik method and we recall some basic concepts and results inthe theory of evolution equations associated with monotone operators which will be used in the nextsections. The ﬁrst problem, investigated in Section 1.2, is related to the theory of electrical and elec-tronics circuits. The mathematical model is so called telegraph system , associated with the initial
conditions and different types of boundary conditions, depending on the physical phenomenon un-
ii

Abstract
der discussion. The cases of algebraic–differential and differential–differential boundary conditions
were studied by the author in her Ph.D thesis. At that time, algebraic–algebraic boundary conditions
(which, in fact, represent nonlinear Ohm’s laws at the ends of the circuit) were left unsolved. This
type of boundary conditions is considered in Section 1.2. The small parameter denoted by ε, appears
in the ﬁrst equation of the system and it represents the speciﬁc inductance. The problem that is inves-tigated is of hyperbolic type and it represents the perturbed model, denoted by P
ε.If we put ε=0,
we get the reduced model, P0,which turns out to be a boundary value problem of parabolic type. It is
noteworthy to mention that such parabolic models, extended to the case of ntelegraph systems, con-
nected by means of some appropriate boundary conditions, can be used for many integrated circuits,in which the distributed inductances of the conductive layers are negligible (see C. A. Marinov-P.Neittaanmaki [73, Sections 5.3. and 6.1.]). However, from a physical standpoint, εis just small, not
equal to zero. That is why we have to ask ourselves whether or not the resulting parabolic modelP
0still describes the physical phenomenon properly. Using the Vishik-Lyusternik method, a formal
asymptotic expansion of order zero for the solution of Pεis obtained. The presence of a corrector
function shows that the reduced model is not valid in all the deﬁnition domain of the solution, DT,a t
least with respect to the norm of uniform convergence in DT(the C–norm). In order to validate the
determined asymptotic expansion we have proven some results concerning the existence, uniquenessand higher regularity of the solutions to problems P
εandP0. Finally, we derived some estimates for
the remainder terms of the expansion.It is important to emphasize the fact that other singularly per-turbed problems such as the case of n, even semilinear, telegraph systems, with various boundary
conditions, can be investigated in a similar manner. The last section of this chapter is dedicated tothe study of some singularly perturbed (with respect to the uniform convergence norm) parabolic–parabolic problems, considered in two subdomains of a given domain, with transmission conditionsat the interface, which represent models for one– dimensional convection–diffusion processes. Inthese cases, the small parameter εrepresenting the diffusion coefﬁcient in a subdomain of the spa-
tial domain. Depending on the chosen boundary conditions (related to the physical phenomenon thatwe wish to describe), we are going to study three types of such problems. Each subsection of Sec-tion 1.3 is dedicated to the study of one of the three problems, which follows the same steps as theone discussed in the previous section. Let us also point out that the stationary cases of the ﬁrst twoproblems (from Subsections 1.3.1 and 1.3.2) were studied by F. Gastaldi-A. Quarteroni [55], as aﬁrst step in the study of more complex problems in Fluid Mechanics. More precisely, F. Gastaldi-A.Quarteroni’s article, reduced models are taken as starting points, while perturbed models are seen astheir elliptical regularization, and can be used to obtain appropriate transmission conditions, whichare necessary in the numerical treatment.
While in Chapter 1 the possibility to replace some singular perturbation problems with the
corresponding reduced models is discussed, in Chapter 2 we aim at reversing the process in thesense that we replace certain problems with singularly perturbed, higher order (with respect to t)
problems, admitting solutions which are more regular and approximate the solutions of the originalproblems. Thus, the ﬁrst part of Chapter 2 is focused on elliptical regularizations (Lions regular-
izations) of the abstract Cauchy problem (P
0),in a Hilbert space H,which consists of equation (E):
u/prime(t)+Au(t)+Bu(t)=f(t),t∈[0,T]and the initial condition (IC): u(0)=u0,where T>0,u0is
a given initial state, f:[0,T]→His a given function, A:D(A)⊂H→His assumed to be a linear,
iii

Abstract
maximal monotone operator and B:H→His a nonlinear operator. We will also consider the equa-
tion(Eε):−εu/prime/prime(t)+u/prime(t)+Au(t)+Bu(t)=f(t),t∈[0,T],where ε>0 is a small parameter, with
which we associate either of the following boundary conditions: (BC.1): u(0)=u0,u(T)=uT,
(BC.2.): u(0)=u0,u/prime(T)=0.We denote by (P.k)εthe problem (Eε)-(BC.k),k=1,2. Both these
problems are elliptic regularizations of the original problem (P0). Such kind of approximate prob-
lems (regularizations) have been suggested by J. L. Lions in his monograph on singular perturbations[69]. Lions’ reasoning was that such regularizations provide more regular solutions with respect to
t, which approximate the solution of the original problem (P
0)asε→0.Due to the additional term
−εu/prime/prime,this method was called by J. L. Lions the method of artiﬁcial viscosity . One other use of it
was previously mentioned, in regard to the numerical treatment of more complex coupled problemsin Fluid Mechanics. Therefore, in Section 2.1 we are going to establish a zeroth order asymptoticexpansion of the solution u
εof(P.1)ε.If the operator Ais a linear maximal operator, and Bis Lips-
chitzian on bounded subsets of Hand monotone or Bis not necessarily monotone but is Lipschitzian
onH, we get the results regarding the existence and uniqueness for the solutions of problems (P.1)ε
and(P0),and in the end we obtain that uε,corrected by a boundary layer function, approximates
the solution of (P0)with respect to the sup norm of C([0,T];H).Asymptotic analysis of problem
(P.2)ε, which is regularly perturbed of order zero with respect to the same norm, unlike (P.1)ε, had
similar results. The hypotheses considered for the two operators here, are as follows: operator Ais
only maximal monotone, while operator Bis Lipschitzian on H. The last part of the chapter presents
asymptotic analysis of some elliptic regularizations of type (P.1)ε, for semiliniar telegraph system
and semilinear heat equations, respectively, with nonlinear boundary conditions. In these cases, op-erator Ais nonlinear; consequently, the abstract results from Section 2.1 are not directly applicable.
This chapter is based on the articles: [19], [20], [9] and [8], which contain substantial generalizationsof the results from [1], [2], [4] and [5], respectively.
Chapter 3 includes the proofs of some maximum principles for certain appropriate P−func-
tions which enable us to study some overdetermined problems and to obtain a Liouville type result.More speciﬁcally, in Section 3.1, we extend some maximum principles obtained by G. Philippin-S.Safoui [82], [83], G. Porru-A. Safoui-S. Vernier-Piro [84] and C. Enache [48], to a class of fullynonlinear elliptic equations, including p-Hessian equations and Weingarten equations. At the end ofthis section we use this new maximum principle in order to obtain some inequalities for the solutionand its gradient. In Section 3.2 we consider a general class of quasilinear anisotropic equations. Weﬁrst derive some maximum principles for two appropriate P−functions. These maximum principles
are then employed to obtain a Liouville-type result and a Serrin–Weinberger-type symmetry result.In the last two sections, using similar ideas we derive certain generalizations of results related tosome free (overdetermined) boundary value problems in electrostatic and potential theory, obtainedby E. Sartori [88] and L.E. Payne-G.A. Philippin [81], respectively. Articles [11], [12] and [13] formthe basis for this chapter.
The project concerning the subsequent development of the author’s academic and scientiﬁc
career are presented in the last chapter of this thesis.
iv

Rezumat
Obiectivul acestei teze de abilitare este prezentarea principalelor rezultate s ¸tiint¸iﬁce, precum s ¸i vi-
itorul plan de cercetare al autoarei. Aceste cercet ˘ariˆın domeniul problemelor singular perturbate au
ˆınceput odat ˘a cu preg ˘atirea tezei de doctorat, cu titlul Analiza asimptotic ˘a a unor probleme la limit ˘a
cu parametru mic , sust¸inut ˘aˆın anul 1998 la Institultul de Matematic ˘a al Acadmiei Rom ˆaneSimion
Stoilow ,s¸i au continuat p ˆanaˆın prezent. La acestea s-au ad ˘augat ˆın ultimii ani cercet ˘arile legate
de principii de maxim pentru P(ayne)-funct ¸ii s¸i aplicat ¸iile acestora ˆın studiul ecuat ¸iilor cu derivate
part¸iale. Rezultatele prezentate sunt din aceste dou ˘a domenii de cercetare, publicate dup ˘a anul 1998,
anul sust ¸inerii tezei de doctorat.
Teoria perturbat ¸iilor singulare este un domeniu matematic foarte vast, cu numeroase subdivi-
ziuni ˆın studiul c ˘arora s-au f ˘acut eforturi considerabile. A. N. Tikhonov ([95], [96], [97]) este cel
care a ˆınceput studiul sistematic al teoriei perturbat ¸iilor singulare. ˆIn anul 1957, M. I. Vishik s ¸i L. A.
Lyusternik [102] au studiat o ecuat ¸ie cu derivate part ¸iale liniar ˘a, introduc ˆand celebra metod ˘a cunos-
cut˘a azi sub numele de metoda Vishik-Lyusternik . Ulterior, o ˆıntreag ˘a literatur ˘a a fost dedicat ˘a
acestui subiect: A. B. Vasilieva-V . F. Butuzov-L. V . Kalachev [100], W. Eckhaus [47], J. L. Lions[69], R. E. O’Malley [78], [79] s ¸i lista poate continua.
Pentru a preciza important
¸a metodei ment ¸ionate anterior vom face c ˆateva preciz ˘ari privind
aplicabilitatea acesteia. Este bine cunoscut faptul c ˘a cele mai multe fenomene din viat ¸a real ˘a pot ﬁ
descrise prin probleme la limit ˘a asociate unor ecuat ¸ii sau sisteme de ecuat ¸ii cu derivate part ¸iale. De
cele mai multe ori, modelul matematic este simpliﬁcat pentru a putea ﬁ studiat. Dac ˘a, de exemplu,
anumit ¸i termeni sunt neglijabili ˆın comparat ¸ i ec ua l t ¸ii, aces ¸tia vor ﬁ omis ¸i. Modelul init ¸ial care
cont¸ine aces ¸ti parametri mici poart ˘a numele de model perturbat , iar modelul simpliﬁcat se va
numi model redus . Ceea ce ne intereseaz ˘a este dac ˘a solut ¸ia acestuia din urm ˘a aproximeaz ˘a solut ¸ia
modelului perturbat. Exist ˘a, astfel, dou ˘a tipuri de perturbat ¸ii, regulate s¸isingulare . Cele regulate
produc variat ¸ii mici ale solut ¸iei, iar cele singulare produc modiﬁc ˘ari semniﬁcative ale solut ¸iei, cel
put¸inˆıntr-o port ¸iune a domeniului de deﬁnit ¸ie al acesteia.
Primul capitol al acestei lucr ˘ari cuprinde analiza asimptotic ˘a a unor probleme la limit ˘a sin-
gular perturbate s ¸i se bazeaz ˘ap eu r m ˘atoarele articole ( ˆın colaborare) ale autoarei: [10], [23], [14],
[17], [16]. ˆIn prima sect ¸iune se introduc c ˆateva deﬁnit ¸ii s¸i rezultate fundamentale speciﬁce metodei
Vishik-Lyusternik s ¸i teoriei ecuat ¸iilor de evolut ¸ieˆın spat ¸ii Hilbert, asociate operatorilor maximali
monotoni, ce vor ﬁ utilizate pe parcursul tezei. Prima problem ˘a, studiat ˘aˆın Sect ¸iunea 1.2, este
legat ˘a de teoria circuitelor electrice s ¸i electronice. Modelul matematic este as ¸a-numitul sistem al
telegraﬁs ¸tilor, c ˘aruia i se asociaz ˘a condit ¸iile init ¸iale s ¸i diverse tipuri de condit ¸ii la limit ˘a,ˆın funct ¸ie
de fenomenul ﬁzic descris. Cazurile condit ¸iilor la limit ˘a de tip algebric-diferent ¸ial sau diferent ¸ial-
diferent ¸ial au fost studiate de c ˘atre autoare ˆın teza de doctorat. La acel moment, condit ¸iile la limit ˘a
v

Rezumat
neliniare, de tip algebric-algebric (care reprezint ˘a, de fapt, legile lui Ohm neliniare la capetele cir-
cuitului) au r ˘amas nesolut ¸ionate. Acest tip de condit ¸ii la limit ˘a este considerat ˆın sect ¸iunea am-
intit˘a. Parametrul mic, notat cu ε,apare ˆın prima ecuat ¸ie a sistemului s ¸i reprezint ˘a inductant ¸a spe-
ciﬁc˘a. Problema prezentat ˘a, care este de tip hiperbolic, este modelul perturbat, pe care l-am nu-
mitPε.Luˆandε=0, obt ¸inem modelul redus, P0,care se dovedes ¸te a ﬁ o problem ˘a la limit ˘ad e
tip parabolic (trebuie precizat faptul c ˘a astfel de modele parabolice, extinse la cazul a nsisteme
ale telegraﬁs ¸tilor, conectate prin condit ¸ii la limit ˘a speciﬁce, se aplic ˘a multor circuite digitale inte-
grate, ˆın care inductant ¸a distribuit ˘a este neglijabil ˘a, cf. C. A. Marinov-P. Neittaanmaki [73, Sections
5.3. and 6.1.]). Totus ¸i, parametrul ε, din punct de vedere ﬁzic, este doar mic, nu egal cu zero. De
aceea trebuie s ˘an e ˆıntreb ˘am dac ˘a solut ¸ia modelului redus aproximeaz ˘a suﬁcient de bine solut¸ia
modelului perturbat. Folosind metoda lui Vishik-Lyusternik, am construit o dezvoltare asimptotic ˘a
de ordinul zero pentru solut ¸ia lui Pε.Prezent ¸a unei funct ¸ii corectoare arat ˘ac˘a modelul redus nu
este de preferat ˆın tot domeniul de deﬁnit ¸ie, notat DT,al solut ¸iilor, cel put ¸inˆın raport cu norma
convergent ¸ei uniforme pe DT.Pentru validarea dezvolt ˘arii asimptotice determinate, am demonstrat
existent ¸a, unicitatea s ¸i regularitatea solut ¸iilor celor dou ˘a modele, perturbat s ¸i redus, iar ˆın ﬁnalul
sect¸iunii am obt ¸inut estim ˘ari ale restului dezvolt ˘ariiˆın raport cu norma convergent ¸ei uniforme pe
DT.Este important s ˘a subliniem faptul c ˘a multe alte probleme singular perturbate, cum ar ﬁ cazul a
nsisteme hiperbolice, chiar semiliniare, ale telegraﬁs ¸tilor, cu diverse condit ¸ii la limit ˘a, pot ﬁ sudiate
ˆıntr-o manier ˘a similar ˘a. Ultima sect ¸iune a acestui capitol este dedicat ˘a studiului unor probleme la
limit ˘a cuplate, considerate ˆın dou ˘a subdomenii ale unui domeniu dat, cu condit ¸ii de transmitere la
interfat ¸˘a, de tip parabolic-parabolic, singular perturbate ( ˆın raport cu norma convergent ¸ei uniforme),
care reprezint ˘a modele ale unor procese de tip convect ¸ie-difuzie 1-dimensionale. ˆIn cazul acestora,
parametrul mic εse datoreaz ˘a faptului c ˘a difuzia este considerat ˘a neglijabil ˘a pe un subdomeniu al
domeniului spat ¸ial.ˆIn funct ¸ie de condit ¸iile la limit ˘a alese (corespunz ˘atoare fenomenului ﬁzic con-
siderat), vom studia trei tipuri de astfel de probleme. Fiecare subsect ¸iune a Sect ¸iunii 1.3 este dedicat ˘a
analizei asimptotice a uneia din cele trei probleme, care urmeaz ˘a aceias ¸i pas ¸i cu cea considerat ˘aˆın
sect¸iune anterioar ˘a. Trebuie s ˘a preciz ˘am faptul c ˘a variantele stat ¸ionare ale primelor dou ˘a probleme
(din Subsect ¸iunile 1.3.1 s ¸i 1.3.2) au fost utilizate de c ˘atre F. Gastaldi-A. Quarteroni [55], ca un prim
pasˆın direct ¸ia studierii unor probleme cuplate mai complexe din Mecanica Fluidelor. Mai exact, ˆın
articolul respectiv, modelele reduse sunt considerate ca probleme de pornire, ˆın timp ce modelele
perturbate sunt privite ca regulariz ˘ari eliptice ale acestora, ﬁind utile pentru obt ¸inerea condit ¸iilor de
transmisie necesare ˆın abordarea numeric ˘a a modelelor reduse.
Dac˘aˆın Captitolul 1 am studiat posibilitatea ˆınlocuirii unor probleme singular perturbate cu
modelele reduse corespunz ˘atoare, ˆın Captitolul 2 ne propunem s ˘a invers ˘am acest procedeu, ˆın sen-
sul c ˘av o m ˆınlocui anumite probleme, cu probleme singular perturbate, de ordin mai mare ( ˆın ra-
port cu t), care admit solut ¸ii mai regulate s ¸i care aproximeaz ˘a solut ¸iile problemelor init ¸iale. Astfel,
prima parte a Capitolul 2 este dedicat ˘a sudiului unor regulariz ˘ari eliptice (sau de tip Lions) ale
problemei Cauchy abstracte (P0),ˆıntr-un spat ¸iu Hilbert H,format ˘a din ecuat ¸ia(E): u/prime(t)+Au(t)+
Bu(t)=f(t),t∈[0,T]s¸i condit ¸ia init ¸ial˘a(IC): u(0)=u0,unde T>0,u0este un element din H,
f:[0,T]→Heste o funct ¸ie dat ˘a,A:D(A)⊂H→Heste un operator liniar, maximal monoton s ¸i
B:H→Heste un operator neliniar deﬁnit pe H. V om considera de asemenea ecuat ¸ia de ordinul al
doilea (Eε):−εu/prime/prime(t)+u/prime(t)+Au(t)+Bu(t)=f(t),t∈[0,T],unde ε>0 este un parametru mic,
vi

