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Elliptic Perturbations od Some Parabolic and Heperbolic Problems (pp.453500) 
$100.00 

Authors: (Narcisa Apreutesei1, Behzad Djafari Rouhani, Department of Mathematics, Technical University ”Gh. Asachi”,Iasi, Romania)

Abstract: The goal of this chapter is to put together some recent results concerning applications
of monotone second order differential equations to singularly perturbed problems
of elliptic  parabolic and elliptic  hyperbolic type. More exactly, the solution v of the
heat equation or of the telegraph system is compared with the solution v" of an elliptic
regularization. This elliptic regularization is a perturbed problem written with the aid
of a small parameter " > 0 . It is a particular case of some second order differential
equations governed by a maximal monotone operator in the Hilbert space L2 (
) . Under
some specific hypotheses, we construct a zero order asymptotic approximation for
v" making use of the boundary layer function method of Vishik and Lyusternik. The
higher order regularity of the solutions to both perturbed and unperturbed problems
is investigated. The order of accuracy of the difference v" − v is also established in
some appropriate function spaces. Thus, the solution v of the heat equation (or telegraph
system) is approximated by the solution v" of its elliptic regularization, which
is a more regular function. This is a motivation for the study of the above mentioned
second order evolution equations associated with monotone operators. This study can
involve different unperturbed problems: semilinear heat equation, linear heat equation
with nonlinear boundary conditions, semilinear telegraph system, nonlinear telegraph
system with nonlinear boundary conditions, etc.
we are concerned with the asymptotic analysis of some partial differential equations representing
the semilinear heat equation with Dirichlet boundary condition, the linear heat
equation with nonlinear boundary conditions, and the semilinear telegraph system with
homogeneous boundary conditions. Denote the three problems by (A0) , (P0) and (T0) ,
respectively and by (A") , (P") , (T") their elliptic regularizations, where " > 0 is a small
parameter. One constructs zero order asymptotic approximations for the solutions of these
problems, employing the boundary layer function method of Vishik and Lyusternik ( [16] ,
[17] , [18] , [19] , [20]). According to this method, one searches the asymptotic approximation
like a sum of the solution of the unperturbed problem (heat equation or telegraph
system) with a boundary layer function, which can be found explicitly. The role of the
boundary layer function is to compensate the difference between the solutions of the two
problems (perturbed and unperturbed) in the neighbourhood of the final point T.
We will prove that the solutions of (A0) , (P0) , (T0) are approximated by the solutions
of the perturbed problems (A") , (P") , (T") respectively, which are smoother functions.
This justifies the advantage of the elliptic regularizations. The perturbed problems can be
written like twopoint boundary value problems. Therefore, the existence of their solutions
is assured by the results from the preliminary section. In each case, we also find the order
of accuracy of the asymptotic approximations.
Problems (A") and (P") are singularly perturbed problems of the ellipticparabolic type,
while (T") is an elliptichyperbolic singularly perturbed problem.
The structure of the chapter is the following. First section is devoted to some basic
notions and results on the existence of the solution to some first order and second order
evolution equations associated to maximal monotone operators in Hilbert spaces.
In the second section, we compare the solution v of the semilinear heat equation with
the solution v" of an elliptic regularization. Some bilocal boundary conditions are attached
to this equation. Under some specific hypotheses, we construct a zero order asymptotic
approximation for the solution v" and find the order of accuracy of the difference v" − v.
The problem is singularly perturbed of ellipticparabolic type. The case of the linear heat
equation with nonlinear boundary conditions is the subject of Section 3. The fourth section
deals with the semilinear telegraph system and its corresponding elliptic regularization.
Thus the singular perturbation problem is of elliptichyperbolic type. Last section presents
briefly the conclusion. 











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