Rezumat
cu condit ¸iile la limit ˘a(BC.1): u(0)=u0,u(T)=uTsau(BC.2.): u(0)=u0,u/prime(T)=0.S˘a not ˘am
cu(P.k)εproblemele (Eε)-(BC.k),k=1,2. Acestea sunt regul ˘ariz˘ari eliptice ale problemei init ¸iale
(P0). Astfel de aproxim ˘ari (regulariz ˘ari) au fost introduse de J. L. Lions ˆın monograﬁa dedicat ˘a
perturbat ¸iilor singulare [69]. Motivat ¸ia lui Lions a fost aceea c ˘a aceste probleme furnizeaz ˘a solut ¸ii
mai regulate ˆın raport cu t,care aproximeaz ˘a solut ¸ia lui (P0)cˆandε→0. Datorit ˘aa d˘aug˘arii termenu-
lui−εu/prime/prime,aceast ˘a metod ˘a introdus ˘a de J. L. Lions este numit ˘as¸imetoda v ˆascozit ˘at¸ii artiﬁciale .O
alt˘a utilizare a acesteia a fost precizat ˘a anterior, referindu-se la tratarea numeric ˘a a unor probleme
din Mecanica Fluidelor. Astfel, ˆın Sect ¸iunea 2.1 vom stabili o dezvoltare asimptotic ˘a de ordinul zero
a solut ¸ieiuεa problemei (P.1)ε, care este singular perturbat ˘aˆın raport cu norma din C([0;T];H).
Dac˘aAeste operator liniar, maximal monoton, s ¸iBeste Lipschitz pe Hsau monoton s ¸i local Lip-
schitz, sunt prezentate rezultate de existent ¸˘as¸i unicitate pentru solut ¸iile problemelor (P.1)εs¸i(P0),
iarˆın ﬁnal este obt ¸inut ˘a estimarea /bardblrε/bardblC([0,T];H)=O(ε1/4),unde rεeste restul dezvolt ˘arii. Rezultate
similare au fost demonstrate s ¸i pentru dezvoltarea asimptotic ˘a a problemei (P.2)ε,care, spre deose-
bire de primul caz, este regulat perturbat ˘a de ordin zero ˆın raport cu norma spat ¸iului C([0,T];H).
Ipotezele considerate asupra celor doi operatori sunt ˆın acest caz urm ˘atoarele: operatorul Aeste
maximal monoton, iar Beste operator Lipschitz pe H. Ultima parte a capitolului cont ¸ine analiza
asimptotic ˘a a unor regulariz ˘ari eliptice de tipul (P.1)εpentru sistemului semiliniar al telegraﬁs ¸tilor
s¸i ecuat ¸ia semiliniar ˘aac ˘aldurii cu condit ¸ii la limit ˘a neliniare, pentru care operatorul Aeste neliniar;
ca urmare, rezultatele abstracte obt ¸inute ˆın Sect ¸iunea 2.1 nu pot ﬁ aplicate direct. Acest capitol se
bazeaz ˘a pe articolele ( ˆın colaborare) ale autoarei: [19], [20], [9] s ¸i [8], reprezent ˆand generaliz ˘ari
substant ¸iale ale unor rezultate obt ¸inute ˆın lucr ˘arile [1], [2], [4], respectiv [5].
Capitolul 3 cuprinde demonstrarea unor principii de maxim pentru anumite P-funct ¸iilor pe
care le vom utiliza ˆın studiul unor probleme supra-determinate, ˆın obt ¸inerea unor teoreme de tip
Liouville sau ˆın deducerea unor inegalit ˘at¸i isoperimetrice. Mai precis, ˆın Sectiunea 3.1 extindem la
o clas ˘a de ecuat ¸ii complet neliniare, care includ clasa ecuat ¸iilor hessiene s ¸i ecuat ¸iilor Weingarten,
anumite principii de maxim obt ¸inute de G. Philippin-S. Safoui [82] s ¸i [83], G. Porru-A. Safoui-S.
Vernier-Piro [84] s ¸i C. Enache [48]. ˆIn ﬁnalul sect ¸iunii folosim acest nou principiu de maxim pentru
a obt¸ine nis ¸te estim ˘ari geometrice apriori pentru suprafet ¸e Weingarten. ˆIn Sect ¸iunea 3.2 se consider ˘a
o clas ˘a general ˘a de ecuat ¸ii anisotropice cvasiliniare. Se construiesc dou ˘a P-funct ¸ii asociate acestora
pentru care se demonstreaz ˘a dou ˘a principii de maxim din care se deduc un rezultat de tip Liouville
s¸i un rezultat de simetrie de tip Serrin −Weinberger. ˆIn ultimele dou ˘a sect ¸iuni, folosind idei similare,
se obt ¸in generaliz ˘ari ale unor rezultate legate de probleme cu frontier ˘a liber ˘a din electrostatic ˘as¸i
respectiv teoria potent ¸ialului, obt ¸inute de E. Sartori [88] s ¸i L.E. Payne-G.A. Philippin [81]. Acest
capitol se bazeaz ˘a pe lucr ˘arile [11], [12] s ¸i [13].
ˆIn ultima parte a tezei prezent ˘am proiectele privind evolut ¸iile ulterioare ale carierei academice
s¸i s¸tiint¸iﬁce ale autoarei.
vii

Contents
1 On Some Singularly Perturbed Problems 1
1.1 Preliminaries ………………………………. 1
1.1.1 Regular and Singular Perturbations ………………… 1
1.1.2 Evolution Equations ……………………….. 5
1.2 Hyperbolic Systems with Algebraic Boundary Conditions ………….. 7
1.2.1 A zeroth order asymptotic expansion ……………….. 9
1.2.2 Existence, uniqueness and regularity of the solutions of problems PεandP0 10
1.2.3 Estimates for the remainder components ………………. 1 7
1.3 On some singularly perturbed, coupled parabolic-parabolic problems ……. 2 1
1.3.1 A ﬁrst order asymptotic expansion for
the solution of problem (P.1)ε…………………… 2 5
1.3.2 A ﬁrst order asymptotic expansion for
the solution of problem (P.2)ε…………………… 3 9
1.3.3 A zeroth order asymptotic expansion for
the solution of problem (P.3)ε…………………… 4 9
2 Elliptic-like Regularizations of Nonlinear Evolution Equations 53
2.1 Asymptotic analysis of problem (P.1)ε…………………… 5 4
2.1.1 Main result ……………………………. 5 5
2.1.2 Applications …………………………… 6 0
2.2 Asymptotic analysis of problem (P.2)ε…………………… 6 5
2.2.1 Main result ……………………………. 6 5
2.2.2 Applications …………………………… 7 0
2.3 Elliptic regularization of the semilinear telegraph system with nonlinear boundary
conditions ………………………………… 7 6
2.4 Elliptic regularization of the semilinear heat equation with nonlinear boundary con-
ditions ………………………………….. 8 3
3 Maximum principles for appropriate P(ayne)-functions. Applications 89
3.1 A maximum principle for some fully nonlinear elliptic equations with applications
to Weingarten hypersurfaces ……………………….. 8 9
3.1.1 A maximum principle for an appropriate P-function …………. 9 1
3.1.2 A priori estimates. Applications to Weingarten hypersurfaces …….. 9 6
viii

Rezumat
3.2 Maximum principles and Liouville type theorems for a general class of quasilinear
anisotropic equations …………………………… 9 9
3.2.1 Maximum principles for appropriate P-functions ………….. 1 0 2
3.2.2 Proof of Theorem 3.2.1 ………………………. 1 0 6
3.2.3 Proof of Theorems 3.2.2 and 3.2.3 …………………. 1 0 8
3.3 On a free boundary problem for a class of anisotropic equations ………. 1 1 1
3.3.1 A maximum principle for an appropriate P-function ………… 1 1 3
3.3.2 The proof of Theorem 3.3.2 ……………………. 1 1 7
3.4 A free boundary problem with multiple boundaries for a general class of anisotropic
equations ………………………………… 1 2 0
3.4.1 Preliminaries …………………………… 1 2 2
3.4.2 The proof of Theorem 3.4.1 ……………………. 1 2 3
4 Further Developments 129Bibliography 130
ix

Chapter 1
On Some Singularly Perturbed Problems
1.1 Preliminaries
1.1.1 Regular and Singular Perturbations
In this subsection we recall some basic concepts about regular and singular perturbations which will
be needed later. A general reference to this topics is W. Eckhaus [47], J. Kervorkian-J. D. Cole [65],J. L. Lions [69], A. B. Vasilieva-V . F. Butuzov-L. V . Kalachev [100].
LetD⊂R
nbe a nonempty open bounded set with a smooth boundary ∂Ω. Consider the fol-
lowing equation, denoted Eε,
Lεu=f(x,ε),x∈D,
where εis a small parameter, 0 <ε/lessmuch1,Lεis a differential operator, and fis a given real-valued
smooth function. If we associate with Eεsome condition(s) for the unknown uon the boundary ∂Ω,
we obtain a boundary value problem Pε. We assume that, for each ε,Pεhas a unique smooth solution
u=uε(x). Our goal is to construct approximations of uεfor small values of ε. The usual norm we
are going to use for approximations is the sup norm, i.e., /bardblg/bardblC(D)=sup{|g(x)|;x∈D},for every
continuous function g:D−→R. We will also use the weaker Lp-norm/bardblg/bardblLp(D)=(/integraltext
D|g|pdx)1/p,
where 1≤p<∞.
In many applications, operator Lεis of the form Lε=L0+εL1,where L0andL1are differential
operators which do not depend on ε.I fL0does not include some of the highest order derivatives of
Lε, then we should associate with L0fewer boundary conditions. So, Pεbecomes
L0u+εL1u=f(x,ε),x∈D,
with the corresponding boundary conditions. Let us also consider the equation, denoted E0,
L0u=f0,x∈D,
where f0(x):=f(x,0), with some boundary conditions, which usually come from the original prob-
lemPε. Let us denote this problem by P0. Some of the original boundary conditions are no longer
necessary for P0. Problem Pεis said to be a perturbed problem (perturbed model ), while problem P0
is called unperturbed (orreduced model ).
1

Section 1.1 Chapter 1
Deﬁnition 1.1.1. Problem Pεis called regularly perturbed with respect to some norm /bardbl·/bardblif there
exists a solution u0of problem P0such that /bardbluε−u0/bardbl− → 0a s ε→0.Otherwise, Pεis said to be
singularly perturbed with respect to the same norm.
In a more general setting, we may consider time tas an additional independent variable for
problem Pεas well as initial conditions at t=0. Moreover, we may consider systems of differential
equations instead of a single equation (see Section 1.2). Note also that the small parameter mayalso occur in the conditions associated with the corresponding system of differential equations. Forexample, we will discuss in Section 1.3 some coupled problems in which the small parameter is alsopresent in transmission conditions. The deﬁnition above also applies to these more general cases.
In order to illustrate this deﬁnition we are going to consider some examples.
Example 1.1.1. LetP
εbe the following Cauchy problem
εdu
dt+ru=f0(t),0<t<T;u(0)=θ,
where ris a positive constant, θ∈Randf0:[0,T]→Ris a given Lipschitzian function.
The solution of this problem is given by uε(t)=θe−rt/ε+1
ε/integraltextt
0f0(s)e−r(t−s)/εds,0≤t≤T,
which can be written as
uε(t)=1
rf0(t)+/parenleftbigg
θ−1
rf0(0)/parenrightbigg
e−rt/ε+rε(t),0≤t≤T,
where rε(t)=−1
r/integraltextt
0f/prime
0(s)e−r(t−s)/εds.
We have |rε(t)|≤L
r/integraltextt
0e−r(t−s)/εds≤L
r2ε,where Lis the Lipschitz constant of f0. Therefore, rε
converges uniformly to zero on [0,T]asεtends to 0. Thus uεconverges uniformly to u0(t)=
(1/r)f0(t)on every interval [δ,T],0<δ<T, but not on the whole interval [0,T]iff0(0)/negationslash=rθ.
Note also that u0is the solution of the (algebraic) equation
ru=f0(t),0<t<T,
which represents our reduced problem. Therefore, if f0(0)/negationslash=rθ, this Pεis a singular perturbation
problem of the boundary layer type with respect to the sup norm. The boundary layer is a smallright vicinity of the point t=0. A uniform approximation of u
ε(t)on[0,T]is the sum u0(t)+/parenleftbig
θ−1
rf0(0)/parenrightbig
e−rt/ε. The function/parenleftbig
θ−1
rf0(0)/parenrightbig
e−rt/εis a boundary layer function, which corrects
the discrepancy between uεandu0within the boundary layer.
Example 1.1.2. InDT={(x,t);0<x<1,0<t<T}we consider the telegraph system
/braceleftBigg
εut+vx+ru=f1(x,t),
vt+ux+gv=f2(x,t),(S)ε
with initial conditions
u(x,0)=u0(x),v(x,0)=v0(x),0<x<1, (IC)ε
2

Section 1.1 Chapter 1
and boundary conditions of the form
/braceleftBigg
r0u(0,t)+v(0,t)=0,
−u(1,t)+f0(v(1,t)) = 0,0<t<T,(BC)ε
where f1,f2:DT→R,f0:R→R,u0,v0:[0,1]→Rare given smooth functions, and r0,r,gare
constants, r0>0,r>0,g≥0. If in the model formulated above and denoted by Pεwe take ε=0,
we obtain the following reduced problem P0:
/braceleftBigg
u=r−1(f1−vx),
vt−r−1vxx+gv=f2−r−1f1xinDT,(S)0
v(x,0)=v0(x),0<x<1, (IC)0
/braceleftBigg
rv(0,t)−r0vx(0,t)+r0f1(0,t)=0,
rf0(v(1,t)) +vx(1,t)−f1(1,t)=0,0<t<T.(BC)0
In this case, the reduced system (S)0consists of an algebraic equation and a differential equation of
the parabolic type, whereas system (S)εis of the hyperbolic type. The initial condition for uis no
longer necessary. We will derive P0later in a justiﬁed manner.
Let us remark that if the solution of Pε, say Uε(x,t)=( uε(x,t),vε(x,t)), would converge uni-
formly in DTto the solution of P0, then necessarily v/prime
0(x)+ru0(x)= f1(x,0),∀x∈[0,1].If this
condition is not satisﬁed then that uniform convergence is not true and, as we will show later, Uεhas
a boundary layer behavior in a neighborhood of the segment {(x,0);0≤x≤1}.
Therefore, this Pεis a singular perturbation problem of the boundary layer type with respect
to the sup norm /bardbl·/bardblC(DT)2. However, using the form of the boundary layer functions which we are
going to determine later, we will see that the boundary layer is not visible in weaker norms, like forinstance/bardbl·/bardbl
C([0,1];Lp(0,T))2,1≤p<∞, and Pεis regularly perturbed in such norms.
Deﬁnition 1.1.2. Letuεbe the solution of some perturbed problem Pεdeﬁned in a domain D.
Consider a function U(x,ε),x∈D1, where D1is a subdomain of D. The function U(x,ε)is called
an asymptotic approximation in D1of the solution uε(x)with respect to the sup norm if
sup
x∈D1/bardbluε(x)−U(x,ε)/bardbl→ 0a s ε→0.
Moreover, if
sup
x∈D1/bardbluε(x)−U(x,ε)/bardbl=O(εk),
then we say that U(x,ε)is an asymptotic approximation of uε(x)inD1with an accuracy of the order
εk.We have similar deﬁnitions with respect to other norms. In the above deﬁnition we have assumed
thatUanduεtake values in Rn, and/bardbl·/bardbldenotes one of the norms of this space.
For a real-valued function E(ε), the notation E(ε)= O(εk)means that |E(ε)|≤Mεkfor some
positive constant Mand for all εsmall enough.
In the following we are going to discuss the celebrated Vishik-Lyusternik method [100] for
the construction of asymptotic approximations for the solutions of singular perturbation problems
3

Section 1.1 Chapter 1
of the boundary layer type. To explain this method we consider the problem used in Example 1.1.2
above. This problem admits a boundary layer near the side {(x,0);0≤x≤1}of the rectangle DT
with respect to the sup norm. For a given xthis has the same form as the equation discussed in
Example 1.1.1, with f0(t):=f1(x,t)−vx(x,t). So we expect to have a singular behavior of the
solution near the value t=0 for all x. We will restrict ourselves to seeking an asymptotic expansion
of the order zero for the solution of Pε, i.e.,
Uε(x,t)=/parenleftbig
uε(x,t),vε(x,t)/parenrightbig
=/parenleftbig
X(x,t),Y(x,t)/parenrightbig
+/parenleftbig
c0(x,τ),d0(x,τ)/parenrightbig
+/parenleftbig
R1ε(x,t),R2ε(x,t)/parenrightbig
,
where/parenleftbig
X(x,t),Y(x,t)/parenrightbig
is the regular term, τ=t/εis the rapid variable for this problem,/parenleftbig
c0(x,τ),
d0(x,τ)/parenrightbig
is the correction (of order zero), and Rε:=/parenleftbig
R1ε,R2ε/parenrightbig
is the remainder (of order zero). The
form of the rapid variable τis also suggested by the analogous problem in Example 1.1.1 above.
We substitute the above expansion into Pεand identify the coefﬁcients of the like powers of ε.O f
course, we distinguish between the coefﬁcients depending on (x,t)and those depending on (x,τ).
We should keep in mind that the remainder components are small as compared to the other terms.So, after substituting the above expansion into (S)
ε, we see that the only coefﬁcient of ε−1in the
second equation is d0τ(x,τ). So, this should be zero, thus d0is a function depending on xonly.
Taking also into account the fact that a boundary layer function should converge to zero as τ→∞,
we infer that d0is identically zero. From the ﬁrst equation of (S)ε, we derive the following boundary
layer equation by identifying the coefﬁcients of ε0:
c0τ(x,τ)+rc0(x,τ)=0.
If we integrate this equation and use the usual condition c0(x,τ)→0a s τ→∞, we get c0(x,τ)=
α(x)e−rτ,where α(x)will be determined later. Applying the identiﬁcation procedure to regular
terms, one can see that (X,Y)satisﬁes the reduced problem P0we have already indicated before. We
have in mind that the remainder components should tend to 0 as ε→0. From the initial condition
foruε, which reads
u0(x)=X(x,0)+α(x)+R1ε(x,0),
we get
u0(x)=X(x,0)+α(x),
which shows how c0(x,τ)compensate for the discrepancy in the corresponding initial condition. In
fact, at this moment c0(x,τ)is completely determined, since the last equation gives
α(x)=u0(x)−1
r/parenleftBig
f1(x,0)−v/prime
0(x)/parenrightBig
.
Let us now discuss the boundary conditions. From the equation
r0/parenleftBig
X(0,t)+c0(0,τ)+R1ε(0,t)/parenrightBig
+Y(0,t)+R2ε(0,t)=0
we derive
r0X(0,t)+Y(0,t)=0,
4

Section 1.1 Chapter 1
plus the condition c0(0,τ)=0, i.e., α(0)=0. Now, let us discuss the nonlinear boundary condition,
which reads
−/parenleftBig
X(1,t)+c0(1,τ)+R1ε(1,t)/parenrightBig
+f0/parenleftbig
Y(1,t)+R2ε(1,t)/parenrightbig
=0.
We obtain c0(1,τ)=0, i.e., α(1)=0. For the regular part we get −X(1,t)+f0(Y(1,t)) = 0.
We have already determined the corrections of the order zero, and it is an easy matter to ﬁnd
the problem satisﬁed by the remainder components R1ε,R2ε. So, at this moment, we have a formal
asymptotic expansion of the order zero. The next step would be to show that the expansion is welldeﬁned, in particular to show that, under some speciﬁc assumptions on the data, both P
εandP0have
unique solutions in some function spaces. Finally, to show that the expansion is a real asymptoticexpansion, it should be proved that the remainder tends to zero with respect to a given norm. In ourapplications (including the above hyperbolic P
ε) we are going to do even more, to establish error
estimates for the remainder components (in most of the cases with respect to the sup norm).
By the same technique, terms of the higher order approximations can be constructed as well.
1.1.2 Evolution Equations
In this subsection we are going to state some basic concepts and results in the theory of evolutionequations associated with monotone operators which will be used in the next sections. The proofsof the theorems will be omitted, but appropriate references will be indicated. For more informationregarding evolution equations in Hilbert spaces, including those associated with monotone operators,we refer the reader to H. Br ´ezis [35], G. Moros ¸anu [77].
LetHbe a real Hilbert space. Its scalar product and norm are again denoted by /angbracketleft·,·/angbracketrightand/bardbl·/bardbl,
respectively. Consider in Hthe following Cauchy problem:
/braceleftBigg
u
/prime(t)+Au(t)=f(t),0<t<T,
u(0)=u0,(1.1.1)
where A:D(A)⊂H→His a nonlinear single-valued operator and f∈L1(0,T;H).In fact, the most
known existence results are valid for multivalued A’s, but we will consider only single-valued A’s.
This is enough for our considerations.
Deﬁnition 1.1.3. A function u∈C([0,T];H)is said to be a strong solution of equation (1.1.1) 1if:
(i)uis absolutely continuous on every compact subinterval of (0,T),
(ii)u(t)∈D(A)for a. e. t∈(0,T),
(iii)usatisﬁes (1.1.1) 1for a. e. t∈(0,T).
If in addition u(0)=u0, then uis called a strong solution of the Cauchy problem (1.1.1).
Deﬁnition 1.1.4. A function u∈C([0,T];H)is called a weak solution of (1.1.1) 1if there exist two
sequences {un}⊂W1,∞(0,T;H)and{fn}⊂L1(0,T;H)such that :
(k)u/prime
n(t)+Aun(t)=fn(t)for a. e. t∈(0,T),n∈N,
(kk)un→uinC([0,T];H)asn→∞,
(kkk)fn→finL1(0,T;H)asn→∞.
Again, if in addition u(0)=u0, then uis called a weak solution of the Cauchy problem (1.1.1).
5

Section 1.1 Chapter 1
Theorem 1.1.1. (see, e.g., G. Moros ¸anu [77, Theorem 2.1, p. 48]) If A:D(A)⊂H→H is a maximal
monotone operator, u 0∈D(A)and f∈W1,1(0,T;H),then the Cauchy problem (1.1.1) has a unique
strong solution u ∈W1,∞(0,T;H).Moreover u (t)∈D(A)for all t∈[0,T],u is differentiable on the
right at every t ∈[0,T),and
d+u
dt(t)+Au(t)=f(t)∀t∈[0,T),
/vextenddouble/vextenddouble/vextenddouble/vextenddoubled+u
dt(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble≤/bardblf(0)−Au
0/bardbl+/integraldisplayt
0/bardblf/prime(s)/bardblds∀t∈[0,T). (1.1.2)
If u and u are the strong solutions corresponding to (u0,f),(u0,f)∈D(A)×W1,1(0,T;H),then
/bardblu(t)−u(t)/bardbl≤/bardbl u0−u0/bardbl+/integraldisplayt
0/bardblf(s)−f(s)/bardblds,0≤t≤T. (1.1.3)
Theorem 1.1.2. (see, e.g., G. Moros ¸anu [77, Theorem 2.1, p. 55]) If A:D(A)⊂H→H is a maximal
monotone operator, u 0∈D(A)and f∈L1(0,T;H),then the Cauchy problem (1.1.1) has a unique
weak solution u ∈C([0,T];H).If u and u are the weak solutions corresponding to (u0,f),(u0,f)∈
D(A)×L1(0,T;H),then u and u still satisfy (1.1.3) .
Remark 1.1.1. Both the above theorems still hold in the case of Lipschitz perturbations, i.e., in the
case in which Ais replaced by A+B, where Ais maximal monotone as before and B:D(B)=H→H
is a Lipschitz operator (see H. Br ´ezis [35, p. 105]). The only modiﬁcations appear in estimates (1.1.2)
and (1.1.3) :
/vextenddouble/vextenddouble/vextenddouble/vextenddoubled
+u
dt(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble≤e
ωt/parenleftBig
/bardblf(0)−Au0/bardbl+/integraldisplayt
0e−ωs/bardblf/prime(s)/bardblds/parenrightBig
,0≤t<T, (1.1.4)
/bardblu(t)−u(t)/bardbl≤eωt/parenleftBig
/bardblu0−u0/bardbl+/integraldisplayt
0e−ωs/bardblf(s)−f(s)/bardblds/parenrightBig
,0≤t≤T, (1.1.5)
where ωis the Lipschitz constant of B.
Theorem 1.1.3. (H. Br ´ezis [35]) If A is the subdifferential of a proper convex lower semicontinuous
function ϕ:H→(−∞,+∞],u0∈D(A)and f∈L2(0,T;H),then the Cauchy problem (1.1.1) has
a unique strong solution u ,such that t1/2u/prime∈L2(0,T;H),t→ϕ(u(t))is integrable on [0,T]and
absolutely continuous on [δ,T],∀δ∈(0,T).If, in addition, u 0∈D(ϕ),then u/prime∈L2(0,T;H),t→
ϕ(u(t))is absolutely continuous on [0,T],andϕ(u(t))≤ϕ(u0)+1
2/integraltextT
0/bardblf(t)/bardbl2dt∀t∈[0,T].
Next, we recall the following theorem due to H. Attouch-A. Damlamian [6] which will be used
in the following:
Theorem 1.1.4. Let A (t)=∂φ(t,·),0≤t≤T , where φ(t,·):H→(−∞,+∞]are all proper, convex,
and lower semicontinuous. Assume further that there exist some positive constants C 1,C2and a
nondecreasing function γ:[0,T]→Rsuch that
φ(t,x)≤φ(s,x)+[γ(t)−γ(s)][φ(s,x)+C1/bardblx/bardbl2+C2], (1.1.6)
for all x∈H,0≤s≤t≤T.Then, for every u 0∈D(φ(0,·))and f∈L2(0,T;H),there exists a
6

Section 1.2 Chapter 1
unique function u ∈W1,2(0,T;H)such that u (0)=u0and
u/prime(t)+A(t)u(t)=f(t)for a. e. t ∈(0,T). (1.1.7)
Moreover, there exists a function h ∈L1(0,T)such that
φ(t,u(t))≤φ(s,u(s)) +/integraldisplayt
sh(r)dr for all 0≤s≤t≤T.
Here, we denote by ∂φ(t,·)the subdifferential of the function ∂φ(t,·).
The following result (see V . Barbu [26, Theorem 3.3, p. 14] ), which will also be used later:
Theorem 1.1.5. Let X be a real reﬂexive Banach space and let u ∈Lp(a,b;X),a,b∈R,a<b,
1<p<∞. Then, the following two conditions are equivalent:
(i) u∈W1,p(a,b;X);
(ii) There exists a constant C >0such that
/integraldisplayb−h
a/bardblu(t+h)−u(t)/bardblp
Xdt≤Chp∀h∈(0,b−a].
If p=1then (i)implies (ii). Moreover, (ii)is true for p =1if one representative of u ∈L1(a,b;X)
is of bounded variation on [a,b], where X is a general Banach space, not necessarily reﬂexive.
The remainder of this section is meant for recalling an existence and regularity result for second
order abstract differential equations (see A. R. Aftabizadeh-N. H. Pavel [3, Theorem 3.2.]).
Speciﬁcally, let us consider in Hthe problem
/braceleftBigg
p(t)u/prime/prime(t)+r(t)u/prime(t)=Au(t)+f(t),0<t<T,
u/prime(0)∈β1(u(0)−a),−u/prime(T)∈β2(u(T)−b).(1.1.8)
Theorem 1.1.6. If A,β1,β2are maximal monotone in H; 0,a,b∈D(A);f∈L2(0,T;H);
/angbracketleftAλx−Aλy,v/angbracketright≥0∀λ>0∀v,x,y∈H,x−y∈D(β1),v∈β1(x−y),
/angbracketleftAλx−Aλy,v/angbracketright≥0∀λ>0∀v,x,y∈H,x−y∈D(β2),v∈β2(x−y),
where A λdenotes the Yosida approximation of A; either D (β1)or D(β2)is bounded; p ,r∈W1,∞(0,T)
and p (t)≥c>0∀t∈[0,T],then problem (1.1.8) has at least one solution u ∈W2,2(0,T;H).
If, in addition, at least one of the operators A ,β1,β2is injective, then the solution is unique.
1.2 Hyperbolic Systems with Algebraic Boundary Conditions
In this section we are interested in an initial-boundary value problems associated with a partial
differential system, known as the telegraph system.
7

Section 1.2 Chapter 1
LetDT:=/braceleftbig
(x,t)∈R2;0<x<1,0<t<T/bracerightbig
, where T>0 is a given time instant. Consider
the telegraph system in DT:/braceleftBigg
εut+vx+ru=f1(x,t),
vt+ux+gv=f2(x,t),(LS)
with the initial conditions:
u(x,0)=u0(x),v(x,0)=v0(x),0≤x≤1, (IC)
and nonlinear algebraic boundary conditions of the form:
/braceleftBigg
u(0,t)+r0(v(0,t)) = 0,
u(1,t)=f0(v(1,t)),0≤t≤T.(BC)
We suppose that f1,f2:DT→R,u0,v0:[0,1]→R,f0,r0:R→Rare known functions while
r,gare given constants, r>0,g≥0,εis a positive small parameter. Problem (LS),(IC),(BC),
which will also be called Pε, is of the hyperbolic type with nonlinear algebraic boundary conditions.
Denote the solution of PεbyUε(x,t)=( uε(x,t),vε(x,t)).
Problem Pεis a model for an electrical circuit with nonlinear resistors at the ends x=0 and
x=1, where urepresents the current ﬂowing in the line, and vis the voltage across the line. The pa-
rameter εrepresents the speciﬁc inductance, while the speciﬁc capacitance, which usually multiplies
vtin the telegraph system, is assumed to be equal to 1 .This does not restrict the generality of the
problem. For more details concerning the physical model, we refer to K. L. Cooke-D. W. Krumme[41]. If we put ε=0,then our problem P
εbecomes a parabolic boundary value problem, say P0.
However, from a physical point of view, the parameter εis just small, not zero. So, we ask ourselves
if, by putting ε=0, the resulting parabolic model P0still describes the physical phenomenon prop-
erly. In other words, the question is whether the solution of the reduced model, P0is ”sufﬁciently”
close to the solution of the hyperbolic model Pε.
Using the Vishik-Lyusternik method, an asymptotic expansion of the solution of Pεwill be
constructed. The presence of some corrector in that asymptotic expansion shows that the reducedparabolic model P
0is not valid in DT, with respect to the norm of uniform convergence in DT. No-
tice that such a parabolic model, extended to the case of ntelegraph systems which are connected
by means of some appropriate boundary conditions, is applicable to on-chip and inter-chip intercon-nections for many digital systems, where the distributed inductances of the conductive layers arenegligible (see C. A. Marinov-P. Neittaanmaki [73]).
Mention should be made of the fact that the model presented above, or similar more complex
models, also describe further physical problems, in particular problems which occur in hydraulics(see V . Barbu [25], V . Hara [59], V . Iftimie [64], I. Straskraba-V . Lovicar [92], V . L. Streeter-E. B.Wylie [93]).
In the ﬁrst subsection we derive formally a zeroth order asymptotic expansion for the solution
of(LS),(IC),(BC)by using the method described in Section 1.1. This problem is singularly per-
turbed of the boundary layer type. We determine the corresponding boundary layer function as wellas the problems satisﬁed by the regular term and by the remainder components.
In the second subsection we are going to recall a result that is already known, related to the
8

Section 1.2 Chapter 1
solution of problem Pε. Also, we are going to prove some results on the existence, uniqueness and
higher regularity of the solution of the reduced problem, under appropriate assumptions on the data.For this purpose we use the general theory of evolution equations, including non-autonomous evo-lution equations. All these results guarantee the fact that our asymptotic expansion is well deﬁned.
Moreover, they are used in the third and last subsection to derive estimates for the remainder
components with respect to the uniform convergence norm. These estimates show that our expansionis a real asymptotic expansion.
1.2.1 A zeroth order asymptotic expansion
Arguing as in Subsection 1.1.1, Example 1.1.2, we see that Uεhas a singular behavior with respect to
the uniform convergence topology within a neighborhood of the segment {(x,0);0≤x≤1}which
is the boundary layer . Having in mind the case of linear boundary conditions investigated in Sub-
section 1.1.1 we are looking for a zeroth order expansion of Uεin the form:
Uε=U0(x,t)+V0(x,τ)+Rε(x,t), (1.2.9)
where:
U0=(X(x,t),Y(x,t))is the zero-th order term of the regular series; V0=(c0(x,τ),d0(x,τ)),τ=t/ε,
is the boundary layer (vector) function; Rε=(R1ε(x,t),R2ε(x,t))is the remainder of the order zero.
By the standard identiﬁcation procedure presented in Subsection 1.1.1, making use of (1.2.9) in (LS)
we can see that XandYsatisfy an algebraic equation and a parabolic equation, respectively:
X=(1/r)(f1−Yx)inDT, (1.2.10)
Yt−(1/r)Yxx+gY=f2−(1/r)f1x,inDT. (1.2.11)
For the boundary functions c0,d0we get
d0≡0,c0(x,τ)=α(x)e−rτ, (1.2.12)
where function αwill be determined below from (IC). For the remainder components we obtain
formally the following system:
/braceleftBigg
εR1εt+R2εx+rR1ε=−εXt,inDT,
R2εt+R1εx+gR2ε=−c0x,inDT.(1.2.13)
Next, from (IC)it follows that
Y(x,0)=v0(x),0≤x≤1, (1.2.14)
α(x)=u0(x)+(1/r)(v/prime
0(x)−f1(x,0)),0≤x≤1, (1.2.15)
R1ε(x,0)=R2ε(x,0)=0,0≤x≤1. (1.2.16)
Finally, substituting (1.2.9) into (BC)we derive
X(0,t)+r0(Y(0,t)) = 0,X(1,t)−f0(Y(1,t)) = 0,0≤t≤T,
9

Section 1.2 Chapter 1
which can be written as (cf. (1.2.10))
⎧
⎨
⎩−(1/r)Yx(0,t)+r0(Y(0,t)) =−(1/r)f1(0,t),
(1/r)Yx(1,t)+f0(Y(1,t)) = ( 1/r)f1(1,t),0≤t≤T.(1.2.17)
Also we obtain
c0(0,τ)=c0(1,τ)=0⇔ α(0)=α(1)=0, (1.2.18)
and the following boundary conditions for the remainder components
⎧
⎨
⎩R1ε(0,t)+r0/parenleftBig
R2ε(0,t)+Y(0,t)/parenrightBig
−r0/parenleftBig
Y(0,t)/parenrightBig
=0,
R1ε(1,t)−f0/parenleftBig
R2ε(1,t)+Y(1,t)/parenrightBig
+f0/parenleftBig
Y(1,t)/parenrightBig
=0,0≤t≤T.(1.2.19)
Summarizing, we see that the components of the regular part satisfy the reduced problem P0, which
is made up by (1.2.10), (1.2.11), (1.2.14) and (1.2.17), while the remainder components satisfy theproblem (1.2.13), (1.2.16), (1.2.19).
Conditions (1.2.18) are needed to eliminate possible discrepancies which may occur due to
c
0at the corner points (x,t)=( 0,0)and(x,t)=( 1,0). As we will see in the next subsection,
these conditions are also compatibility conditions that we need to derive results on the existence,uniqueness and smoothness of the solutions of the problems involved in our asymptotic analysis.
1.2.2 Existence, uniqueness and regularity of the solutions of problems PεandP0
Let us start with problem Pε. This problem has been investigated by V .-M. Hokkanen-G. Moros ¸anu
(see [63, Problem (5.2.1)-(5.2.4)]), so on account of [63, Theorem 5.2.1], we have:
Theorem 1.2.1. Assume that r ,g are some nonnegative constants ;
f1,f2∈C1([0,T];C[0,1]);r0,f0∈C1(R),r/prime
0≥0,f/prime
0≥0; (1.2.20)
u0,v0∈C1[0,1]and satisfy the zeroth order compatibility conditions:
/braceleftBigg
u0(0)+r0(v0(0)) = 0,
u0(1)−f0(v0(1)) = 0,(1.2.21)
and the following ﬁrst order compatibility conditions
/braceleftBigg
f1(0,0)−v/prime
0(0)+εr/prime
0(v0(0))[f2(0,0)−u/prime
0(0)] = 0,
f1(1,0)−v/prime
0(1)−εf/prime
0(v0(1))[f2(1,0)−u/prime
0(1)] = 0.(1.2.22)
Then, the solution (uε,vε)of problem P εbelongs to C1(DT)2.
The rest of this subsection is focused on the reduced problem P0,which consists of the algebraic
equation (1.2.10) and the boundary value problem (1.2.11), (1.2.14), (1.2.17). The results whichfollows were derived from L. Barbu-G. Moros¸anu-L. W. Wendland [23].
10

Section 1.2 Chapter 1
Let us consider the Hilbert space H0:=L2(0,1)endowed with the usual scalar product and the
associated norm denoted by /bardbl·/bardbl 0.
We deﬁne the operator A(t):D(A(t))⊂H0→H0,
D(A(t)) ={p∈H2(0,1);r−1p/prime(0)+σ1(t)=r0(p(0)),
−r−1p/prime(1)+σ2(t)=f0(p(1))},A(t)p=−(1/r)p/prime/prime+gp,
where σ1(t)=−r−1f1(0,t),σ2(t)=r−1f1(1,t).
Denoting /tildewidey(t)=Y(·,t),/tildewideh(t)=h(·,t)=( f2−r−1f1x)(·,t),0<t<T,it is obvious that problem
(1.2.11), (1.2.14), (1.2.17) can be expressed as the following Cauchy problem in H0:
/braceleftBigg
/tildewidey/prime(t)+A(t)/tildewidey(t)=/tildewideh(t),0<t<T,
/tildewidey(0)=v0.(1.2.23)
Since we plan to prove high regularity properties of the solution of the reduced problem, we note
that by formal differentiation with respect to tof problem (1.2.11), (1.2.14), (1.2.17) we obtain the
following problem:
⎧
⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩z
t−r−1zxx+gz=htinDT,
z(x,0)=z0(x),0≤x≤1,
r−1zx(0,t)+σ/prime
1(t)=α(t)z(0,t),
−r−1zx(1,t)+σ/prime
2(t)=β(t)z(1,t),0≤t≤T,(1.2.24)
where z=Yt,z0=h(0)−A(0)v0,α=r/prime
0(Y(0,·)),β=f/prime
0(Y(1,·)). We see that this problem can be
expressed as the following Cauchy problem in the Hilbert space H0:
/braceleftBigg
/tildewidez/prime(t)+A1(t)/tildewidez(t)=ht(·,t),0<t<T,
/tildewidez(0)=z0,(1.2.25)
with/tildewidez(t):=z(·,t)andA1(t):D(A1(t))⊂H0→H0,
D(A1(t)) ={p∈H2(0,1);r−1p/prime(0)+σ/prime
1(t)=α(t)p(0),
−r−1p/prime(1)+σ/prime
2(t)=β(t)p(1)},A1(t)p=−r−1p/prime/prime+gp.
We continue with a result about the existence, uniqueness, and regularity of the solution to problem
(1.2.23).
Theorem 1.2.2. Assume that
h∈W2,2(0,T;L2(0,1)),σ1,σ2∈H2(0,T); (1.2.26)
r>0,g≥0,r0,f0∈C2(R),r/prime
0,f/prime
0≥0; (1.2.27)
v0∈D(A(0)),z0:=h(0)−A(0)v0∈D(A1(0)). (1.2.28)
11

Section 1.2 Chapter 1
Then, problem (1.2.23) has a unique strong solution /tildewidey(t)=Y(·,t), and
Y∈W2,2(0,T;H1(0,1))/intersectiondisplay
W1,2(0,T;H2(0,1)).
Proof. First we note that under assumptions (1.2.27), operator A(t)deﬁned above is maximal mono-
tone for all t∈[0,T](see, e. g., L. Barbu-G. Moros ¸anu [18, Example 1 and Proposition 2.0.13, p.
24]). In addition, A(t)is the subdifferential of the function φ(t,·):H0→(−∞,+∞],
φ(t,p)=⎧
⎪⎪⎪⎨
⎪⎪⎪⎩1
2r/integraltext1
0p/prime(x)2dx+g
2/integraltext1
0p2(x)dx+j1(p(0)) + j2(p(1))
−σ1(t)p(0)−σ2(t)p(1),ifp∈H1(0,1),
+∞,otherwise .
where j1,j2are primitives for r0,f0,respectively. For every t∈[0,T], the effective domain D(φ(t,·))
=H1(0,1).Thus, D(φ(t,·))does not depend on t. Let us show that condition (1.1.6) of Theo-
rem 1.1.4 is satisﬁed. Indeed, for every p∈H1(0,1)and 0≤s≤t≤T,we have
φ(t,p)−φ(s,p)≤(|p(0)|+|p(1)|)/integraldisplayt
s(|σ/prime
1(τ)|+|σ/prime
2(τ)|)dτ. (1.2.29)
Since j1,j2are bounded from below by some afﬁne functions, we have
φ(s,p)≥1
2r/bardblp/prime/bardbl2
0−C1|p(0)|−C2|p(1)|−C3,∀s∈[0,T],p∈H1(0,1),
where C1,C2,C3are some positive constants. Thus, it is easily seen that
|p(0)|+|p(1)|≤φ(s,p)+M1/bardblp/bardbl2
0+M2,
where M1,M2are positive constants, and therefore, (1.2.29) implies (1.1.6) with γ(t)=/integraltextt
0(|σ/prime
1(τ)|
+|σ/prime
2(τ)|)dτ.We have used the following obvious inequality
p2(x)≤(1+η)/bardblp/bardbl2
0+η−1/bardblp/prime/bardbl2
0,∀x∈[0,1],η>0,p∈H1(0,1). (1.2.30)
So, according to Theorem 1.1.4, problem (1.2.23) has a unique strong solution /tildewidey(t):=Y(·,t),
Y∈W1,2(0,T;H0)/intersectiondisplay
L2(0,T;H2(0,1)).
Now, we are going to prove that Y∈W1,2(0,T;H1(0,1)).We start from the obvious inequalities
1
2d
dt/bardbl/tildewidey(t+δ)−/tildewidey(t)/bardbl2
0+1
r/bardblYx(·,t+δ)−Yx(·,t)/bardbl2
0≤|σ1(t+δ)−σ1(t)|·|Y(0,t+δ)−Y(0,t)|
+|σ2(t+δ)−σ2(t)|·|Y(1,t+δ)−Y(1,t)+/bardbl/tildewideh(t+δ)−/tildewideh(t)/bardbl0·/bardbl/tildewidey(t+δ)−/tildewidey(t)/bardbl0,
for a. e. 0 ≤t≤t+δ≤T, and
1
2d
dδ/bardbl/tildewidey(δ)−v0/bardbl2
0+1
r/bardblYx(·,δ)−v/prime
0/bardbl20≤|σ1(δ)−σ1(0)|·|Y(0,δ)−v0(0)|
+|σ2(δ)−σ2(0)|·|Y(1,δ)−v0(1)|+/bardbl/tildewideh(δ)−A(0)v0/bardbl0·/bardbl/tildewidey(δ)−v0/bardbl0,
12

Section 1.2 Chapter 1
for a. e. δ∈(0,T). Since H1(0,1)is continuously embedded into C[0,1], we infer from the above
inequalities the following estimates
1
2d
dt/bardbl/tildewidey(t+δ)−/tildewidey(t)/bardbl2
0+1
r/bardblYx(·,t+δ)−Yx(·,t)/bardbl2
0
≤C4/parenleftBig
|σ1(t+δ)−σ1(t)|2+|σ2(t+δ)−σ2(t)|2/parenrightBig
+1
2r/bardblYx(·,t+δ)−Yx(·,t)/bardbl2
0+/parenleftBig
/bardbl/tildewideh(t+δ)−/tildewideh(t)/bardbl0
+C5(|σ1(t+δ)−σ1(t)|+|σ2(t+δ)−σ2(t)|)/parenrightBig
/bardbl/tildewidey(t+δ)−/tildewidey(t)/bardbl0,
for a. e. 0 ≤t≤t+δ≤T, and
1
2d
dδ/bardbl/tildewidey(δ)−v0/bardbl2
0+1
r/bardblYx(·,δ)−v/prime
0/bardbl20≤1
2r/bardblYx(·,δ)−v/prime
0/bardbl20
+C6/parenleftBig
|σ1(δ)−σ1(0)|2+|σ2(δ)−σ2(0)|2/parenrightBig
+/parenleftBig
/bardbl/tildewideh(δ)−A(0)v0/bardbl0
+C7(|σ1(δ)−σ1(0)|+|σ2(δ)−σ2(0)|)/parenrightBig
/bardbl/tildewidey(δ)−v0/bardbl0,
for a. e. δ∈(0,T), where C4,C5,C6,C7are some positive constants. Integration over [0,T−δ]and
[0,δ]of the above inequalities leads us to
1
2r/integraldisplayT−δ
0/bardblYx(·,t+δ)−Yx(·,t)/bardbl2
0≤1
2/bardbl/tildewidey(δ)−v0/bardbl2
0
+/integraldisplayT−δ
0/bracketleftBig
C4/parenleftBig
|σ1(t+δ)−σ1(t)|2+|σ2(t+δ)−σ2(t)|2/parenrightBig
+/parenleftBig
C5(|σ1(t+δ)−σ1(t)|+|σ2(t+δ)−σ2(t)|)
+/bardbl/tildewideh(t+δ)−/tildewideh(t)/bardbl0/parenrightBig
/bardbl/tildewidey(t+δ)−/tildewidey(t)/bardbl0/bracketrightBig
dt,(1.2.31)
for all δ∈(0,T], and
1
2/bardbl/tildewidey(δ)−v0/bardbl2
0≤/integraldisplayδ
0/bracketleftBig
C6/parenleftBig
|σ1(s)−σ1(0)|2+|σ2(s)−σ2(0)|2/parenrightBig
+/parenleftBig
C7(|σ1(s)−σ1(0)|+|σ2(s)−σ2(0)|)+/bardbl/tildewideh(s)−A(0)v0/bardbl0/parenrightBig
/bardbl/tildewidey(s)−v0/bardbl0/bracketrightBig
ds,(1.2.32)
for all δ∈(0,T].Applying Gronwall’s lemma (see e.g. H. Br ´ezis, [35, Lemma A.5, p. 157] ) to
inequality (1.2.32) and using σ1,σ2∈H2(0,T),/tildewideh,/tildewidey∈W1,2(0,T;H0), we infer that
/bardbl/tildewidey(δ)−v0/bardbl0≤C8δ,for all δ∈(0,T],
and then, by (1.2.31), we get
/integraldisplayT−δ
0/bardblYx(·,t+δ)−Yx(·,t)/bardbl2
0dt≤C9δ2,
from which it follows that Y∈W1,2(0,T;H1(0,1))(see Theorem 1.1.5).
Now, let us prove that Y∈W2,2(0,T;H1(0,1)). Set V:=H1(0,1)and denote its dual by V∗.
We will prove that Ytsatisﬁes problem (1.2.24) derived by formal differentiation with respect to tof
13

Section 1.2 Chapter 1
problem (1.2.23). First, let us check that Yt∈W1,2(0,T;V∗).Note that
/angbracketleftϕ,Yt(·,t+δ)−Yt(·,t)/angbracketright=/angbracketleftϕ,r−1(Yxx(·,t+δ)−Yxx(·,t))/angbracketright
−g/angbracketleftϕ,Y(·,t+δ)−Y(·,t)/angbracketright+/angbracketleftϕ,h(·,t+δ)−h(·,t)/angbracketright(1.2.33)
for a. e. t∈(0,T−δ)and all ϕ∈V,where/angbracketleft·,·/angbracketrightdenotes the pairing between VandV∗. Integration
by parts in (1.2.33) yields
/angbracketleftϕ,Yt(·,t+δ)−Yt(·,t)/angbracketright+/angbracketleftϕ/prime,r−1(Yx(·,t+δ)−Yx(·,t))/angbracketright
+[ (r0(Y(0,t+δ))−r0(Y(0,t))−(σ1(t+δ)−σ1(t))]ϕ(0)
+[ (f0(Y(1,t+δ))−f0(Y(1,t)))−(σ2(t+δ)−σ2(t))]ϕ(1)
+g/angbracketleftϕ,Y(·,t+δ)−Y(·,t)/angbracketright=/angbracketleftϕ,h(·,t+δ)−h(·,t)/angbracketright
for a. e. t∈(0,T−δ),∀ϕ∈V.This equation together with (1.2.26) and (1.2.27) implies
/bardblYt(·,t+δ)−Yt(·,t)/bardblV∗≤C10(/bardbl/tildewidey(t+δ)−/tildewidey(t)/bardblV+/bardblh(·,t+δ)−h(·,t)/bardbl0+δ),
where C10is a positive constant. Thus,
/integraldisplayT−δ
0/bardblYt(·,t+δ)−Yt(·,t)/bardbl2
V∗dt≤C11δ2,∀δ∈(0,T],
and so it follows that Yt∈W1,2(0,T;V∗)(see Theorem 1.1.5). Therefore, one can differentiate with
respect to tthe equation in Y:
/angbracketleftϕ,Yt(·,t)/angbracketright+/angbracketleftϕ/prime,r−1Yx(·,t)/angbracketright+g/angbracketleftϕ,Y(·,t)/angbracketright+(r0(Y(0,t))−σ1(t))ϕ(0)
+(f0(Y(1,t))−σ2(t))ϕ(1)=/angbracketleftϕ,h(·,t)/angbracketright,∀ϕ∈V,
thus obtaining
/angbracketleftϕ,zt(·,t)/angbracketright+/angbracketleftϕ/prime,r−1zx(·,t)/angbracketright+g/angbracketleftϕ,z(·,t)/angbracketright+(α(t)z(0,t)−σ/prime
1(t))ϕ(0)
+(β(t)z(1,t)−σ/prime
2(t))ϕ(1)=/angbracketleftϕ,ht(·,t)/angbracketright,∀ϕ∈V,(1.2.34)
where z=Yt,α(t)=r/prime
0(Y(0,t)),β(t)=f/prime
0(Y(1,t)).Obviously, α,β∈H1(0,T),α≥0,β≥0, and
z(·,0)=z0. (1.2.35)
One can see that zis the unique solution of problem (1.2.34), (1.2.35). Indeed, if we choose z0=
0,σ/prime
1=σ/prime
2=0,ϕ=z(·,t)andht≡0 in (1.2.34), (1.2.35), we see that
d
dt/bardblz(·,t)/bardbl2
0≤0 for a.e.t∈(0,T)⇒z≡0.
Thus z:=Ytis a variational solution of problem (1.2.24), which can be also written in the form of
the Cauchy problem (1.2.25).
In fact, problem (1.2.25) has a unique strong solution. To show this, we notice that operator A1(t)is
14

Section 1.2 Chapter 1
a maximal monotone operator for all t∈[0,T].Moreover, A1(t)is the subdifferential of the function
φ1(t,·):H0→(−∞,+∞],
φ1(t,p)=⎧
⎪⎪⎨
⎪⎪⎩1
2r/integraltext1
0p/prime(x)2dx+g
2/integraltext1
0p(x)2dx+α(t)
2p(0)2+β(t)
2p(1)2
−σ/prime
1(t)p(0)−σ/prime
2(t)p(1),ifp∈H1(0,1),
+∞,otherwise .
For every t∈[0,T],D(φ1(t,·)) = H1(0,1).Let us show that condition (1.1.6) of Theorem 1.1.4 is
satisﬁed by φ1. Indeed, for every p∈H1(0,1)and 0≤s≤t≤T,we have
φ1(t,p)−φ1(s,p)≤1
2(p(0)2+p(1)2)/integraldisplayt
s(|α/prime|+|β/prime|)(τ)dτ
+(|p(0)|+p(1)|)/integraldisplayt
s(|σ/prime
1|+|σ/prime
2|)(τ)dτ.(1.2.36)
Since
φ1(s,p)≥1
2r/bardblp/prime/bardbl2
0−C12|p(0)|−C13|p(1)|−C14,
where C12,C13,C14are some positive constants, one can easily derive (1.1.6) from (1.2.36). So,
according to Theorem 1.1.4, problem (1.2.25) has a unique strong solution /tildewidez(t):=z(·,t),
z∈W1,2(0,T;H0)/intersectiondisplay
L2(0,T;H2(0,1)),
and there exists a function ϑ∈L1(0,T)such that
φ1(t,/tildewidez(t))≤φ1(0,/tildewidez(0)) +/integraldisplayt
0ϑ(s)ds,∀t∈[0,T],
which implies
1
2r/integraldisplay1
0z2
x(x,t)dx≤C15+σ/prime
1(t)z(0,t)+σ/prime
2(t)z(1,t)
≤C16+C17/bardblz(·,t)/bardbl2
0+1
4r/bardblzx(·,t)/bardbl2
0,∀t∈[0,T],
from which we get /bardblzx(·,t)/bardbl0≤C18.Therefore, z∈L∞(0,T;H1(0,1)).We have used σ1,σ2∈
H2(0,T),z∈W1,2(0,T;L2(0,1))as well as inequality (1.2.30). As z=Yt,we have already proved
that
Y∈W2,2(0,T;L2(0,1))/intersectiondisplay
W1,2(0,T;H2(0,1))/intersectiondisplay
W1,∞(0,T;H1(0,1)).
It remains to show that Ytt=zt∈L2(0,T;H1(0,1)).To this purpose one can apply a reasoning
similar to that used in the ﬁrst part of this proof. We need only some slight modiﬁcations which weare going to point out. Starting from equations (1.2.25)
1and
(z−z0)t−r−1(zxx−z/prime/prime
0)+g(z−z0)=ht−A1(0)z0,
15

Section 1.2 Chapter 1
one gets the following estimates
1
2d
dt/bardbl/tildewidez(t+δ)−/tildewidez(t)/bardbl2
0+1
r/bardblzx(·,t+δ)−zx(·,t)/bardbl2
0
≤/parenleftBig
|σ/prime
1(t+δ)−σ/prime
1(t)|+|z(0,t)|·|α(t+δ)−α(t)|/parenrightBig
|z(0,t+δ)−z(0,t)|
+/parenleftBig
|σ/prime
2(t+δ)−σ/prime
2(t)|+|z(1,t)|·|β(t+δ)−β(t)|/parenrightBig
|z(1,t+δ)−z(1,t)|
+/bardbl/tildewideh/prime(t+δ)−/tildewideh/prime(t)/bardbl0·/bardbl/tildewidez(t+δ)−/tildewidez(t)/bardbl0,
for a. e. 0 ≤t≤t+δ≤T, and
1
2d
dδ/bardbl/tildewidez(δ)−z0/bardbl2
0+1
r/bardblzx(·,δ)−z/prime
0/bardbl20
≤/parenleftBig
|σ/prime
1(δ)−σ/prime
1(0)|+|z0(0)|·|α(δ)−α(0)|/parenrightBig
×|z(0,δ)−z0(0)|+/parenleftBig
|σ/prime
2(δ)−σ/prime
2(0)|+|z0(1)|·|β(δ)−β(0)|/parenrightBig
×|z(1,δ)−z0(1)|+/bardbl/tildewideh/prime(δ)−A1(0)z0/bardbl0·/bardbl/tildewidez(δ)−z0(0)/bardbl0,
for a. e. δ∈(0,T). We have used the conditions α≥0,β≥0.
Asz(0,·),z(1,·)∈L∞(0,T),one can use the fact that H1(0,1)is continuously embedded into C[0,1]
to derive the following estimates
1
2d
dt/bardbl/tildewidez(t+δ)−/tildewidez(t)/bardbl2
0+1
r/bardblzx(·,t+δ)−zx(·,t)/bardbl2
0≤1
2r/bardblzx(·,t+δ)−zx(·,t)/bardbl2
0
+C18/parenleftBig
(σ/prime
1(t+δ)−σ/prime
1(t))2+(σ/prime
2(t+δ)−σ/prime
2(t))2+(α(t+δ)−α(t))2+(β(t+δ)−β(t))2/parenrightBig
+/parenleftBig
C19(|σ/prime
1(t+δ)−σ/prime
1(t)|+|σ/prime
2(t+δ)−σ/prime
2(t)|
+|α(t+δ)−α(t)|+|β(t+δ)−β(t)|)+/bardbl/tildewideht(t+δ)−/tildewideht(t)/bardbl0/parenrightBig
/bardbl/tildewidez(t+δ)−/tildewidez(t)/bardbl0,
(1.2.37)
for a. e. 0 ≤t≤t+δ≤T, and
1
2d
dδ/bardbl/tildewidez(δ)−z0/bardbl2
0+1
r/bardblzx(·,δ)−z/prime
0/bardbl20≤1
2r/bardblzx(·,δ)−z/prime
0/bardbl20
+C20/parenleftBig
(σ/prime
1(δ)−σ/prime
1(0))2+(σ/prime
2(δ)−σ/prime
2(0))2+(α(δ)−α(0))2+(β(δ)−β(0))2/parenrightBig
+/parenleftBig
C21(|σ/prime
1(δ)−σ/prime
1(0)|+|σ/prime
2(δ)−σ/prime
2(0)|+|α(δ)−α(0)|+|β(δ)−β(0)|)
+/bardbl/tildewideh/prime(δ)−A1(z0)/bardbl0/parenrightBig
/bardbl/tildewidez(δ)−z0/bardbl0,(1.2.38)
for a. e. δ∈(0,T), where C18,···,C21are some positive constants.
Note that every ζ∈H1(0,T)satisﬁes the following estimate
|ζ(s1)−ζ(s2)|=/vextendsingle/vextendsingle/vextendsingle/integraldisplays2
s1ζ/prime(τ)dτ/vextendsingle/vextendsingle/vextendsingle≤/bardblζ/prime/bardblL2(0,T)·|s1−s2|1/2∀s1,s2∈[0,T].
16

Section 1.2 Chapter 1
Thus, since α,β,σ/prime
1,σ/prime
2∈H1(0,T)and/tildewideh/prime,/tildewidez∈W1,2(0,T;H0),we can derive from (1.2.38)
/bardbl/tildewidez(δ)−z0/bardbl2
0≤C22δ2+C23/integraldisplayδ
0/bardbl/tildewidez(s)−z0(0)/bardbl0ds.
Then, Gronwall’s lemma yields /bardbl/tildewidez(δ)−z0/bardbl0≤C24δ. This together with (1.2.37) implies the desired
conclusion, i.e., z∈W1,2(0,T;H1(0,1)).
Summarizing what we have done so far, we can state the following concluding result:
Corolary 1.2.1. Assume that
r>0,g≥0,r0,f0∈C2(R),r/prime
0≥0,f/prime
0≥0;f1,f2are sufﬁciently regular;
u0∈H2(0,1),v0∈H4(0,1)and the following compatibility conditions are fulﬁlled
/braceleftBigg
u0(0)+r0(v0(0)) = 0,
u0(1)−f0(v0(1)) = 0,
⎧
⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩f
1(0,0)−ru0(0)−v/prime
0(0)=0,
r/prime
0(v0(0))(f2(0,0)−gv0(0)−u/prime
0(0)) = 0,
f1(1,0)−ru0(1)−v/prime
0(1)=0,
f/prime
0(v0(1))(f2(1,0)−gv0(1)−u/prime
0(1)) = 0,
⎧
⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩r
−1v(3)
0(0)−gv/prime
0(0)+f2x(0,0)−r−1f1xx(0,0)−rr/prime
0(v0(0))×
×/parenleftbig
r−1v/prime/prime
0(0)−gv0(0)+f2(0,0)−r−1f1x(0,0)/parenrightbig
=f1t(0,0),
r−1v(3)
0(1)−gv/prime
0(1)+f2x(1,0)−r−1f1xx(1,0)+rf/prime
0(v0(1))×
×/parenleftbig
r−1v/prime/prime
0(1)−gv0(1)+f2(1,0)−r−1f1x(1,0)/parenrightbig
=f1t(1,0).
Then, all the conclusions of Theorems 1.2.1 and 1.2.2 hold.
Remark 1.2.1. It should be noted that in the statement of the above corollary, the compatibility
conditions are independent of ε. Note also that (1.2.21) ,(1.2.22) as well as the compatibility condi-
tions required for the reduced problem P 0form a compatible system of conditions. These conditions
include in particular our previous conditions (1.2.18) , which were required to eliminate possible
discrepancies caused by correction c 0at the corner points (x,t)=( 0,0)and(x,t)=( 1,0).
1.2.3 Estimates for the remainder components
The following Theorem are based on the result form [10, Section 4].
Under the assumptions of Corollary 1.2.1, our expansion (1.2.9) is well deﬁned, in the sense
that all its terms exist. We are now going to show it is a real asymptotic expansion, that is theremainder tends to zero with respect to the sup norm.
Theorem 1.2.3. Assume that all the assumptions of Corollary 1.2.1 are fulﬁlled. Then, for every ε>
0,the solution of problem P
εadmits an asymptotic expansion of the form (1.2.9) and the following
estimates hold /bardblR1ε/bardblC(DT)=O(ε1/8),/bardblR2ε/bardblC(DT)=O(ε3/8).
17

Section 1.2 Chapter 1
Proof. Throughout this proof we denote by K1,K2,… some positive constants which depend on the
data, but are independent of ε.By Corollary 1.2.1, problem (1.2.13), (1.2.16),(1.2.19) has a unique
smooth solution (R1ε,R2ε).
In order to establish the desired estimates, we consider the Hilbert space H:=L2(0,1)2, en-
dowed with the scalar product /angbracketleftp,q/angbracketright:=ε/integraltext1
0p1(x)q1(x)dx+/integraltext1
0p2(x)q2(x)dx,p=(p1,p2),q=
(q1,q2)∈H.Denote by /bardbl·/bardbl the corresponding Hilbertian norm. Also, we consider the operator
Bε(t):D/parenleftbig
Bε(t)/parenrightbig
⊂H→H,
D/parenleftbig
Bε(t)/parenrightbig
=/braceleftBig
(p,q)∈/parenleftbig
H1(0,1)/parenrightbig2,p(0)+r0/parenleftbig
q(0)+Y(0,t)/parenrightbig
=r0/parenleftbig
Y(0,t)/parenrightbig
,
p(1)−f0/parenleftbig
q(1)+Y(1,t)/parenrightbig
=−f0/parenleftbig
Y(1,t)/parenrightbig/bracerightBig
,
Bε(t)(p,q)=/parenleftbig
ε−1q/prime+rε−1p,p/prime+gq/parenrightbig
for all t∈[0,T].
Obviously, problem (1.2.13), (1.2.16), (1.2.19) can be written as the following Cauchy problem in
H: /braceleftBigg
R/prime
ε(t)+Bε(t)Rε(t)=Fε(t),0<t<T,
Rε(0)=0,(1.2.39)
where Rε(t):=(R1ε(·,t),R2ε(·,t)),Fε(t):=/parenleftbig
−Xt(·,t),−c0x(·,τ)/parenrightbig
,0<t<T.
Taking the scalar product in Hof (1.2.39) 1andRε(t)and integrating the resulting equation
over [0,t],w eg e t
1
2/bardblRε(t)/bardbl2+/integraldisplayt
0/angbracketleftBε(s)Rε(s),Rε(s)/angbracketrightds=/integraldisplayt
0/angbracketleftFε(s),Rε(s)/angbracketrightds (1.2.40)
for all t∈[0,T].We have denoted by /bardbl·/bardbl0the usual norm of L2(0,1).An easy computation involving
the obvious inequalities r/prime
0,f/prime
0≥0 shows that
/angbracketleftBε(s)Rε(s),Rε(s)/angbracketright≥r/vextenddouble/vextenddoubleR1ε(·,s)/vextenddouble/vextenddouble2
0+g/vextenddouble/vextenddoubleR2ε(·,s)/vextenddouble/vextenddouble2
0for all s∈[0,T]. (1.2.41)
Therefore,
1
2/bardblRε(t)/bardbl2≤/integraldisplayt
0/bardblFε(s)/bardbl·/bardblRε(s)/bardbldsfor all t∈[0,T].
By Gronwall’s lemma, it follows that
/bardblRε(t)/bardbl≤/integraldisplayt
0/bardblFε(s)/bardbldsfor all t∈[0,T]. (1.2.42)
On the other hand,
/bardblFε(s)/bardbl2=ε/integraldisplay1
0Xs(x,s)2dx+/integraldisplay1
0c0x/parenleftbig
x,s/ε/parenrightbig2dx≤K1ε
for all s∈[0,T]. Thus,/integraldisplayT
0/bardblFε(s)/bardblds=O(ε1/2). (1.2.43)
Now, it is easily seen that (1.2.40), (1.2.41), (1.2.42) and (1.2.43) imply
/vextenddouble/vextenddoubleRε(t)/vextenddouble/vextenddouble2=ε/vextenddouble/vextenddoubleR1ε(·,t)/vextenddouble/vextenddouble2
0+/vextenddouble/vextenddoubleR2ε(·,t)/vextenddouble/vextenddouble2
0≤K2εfor all t∈[0,T], (1.2.44)
18

Section 1.2 Chapter 1
/integraldisplayT
0/parenleftbig
r/vextenddouble/vextenddoubleR1ε(·,s)/vextenddouble/vextenddouble2
0+g/vextenddouble/vextenddoubleR2ε(·,s)/vextenddouble/vextenddouble2
0/parenrightbig
ds=O(ε). (1.2.45)
In order to continue the proof, we need the following auxiliary result:
Lemma 1.2.1. If the assumptions of Theorem 1.2.3 hold, then /bardblvε/bardblC(DT)=O(1).
Proof. We deﬁne the operator B1ε:D(B1ε)⊂H→H,
D(B1ε)=/braceleftbig
(p,q)∈H1(0,1)2,p(0)+r0/parenleftbig
q(0)/parenrightbig
=0,p(1)−f0/parenleftbig
q(1)/parenrightbig
=0/bracerightbig
,
B1ε(p,q)=/parenleftbig
ε−1(q/prime+rp),p/prime+gq/parenrightbig
.
Obviously, problem Pεcan be written as the following Cauchy problem in H:
/braceleftBigg
U/prime
ε(t)+B1εUε(t)=Gε(t),0<t<T,
Uε(0)=( u0,v0),(1.2.46)
where Uε(t):=/parenleftbig
uε(·,t),vε(·,t)/parenrightbig
,Gε(t):=/parenleftbig
ε−1f1(·,t),f2(·,t)/parenrightbig
,0<t<T.From Theorem 1.1.1 the
strong solution Uεof the Cauchy problem (1.2.46) satsﬁes the inequality:
/bardblU/prime
ε(t)/bardbl≤/bardbl Gε(0)−B1ε(u0,v0)/bardbl+/integraldisplayt
0/bardblG/prime
ε(s)/bardblds≤K3ε−1/2
for all t∈[0,T], which implies
ε/bardbluεt(·,t)/bardbl2
0+/bardblvεt(·,t)/bardbl2
0≤K2
3ε−1∀t∈[0,T]. (1.2.47)
On the other hand, by (1.2.44) we obtain the estimates:
/bardblvε(·,t)/bardbl0≤/bardblY(·,t)/bardbl0+/bardblR2ε(·,t)/bardbl0≤K4, (1.2.48)
/bardbluε(·,t)/bardbl0≤/bardblX(·,t)/bardbl0+/bardblc0(·,τ)/bardbl0+/bardblR1ε(·,t)/bardbl0≤K5, (1.2.49)
for all t∈[0,T].By making use of (1.2.47) and (1.2.49), we derive from the ﬁrst equation of (LS)
/bardblvεx(·,t)/bardbl0≤K6,0≤t≤T.
Together with (1.2.48), this last inequality shows that vε(·,t)is uniformly bounded with respect to
ε>0 and t∈[0,T]inH1(0,1).Since H1(0,1)is continuously embedded into C[0,1], our conclusion
follows.
Let us continue the proof of the theorem. We are going to prove some estimates for R1εt,R2εt.
Denote R/prime
ε(t)=/parenleftBig
R1εt/parenleftbig
·,t/parenrightbig
,R2εt/parenleftbig
·,t/parenrightbig/parenrightBig
and write system (1.2.13) in t,t+δ∈[0,T],δ>0, then
subtract one system from the other, and ﬁnally take the scalar product in Hof the resulting system
with Rε(t+δ)−Rε(t).Thus, we get
1
2d
dt/bardblRε(t+δ)−Rε(t)/bardbl2+r/bardblR1ε(·,t+δ)−R1ε(·,t)/bardbl2
0+E1ε(t,δ)+E2ε(t,δ)
≤/bardblFε(t+δ)−Fε(t)/bardbl·/bardbl Rε(t+δ)−Rε(t)/bardbl(1.2.50)
19

Section 1.2 Chapter 1
for all 0≤t<t+δ≤T, where
E1ε(t,δ)=/bracketleftBig/parenleftbig
f0(vε(1,t+δ))−f0(Y(1,t+δ))/parenrightbig
−/parenleftbig
f0(vε(1,t))−f0(Y(1,t))/parenrightbig/bracketrightBig
×/bracketleftBig
R2ε(1,t+δ)−R2ε(1,t)/bracketrightBig
,
E2ε(t,δ)=/bracketleftBig/parenleftbig
r0(vε(0,t+δ))−r0(Y(0,t+δ))/parenrightbig
−/parenleftbig
r0(vε(0,t))−r0(Y(0,t))/parenrightbig/bracketrightBig
×/bracketleftBig
R2ε(0,t+δ)−R2ε(0,t)/bracketrightBig
.
Integration of (1.2.50) over [0,t]yields
1
2/bardblRε(t+δ)−Rε(t)/bardbl2+r/integraldisplayt
0/bardblR1ε(·,s+δ)−R1ε(·,s)/bardbl2
0ds
+/integraldisplayt
0/parenleftbig
E1ε(s,δ)+E2ε(s,δ)/parenrightbig
ds≤1
2/bardblRε(δ)/bardbl2
+/integraldisplayt
0/bardblFε(s+δ)−Fε(s)/bardbl·/bardbl Rε(s+δ)−Rε(s)/bardblds(1.2.51)
for all 0≤t<t+δ≤T.
Now, we divide (1.2.51) by δ2. Then, if we take into account that Rε∈C1([0,T];H),r0,f0∈C2(R),
Fε∈W1,2(0,T;H),and let δ→0, we infer that
1
2/bardblR/prime
ε(t)/bardbl2+r/integraldisplayt
0/bardblR1εt(·,s)/bardbl2
0ds≤K7+1
2/bardblR/prime
ε(0)/bardbl2+/integraldisplayt
0/bardblF/prime
ε(s)/bardbl·/bardbl R/prime
ε(s)/bardblds (1.2.52)
for all t∈[0,T].We have used the following inequality
lim
δ→0/integraldisplayt
0E1ε(s,δ)
δ2ds=/integraldisplayt
0d
ds/bracketleftBig/parenleftbig
f0(vε(1,s))−f0(Y(1,s))/parenrightbig/bracketrightBig
·d
ds/parenleftbig
R2ε(1,s)/parenrightbig
ds
=/integraldisplayt
0/parenleftbig
f/prime
0(vε(1,s))vεs(1,s)−f/prime
0(Y(1,s))Ys(1,s)/parenrightbig/parenleftbig
vεs(1,s)−Ys(1,s)/parenrightbig
ds
≥−/integraldisplayt
0/parenleftbig
f/prime
0(vε(1,s))vεs(1,s)Ys(1,s)+f/prime
0(Y(1,s))vεs(1,s)Ys(1,s)/parenrightbig
ds
=−f0(vε(1,s))Ys(1,s)/vextendsingle/vextendsingle/vextendsinglet
0−f/prime
0(Y(1,s))Ys(1,s)vε(1,s)/vextendsingle/vextendsingle/vextendsinglet
0
+/integraldisplayt
0f0(vε(1,s))Yss(1,s)ds+/integraldisplayt
0d2
ds2/parenleftBig
f0(Y(1,s))/parenrightBig
vε(1,s)ds≥−K8
for all t∈[0,T], as well as the similar one
lim
δ→0/integraldisplayt
0E2ε(s,δ)
δ2ds≥−K9for all t∈[0,T].
For the last two inequalities we have used Lemma 1.2.1. Now, if we combine inequality (1.2.52) and
the obvious estimates
/bardblR/prime
ε(0)/bardbl=/bardblFε(0)/bardbl≤K10,/integraldisplayt
0/bardblF/prime
ε(s)/bardblds≤K11for all t∈[0,T],
we get by Gronwall’s lemma
/bardblR/prime
ε(t)/bardbl2=ε/bardblR1εt(·,t)/bardbl2
0+/bardblR2εt(·,t)/bardbl2
0≤K12, (1.2.53)
20

Section 1.3 Chapter 1
for all t∈[0,T].Therefore,
ε/bardblR1εt(·,t)/bardbl2
0≤K12,/bardblR2εt(·,t)/bardbl0≤K12, (1.2.54)
for all t∈[0,T].Now, combining (1.2.52) with (1.2.53) we ﬁnd
/bardblR1εt/bardblL2(DT)≤K13. (1.2.55)
From (1.2.45) and (1.2.54), we derive
/bardblR1ε(·,t)/bardbl2
0=2/integraldisplayt
0/angbracketleftR1εs(·,s),R1ε(·,s)/angbracketright0ds≤2/integraldisplayt
0/bardblR1εs(·,s)/bardbl0·/bardblR1ε(·,s)/bardbl0ds
≤2/bardblR1εt/bardblL2(DT)·/bardblR1ε/bardblL2(DT)=O(ε1/2)∀t∈[0,T]. (1.2.56)
Using (1.2.13), (1.2.44), (1.2.54) and (1.2.56), we obtain
/bardblR1εx(·,t)/bardbl0≤K14,/bardblR2εx(·,t)/bardbl0≤K15ε1/4for all t∈[0,T]. (1.2.57)
By the mean value theorem, for every t∈[0,T]andε>0 there exists a point xtε∈[0,1]such that
/bardblR1ε(·,t)/bardbl2
0=R1ε(xtε,t)2. Since
R1ε(x,t)2=R1ε(xtε,t)2+2/integraldisplayx
xtεR1εξ(ξ,t)R1ε(ξ,t)dξ≤R1ε(xtε,t)2+2/bardblR1ε(·,t)/bardbl0·/bardblR1εx(·,t)/bardbl0,
we obtain by (1.2.56) and (1.2.57) that R1ε(x,t)2≤K16ε1/4for all (x,t)∈DT.
Similarly, we can show that R2ε(x,t)2≤K17ε3/4for all (x,t)∈DT.The proof is complete.
Remark 1.2.2. We suspect that the above estimates could be proved under weaker assumptions on
the data. Also, estimates in weaker norms are expected under even more relaxed assumptions on
the data, including less compatibility. Furthermore, one may investigate only simple convergence tozero of the remainder components with respect to different norms. From a practical point of view, itis important to relax our requirements. This seems to be possible, at the expense of getting weakerapproximation results. Note that the set of regular data (u
0,v0,f1,f2), as required in Corollary
1.2.1, is dense in the space V :=L2(0,1)×H1(0,1)×H1(DT)×L2(DT). It is easily seen that for
(u0,v0,f1,f2)∈V , both problems P εand P 0have unique weak solutions (i.e., limits of strong so-
lutions in C ([0,T];L2(0,1))2, and L2(DT)×{C([0,T];L2(0,1))∩L2(0,T;H1(0,1))}, respectively),
provided that f 0and r 0are smooth nondecreasing functions. This remark could be a starting point
in proving further approximation results.
1.3 On some singularly perturbed, coupled parabolic-parabolic prob-
lems
In the following we are particularly interested in coupled problems in which a small parameter
is present. They are mathematical models for diffusion-convection reaction processes. For speciﬁc
21

Section 1.3 Chapter 1
problems describing heat or mass transfer, we refer to A. K. Datta [45] (see also the references
therein).
Let us consider in the rectangle QT=(a,c)×(0,T),a<c,0<T, the following system of
parabolic equations
/braceleftBigg
ut+(−εux+α1(x)u)x+β1(x)u=f(x,t)inQ1T,
vt+(−μ(x)vx+α2(x)v)x+β2(x)v=g(x,t)inQ2T,(Sε)
with which we associate initial conditions
u(x,0)=u0(x),a≤x≤b;v(x,0)=v0(x),b≤x≤c, (IC)
transmissions conditions at x=b
/braceleftBigg
u(b,t)=v(b,t),
−εux(b,t)+α1(b)u(b,t)=−μ(b)vx(b,t)+α2(b)v(b,t),0≤t≤T,(TCε)
as well as one of the following types of boundary conditions
u(a,t)=v(c,t)=0,0≤t≤T; (BC.1)
ux(a,t)=v(c,t)=0,0≤t≤T; (BC.2)
u(a,t)=0,−vx(c,t)=γ(v(c,t)),0≤t≤T, (BC.3)
where Q1T=(a,b)×(0,T),Q2T=(b,c)×(0,T),b∈R,a<b<c,γis a given nonlinear function
andεis a small parameter, 0 <ε/lessmuch1.
The following general assumptions will be required:
(i1)α1∈H1(a,b),β1∈L2(a,b),(1/2)α/prime
1+β1≥C1a. e. on (a,b),for some constant C1;
(i2)α2∈H1(b,c),β2∈L2(b,c),(1/2)α/prime
2+β2≥C2a. e. on (b,c),for some constant C2;
(i3)μ∈H1(b,c),μ(x)≥μ0>0;
(i4)f:Q1T→R,g:Q2T→R,u0:[a,b]→R,v0:[b,c]→R;
(i5)α1>0o n [a,b];o r
(i/prime
5)α1<0o n [a,b];
(i6)γ:D(γ)=R→Ris a continuous nondecreasing function.
Denote by (P.k)εthe problem which consists of (Sε),(IC),(TCε),(BC.k), for k=1,2,3.
Let us recall some things about heat transfer. We know that conductive heat transfer is nothing
else but the movement of thermal energy through the corresponding medium (material) from themore energetic particles to the others. Of course, the local temperature is given by the energy ofthe molecules situated at that place. So, thermal energy is transferred from points with higher tem-perature to other points. If the temperature of some area of the medium increases, then the randommolecular motion becomes more intense in that area. Thus a transfer of thermal energy is produced,which is called heat diffusion. The corresponding conductive heat ﬂux, denoted q
1(x,t), is given by
the Fourier’s rate law
q1=−kTx,
22

Section 1.3 Chapter 1
where kis the thermal conductivity of the medium, and T=T(x,t)is the temperature at point x,a t
time instant t. We will assume that kdepends on xonly.
On the other hand, convective heat transfer is a result of the bulk ﬂow (or net motion) in the
medium, if present. The heat ﬂux due to convection, say q2(x,t),i sg i v e nb y
q2=αρcp(T−T0),
where αis the velocity of the medium in the xdirection, ρis the density, cpis the speciﬁc heat of
the material, and T0is a reference temperature.
Therefore, the total ﬂux is
q=q1+q2=−kTx+αρcp(T−T0).
We will assume that αdepends on xonly.
The ﬁrst law of thermodynamics (conservation of energy) gives in a standard manner (see, e.g., A.K. Datta [45, pp. 29-31]) the well known heat equation
(kT
x)x−ρcp(αT)x+Q=ρcpTt,
where Qis the rate of generated heat per unit volume. By generation we mean the transformation of
energy from one form (e.g., mechanical, electrical, etc.) into heat. The right hand side of the aboveequation represents the stored heat. In other words, the heat equation looks like
T
t+(−μTx+αT)x=S(x,t),where μ=k
ρcp,S=Q
ρcp.
Recall that μis called thermal diffusivity. In our model we do not take into account radiation which
is due to the presence of electromagnetic waves. Note that the same partial differential equation is amodel for mass transfer, but in this case T(x,t)represents the mass density of the material at point
x, at time t. Instead of the Fourier law, a similar law is available in this case, which is called Fick’s
law of mass diffusion. An additional term depending on the density may appear in the equation dueto reaction. Again, both the diffusion coefﬁcient μand the velocity ﬁeld αare assumed to depend
onxonly.
Now, let us assume that the diffusion in the subinterval [a,b]is negligible. Thus, as an approx-
imation of the physical process, we set in our model μ(x)=εforx∈[a,b]. On the other hand, we
assume that μis sizeable in [b,c]. Denoting by uandvthe restrictions of T(which means tem-
perature or mass density) to [a,b]and[b,c], we get the above system (S
ε). Of course, a complete
mathematical model should include initial conditions, boundary conditions, as well as transmission
conditions at x=b, as formulated above. Concerning our transmission conditions (TCε), they are
naturally associated with system (Sε). Indeed, they express the continuity of the solution as well as
the conservation of the total ﬂux at x=b. The most important boundary conditions which may occur
in applications are the following:
1.) The temperature (or mass density) is known on a part of the boundary of the domain or on
the whole boundary, in our case at x=aand/or x=c. For example, let us suppose that u(a,t)=
s1(t),u(c,t)=s2(t),0≤t≤T, where s1,s2are two given smooth functions. In this case, one can
reduce the problem to another similar problem in which we have homogeneous Dirichlet boundary
23

Section 1.3 Chapter 1
conditions at x=aandx=c. Therefore, the new problem is of the form (P.2)ε. Note however that
the right hand side of the ﬁrst partial differential equation of the new system includes an O(ε)-term.
But the treatment is basically the same as for the case in which εis not present there.
2.) Sometimes the heat ﬂux is speciﬁed on a part of the boundary. In this case, we have a
boundary condition of the following form, say at x=a,
−k(a)ux(a,t)=s3(t).
Let us also assume that a (possibly nonhomogeneous) Dirichlet boundary condition is satisﬁed at
x=c. Then, as before, the corresponding coupled problem may be transformed into a problem of
the type (P.2)ε. Again, a term depending on εoccurs in the right hand side of the ﬁrst equation of
system (Sε), but this does not essentially change the treatment.
If the boundary point x=ais highly insulated, then s3(t)=0, and so we do not need to change u.
3.) The most usual situation is that in which heat (or mass) conducted out of the boundary is
convected away by the ﬂuid. Let us suppose this happens at x=c. The balance equation will be of
the following Robin’s type
−k(c)vx(c)=h[v(c,t)−v∗],
where his the transfer coefﬁcient, and v∗is a constant representing the temperature (or mass density)
of the environment. Thus, if we assume a Dirichlet condition at x=a, we have a coupled problem
of the form (P.3)ε. Sometimes, the above boundary balance equation is nonlinear.
Certainly, simpler models corresponding to ε=0 are preferred. But taking ε=0i n(Sε)means
a dramatic change of the equation (from parabolic to hyperbolic). On the other hand, it is physicallyobvious that the ﬂux remains continuous at x=b.This fact should be reﬂected in the corresponding
reduced (unperturbed) models. Once we derive such reduced models (this is done below), we askourselves whether these simpler models have solutions which are close enough to the solutions ofthe original models (P.k)
ε,k=1,2,3 (that is more realistic since the diffusion in [a,b]is just small,
not completely absent). Therefore a mathematical analysis of such problems is extremely important.In particular, this is very useful for the numerical solution. Let us also point out that F. Gastaldi-A.Quarteroni [55] discussed the coupling of parabolic and hyperbolic systems as a ﬁrst step in the
numerical treatment of the Navier– Stokes/Euler coupling which is a key issue in ComputationalFluid Dynamics.
This section contains three subsections. Each subsection addresses one of the three problems
(P.k)
ε,k=1,2,3.
In subsection 1.3.1, problem (P.1)εis investigated, under hypotheses (i1)−(i5).This problem
is singularly perturbed with respect to the uniform norm, with a boundary layer located on the leftside of the line segment {(b,t);t∈[0,T]}. A ﬁrst order asymptotic expansion of the solution will
be constructed. To validate this expansion, we will prove existence and regularity results for theperturbed problem as well as for the problems satisﬁed by the two terms of the regular series whichare present in our expansion. In addition, we will obtain estimates for the remainder components,with respect to the uniform convergence norm.
In the second subsection, we investigate problem (P.2)
ε,under the same requirements, except
for(i5)which is replaced by (i/prime
5).For the solution of this problem, we construct an asymptotic
24

Section 1.3 Chapter 1
expansion of the order one with respect to the uniform norm. Note that (P.2)εis regularly perturbed
of order zero with respect to this norm, so there is no correction in the asymptotic expansion. Butthe problem is singularly perturbed of order one, with respect to the same norm. Indeed, in the ﬁrstorder asymptotic expansion we will construct, a ﬁrst order boundary layer function will be present,corresponding to a boundary layer located near the right side of the line segment {(a,t);t∈[0,T]}.
In the third and ﬁnal subsection we deal with problem (P.3)
ε,which is nonlinear due to the
nonlinear boundary condition at x=c. We restrict ourselves to the construction of a zeroth order
asymptotic expansion, under assumptions (i1)−(i6).As expected, (P.3)εis singularly perturbed
with respect to the uniform norm. We again perform a detailed asymptotic analysis, which is a bitmore difﬁcult due to the nonlinear character of the problem.
Throughout this section we will denote by /bardbl·/bardbl
1and/bardbl·/bardbl 2the norms of L2(a,b)andL2(b,c),
respectively.
1.3.1 A ﬁrst order asymptotic expansion for
the solution of problem (P.1)ε
In this subsection we examine problem (P.1)εformulated above, for which assumptions (i1)−(i5)
are required. This problem is singularly perturbed with respect to the uniform norm, with a boundarylayer in the vicinity of the line segment Σ={(b,t);t∈[0,T]}as it can easily been seen (see also L.
Barbu-G. Moros ¸anu [18, Subsection 7.1.1] for the stationary case of the problem).
Formal expansion
Let us denote by U
ε:=/parenleftbig
uε(x,t),vε(x,t)/parenrightbig
the solution of problem (P.1)ε. In the following we derive a
ﬁrst order asymptotic expansion of this solution following the classical perturbation theory presentedin Subsection 1.1.1. Thus, we seek the solution of our problem in the form
/braceleftBigg
u
ε(x,t)=X0(x,t)+εX1(x,t)+i0(ξ,t)+εi1(ξ,t)+R1ε(x,t),
vε(x,t)=Y0(x,t)+εY1(x,t)+R2ε(x,t),(1.3.58)
where:
ξ:=ε−1(b−x)is the fast variable associated with the left side of Σ;/parenleftbig
Xk(x,t),Yk(x,t)/parenrightbig
,k=0,1,are
the ﬁrst two regular terms; ik(ξ,t),k=0,1, are the transition layer corrections;/parenleftbig
R1ε(x,t),R2ε(x,t)/parenrightbig
is the ﬁrst order remainder.
As usual, we replace (1.3.58) into (Sε)and get
/braceleftBigg
Xkt+(α1Xk)x+β1Xk=/tildewidefkinQ1T,
Ykt+(−μYkx+α2Yk)x+β2Yk=/tildewidegkinQ2T,k=0,1,(1.3.59)
where
/tildewidefk(x,t)=/braceleftBigg
f(x,t),k=0,
X0xx(x,t),k=1,/tildewidegk(x,t)=/braceleftBigg
g(x,t),k=0,
0,k=1.
25

Section 1.3 Chapter 1
The transitions layer functions i0,i1satisfy the equations
⎧
⎨
⎩i0ξξ(ξ,t)+α1(b)i0ξ(ξ,t)=0,
i1ξξ(ξ,t)+α1(b)i1ξ(ξ,t)=I1(ξ,t),t∈[0,T],ξ≥0,
where I1(ξ,t)=i0t(ξ,t)+ξα/prime
1(b)i0ξ(ξ,t)+(α/prime
1(b)+β1(b))i0(ξ,t). By easy computations, we de-
rive
i0(ξ,t)=θ0(t)e−α1(b)ξ,i1(ξ,t)=θ1(t)e−α1(b)ξ
+ξe−α1(b)ξ
α1(b)/parenleftbigg
−θ/prime
0(t)−β1(b)θ0(t)+α1(b)α/prime
1(b)
2θ0(t)ξ/parenrightbigg
,(1.3.60)
where θ0,θ1are as yet undetermined functions.
For the components of the remainder, R1ε,R2ε, we derive the system
/braceleftBigg
R1εt+(−εT1εx+α1R1ε)x+β1R1ε=hεinQ1T,
R2εt+/parenleftbig
−μR2εx+α2R2ε/parenrightbig
x+β2R2ε=0i n Q2T,(1.3.61)
where T1ε=uε−X0−i0−εi1=εX1+R1ε,
hε(x,t)=−εi1t(ξ,t)+[ε−1/parenleftbig
α1(x)−α1(b)) +ξα/prime
1(b)]i0ξ(ξ,t)
+(α1(x)−α1(b))i1ξ(ξ,t)−(α/prime
1(x)−α/prime
1(b))i0(ξ,t)
−(β1(x)−β1(b))i0(ξ,t)−ε(β1(x)+α/prime
1(x))i1(ξ,t)inQ1T.
Next, from (IC)and(BC.1)one gets
Xk(x,0)=/braceleftBigg
u0(x),k=0,
0,k=1,Yk(x,0)=/braceleftBigg
v0(x),k=0,
0,k=1,(1.3.62)
i0(ξ,0)+εi1(ξ,0)=0∀ε>0⇐⇒ θ0(0)=θ/prime
0(0)=θ1(0)=0, (1.3.63)
/braceleftBigg
R1ε(x,0)=0,a≤x≤b,
R2ε(x,0)=0,b≤x≤c,(1.3.64)
Xk(a,t)=Yk(c,t)=0,0≤t≤T,k=0,1, (1.3.65)
/braceleftBigg
R1ε(a,t)=Pε(ξ(a),t),
R2ε(c,t)=0,0≤t≤T,(1.3.66)
where ξ(a)=ε−1(b−a),Pε(ζ,t)=−i0/parenleftbig
ζ,t/parenrightbig
−εi1/parenleftbig
ζ,t/parenrightbig
.
Finally, on account of (TCε)we ﬁnd
θk(t)=Yk(b,t)−Xk(b,t),0≤t≤T,k=0,1, (1.3.67)
−μ(b)Y0x(b,t)=α1(b)X0(b,t)−α2(b)Y0(b,t),0≤t≤T, (1.3.68)
−μ(b)Y1x(b,t)+α2(b)Y1(b,t)=α1(b)X1(b,t)−X0x(b,t)−α1(b)−1(θ/prime
0(t)+β1(b)θ0(t)),(1.3.69)
26

Section 1.3 Chapter 1
⎧
⎪⎪⎨
⎪⎪⎩R
1ε(b,t)=R2ε(b,t),
−εT1εx(b,t)+α1(b)R1ε(b,t)
=−μ(b)R2εx(b,t)+α2(b)R2ε(b,t),0≤t≤T.(1.3.70)
In conclusion, from what we have done so far we see that the components of the zeroth order regular
term satisfy the reduced problem, say (P.1)0, which consists of (1.3.59) k=0, (1.3.62) k=0, (1.3.65) k=0
and (1.3.68), while the components of the ﬁrst order regular term satisfy the problem (P.1)1which
is made up by (1.3.59) k=1, (1.3.62) k=1, (1.3.65) k=1and (1.3.69). The remainder components satisfy
the problem (1.3.61), (1.3.64), (1.3.66) and (1.3.70).
Existence, uniqueness and regularity of the solutions
of problems (P.1)ε,(P.1)0and(P.1)1
In order to investigate problem (P.1)ε, we choose as a basic setup the Hilbert space H:=L2(a,b)×
L2(b,c),endowed with the usual scalar product, denoted /angbracketleft·,·/angbracketright, and the corresponding induced norm,
denoted/bardbl·/bardbl.This problem can be expressed as the Cauchy problem in H:
/braceleftBigg
W/prime
ε(t)+JεWε(t)=F(t),0<t<T,
Wε(0)=W0,(1.3.71)
where Jε:D(Jε)⊂H→H,
D(Jε):=/braceleftBig
(h,k)∈H2(a,b)×H2(b,c),h(b)=k(b),
h(a)=k(c)=0,εh/prime(b)−α1(b)h(b)=μ(b)k/prime(b)−α2(b)k(b)/bracerightBig
,
Jε(h,k):=/parenleftbig
(−εh/prime+α1h)/prime+β1h,(−μk/prime+α2k)/prime+β2k/parenrightbig
,
Wε(t):=/parenleftbig
uε(·,t),vε(·,t)/parenrightbig
,W0:=/parenleftbig
u0,v0),F(t):=/parenleftbig
f(·,t),g(·,t)/parenrightbig
.
Regarding operator Jε, one can prove that
Lemma 1.3.1. Assume that (i1)−(i3)and(i5)are satisﬁed. Then, there is a positive number ω,
independent of ε, such that J ε+ωI is maximal monotone, where I is the identity of H.
Proof. Obviously, Jεis well deﬁned and linear. To prove the monotonicity of Jε+ωIfor a suitable
ω, we can see that for (h,k)∈D(Jε)
/angbracketleftJε/parenleftbig
(h,k)/parenrightbig
+ω(h,k),(h,k)/angbracketright=−ε/integraldisplayb
ah/prime/primehdx+/integraldisplayb
a(α1h)/primehdx+/integraldisplayb
a(ω+β1)h2dx
−/integraldisplayc
b(μk/prime)/primekdx+/integraldisplayc
b(α2k)/primekdx+/integraldisplayc
b(ω+β2)k2dx
=ε/integraldisplayb
a(h/prime)2dx+/integraldisplayc
bμ(k/prime)2dx+[α(b)]
2k(b)2
+/integraldisplayb
a/parenleftbigg
ω+β1+α/prime
1
2/parenrightbigg
h2dx+/integraldisplayc
b/parenleftbigg
ω+β2+α/prime
2
2/parenrightbigg
k2dx,(1.3.72)
where [α(b)]:=α2(b)−α1(b).I f[α(b)]≥0, there is an ω>0 big enough which makes Jε+ωI
27

Section 1.3 Chapter 1
monotone. If [α(b)]<0, keeping in mind that k(c)=0,we have
[α(b)]
2k(b)2≥[α(b)]
2/parenleftbigg
δ/integraldisplayc
bk2dx+1
δ/integraldisplayc
b(k/prime)2dx/parenrightbigg
∀δ>0.
Taking δ=−[α(b)]/μ0in this inequality and using (1.3.72) we obtain
/angbracketleftJε/parenleftbig
(h,k)/parenrightbig
+ω(h,k),(h,k)/angbracketright≥ε/integraldisplayb
a(h/prime)2dx+μ0
2/integraldisplayc
b(k/prime)2dx
+/integraldisplayb
a/parenleftbigg
ω+β1+α/prime
1
2/parenrightbigg
h2dx+/integraldisplayc
b/parenleftbigg
ω+β2+α/prime
2
2−[α(b)]2
2μ0/parenrightbigg
k2dx≥0,
forωbig enough. Now, we are going to show that operator Jε+ωIis maximal monotone or, equiv-
alently (see, e. g., V . Barbu [26, Theorem 1.2, p. 43] ), for all (f1,f2)∈H,there exists (h,k)∈D(Jε),
such that (h,k)+(Jε+ωI)(h,k)=( f1,f2),that is, the following problem
⎧
⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩(−εh
/prime+α1h)/prime+(β1+ω+1)h=f1inL2(a,b),
(−μk/prime+α2k)/prime+(β2+ω+1)k=f2inL2(b,c),
h(a)=k(c)=0,h(b)=k(b),
−εh/prime(b)+α1(b)h(b)=−μ(b)k/prime(b)+α2(b)k(b)
has a solution (h,k)∈H2(a,b)×H2(b,c).This fact follows as in the proof of [18, Theorem 7.1.1,
p. 124], if ωis sufﬁciently large.
As far as problem (P.1)εis concerned, we have the following result
Theorem 1.3.1. Assume that (i1)−(i5)are fulﬁlled and F ∈W1,1(0,T;H),W0∈D(Jε).Then prob-
lem(1.3.71) has a unique strong solution W εwhich belongs to
C1([0,T];H)/intersectiondisplay
W1,2(0,T;H1(a,b)×H1(b,c))/intersectiondisplay
C([0,T];H2(a,b)×H2(b,c)).
If, in addition,
F∈W2,1(0,T;H); (1.3.73)
W0∈D(Jε),F(0)−JεW0∈D(Jε), (1.3.74)
then W εbelongs to
C2([0,T];H)/intersectiondisplay
W2,2(0,T;H1(a,b)×H1(b,c))/intersectiondisplay
C1([0,T];H2(a,b)×H2(b,c)).
Proof. By Lemma 1.3.1, Jε+ωIis maximal monotone for some ω>0. Therefore, according to
M. Ahsan-G. Moros ¸anu [1, Theorem 2.10, p. 762], problem (1.3.71) has a unique strong solution
Wε∈C1([0,T];H). From
ε/integraldisplayb
auεx(x,t)2dx+μ0
2/integraldisplayc
bvεx(x,t)2dx≤/angbracketleftJεWε(t),Wε(t)/angbracketright+ω/bardblWε(t)/bardbl2
=/angbracketleftF(t)−W/prime
ε(t),Wε(t)/angbracketright+ω/bardblWε(t)/bardbl2≤const.
28

Section 1.3 Chapter 1
for all t∈[0,T],we infer that uεx∈C/parenleftbig
[0,T];L2(a,b)/parenrightbig
,vεx∈C/parenleftbig
[0,T];L2(b,c)/parenrightbig
.
It follows from (Sε)1thatuεxx∈C/parenleftbig
[0,T];L2(a,b)/parenrightbig
,that is uε∈C/parenleftbig
[0,T];H2(a,b)/parenrightbig
.
Now, using (Sε)2, we ﬁnd that (μvεx)x∈C/parenleftbig
[0,T];L2(b,c)/parenrightbig
.Since μ∈H1(b,c),μ>0,we have
(μvεx)∈C/parenleftbig
[0,T];H1(b,c)/parenrightbig
⇒vε∈C/parenleftbig
[0,T];H2(b,c)/parenrightbig
.
Let us now prove that Wε∈W1,2(0,T;H1(a,b)×H1(b,c)). By a standard calculation we ﬁnd
1
2d
dt/bardblWε(t+h)−Wε(t)/bardbl2+ε/integraldisplayb
a(uεx(x,t+h)−uεx(x,t))2dx
+μ0
2/integraldisplayc
b(vεx(x,t+h)−vεx(x,t))2dx
≤ω/bardblWε(t+h)−Wε(t)/bardbl2+/bardblWε(t+h)−Wε(t)/bardbl·/bardbl F(t+h)−F(t)/bardbl.
By integration over [0,T−h]this implies
ε/integraldisplayT−h
0/bardbluεx(·,t+h)−uεx(·,t)/bardbl2
1dt+μ0
2/integraldisplayT−h
0/bardblvεx(·,t+h)−vx(·,t)/bardbl2
2dt≤Ch2,
for some C>0. We have used our condition F∈W1,1(0,T;H)as well as the fact that Wε∈
C1([0,T];H). By virtue of Theorem 1.1.5 this last inequality gives
uεx∈W1,2(0,T;L2(a,b)),vεx∈W1,2(0,T;L2(b,c)),
so the ﬁrst part of theorem is proved.
In what follows we suppose that (1.3.73) and (1.3.74) hold. One may readily check that W/prime
εis
the strong solution of the following Cauchy problem in H
/braceleftBigg
W/prime
ε(t)+JεWε(t)=F/prime(t),0<t<T,
Wε(0)=F(0)−W0.
Therefore, using the ﬁrst part of the proof we can see that W/prime
εbelongs to
C1([0,T];H)/intersectiondisplay
W1,2(0,T;H1(a,b)×H1(b,c))/intersectiondisplay
C([0,T];H2(a,b)×H2(b,c)).
Remark 1.3.1. It is important to note that our asymptotic analysis works if Theorem 1.3.1 is valid for
allε>0. Fortunately, assumptions (1.3.73) and(1.3.74) hold if the following sufﬁcient assumptions
(independent of ε) are fulﬁlled:
⎧
⎪⎪⎨
⎪⎪⎩f∈W2,1(0,T;L2(a,b)),g∈W2,1(0,T;L2(b,c)),
f(·,0)∈H2(a,b),g(·,0)∈H2(b,c),α1∈H3(a,b),α2∈H3(b,c)
β1∈H2(a,b),β2∈H2(b,c),μ∈H3(b,c),u0∈H4(a,b),v0∈H4(b,c),
29

Section 1.3 Chapter 1
⎧
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩u
0(b)=v0(b),u/prime
0(b)=0,−μ(b)v/prime
0(b)=α1(b)u0(b)−α2(b)v0(b),
u0(a)=v0(c)=0,u0/prime/prime(a)=0,f(a,0)=α1(a)u/prime
0(a),
g(c,0)+(μv/prime
0)/prime(c)=α2(c)v/prime
0(c),u0/prime/prime(b)=0,
f(b,0)−(α1u0)/prime(b)−(β1u0)(b)
=g(b,0)+(μv/prime
0)/prime(b)−(α2v0)/prime(b)−(β2v0)(b),
u0(3)(b)=0,fx(b,0)=( α1u0)/prime/prime(b)+(β1u0)/prime(b),
−μ(b)[gx(b,0)+(μv/prime
0)/prime/prime(b)−(α2v0)/prime/prime−(β2v0)/prime(b)]
=(α1(b)−α2(b))[f(b,0)−(α1u0)/prime(b)−(β1u0)(b)].(1.3.75)
We continue with problems (P.1)0and(P.1)1. Our aim is to obtain existence, uniqueness and
sufﬁcient regularity for the solutions of these problems, which will allow us to validate our asymp-totic expansion and, even more, to obtain estimates for the remainder components.We start with (P.1)
1for which we need a solution (X1,Y1)satisfying
X1∈W1,2(0,T;H1(a,b)),Y1∈W1,2(0,T;H1(b,c)).
In order to homogenize the boundary conditions at b,w es e t
X1(x,t)=X1(x,t)+B(t)x+B1(t),(x,t)∈Q1T,
whereB(t)=ρ(t)/[α
1(b)(a−b)],B1(t)=−aB(t),ρ(t)=X0x(b,t)+α1(b)−1(θ/prime
0(t)+β1(b)θ0(t)),0≤t≤T.
A straightforward computation shows that (X1,Y1)satisﬁes the problem
⎧
⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎩X1t+(α1X1)x+β1X1=f1inQ1T,
Y1t−(μY1x−α2Y1)x+β2Y1=/tildewideg1inQ2T,
X1(x,0)=σ(x),x∈[a,b],Y(x,0)=0,x∈[b,c],
X1(a,t)=0,Y1(c,t)=0,
−μ(b)Y1x(b,t)+α2(b)Y1(b,t)=α1(b)X1(b,t),0<t<T,(1.3.76)
where σ(x)=B(0)(x−a),f1(x,t)=/tildewidef1(x,t)+B/prime(t)x+B/prime
1(t)+B(t)α1(x)+(α/prime
1(x)+β1(x))(B(t)x+
B1(t)).
Now, we associate with this problem the following Cauchy problem in H
/braceleftBigg
Z/prime
1(t)+A1Z1(t)=F1(t),0<t<T,
Z1(0)=z1,(1.3.77)
30

Copyright Notice

© Licențiada.org respectă drepturile de proprietate intelectuală și așteaptă ca toți utilizatorii să facă același lucru. Dacă consideri că un conținut de pe site încalcă drepturile tale de autor, te rugăm să trimiți o notificare DMCA.

Acest articol: New results and methods in the study of some models described by partial differential equations Cosma (Barbu) Elena Luminit ¸a Faculty of Mathematics… [619781] (ID: 619781)

Dacă considerați că acest conținut vă încalcă drepturile de autor, vă rugăm să depuneți o cerere pe pagina noastră Copyright Takedown.

New results and methods in the study of some models described by partial differential equations Cosma (Barbu) Elena Luminit ¸a Faculty of Mathematics… [619781]

Copyright Notice

Lect.univ.dr. Lidia Oana STOICAN Student: Geta Alina PAVEL (HOTNOG) FOCȘ ANI 2018 UNIVERSITATEA DIN BUCUREȘ TI Facultatea de Psihologie și Științele… [617418]

Fakultät für Wirtschaftswissenschaften und Unternehmensführung Unternehmensfü hrung in deutscher Sprache Bachelorarbeit Absolvent , Andrei BLAGA… [608050]

Specializarea: Informatică Economică [601554]

CARACTERISTICI MINERALOGICE,PETROGRAFICE ȘI DOMENII DE UTILIZARE ALE ANDEZITULUI DE PIETROASA, JUDEȚUL HUNEDOARA [301889]

Proiectarea unei secții de fabricare a produsului baton cu fibre cu [605333]

Raport De Cercetare Nr 2 Sonya Tidorescu [630341]

Copyright Notice

Similar Posts