Reaction-diffusion equations in a noncylindrical thin domain
© Pereira and Silva; licensee Springer. 2013
Received: 1 July 2013
Accepted: 25 October 2013
Published: 20 November 2013
In this paper we are concerned with nonlinear reaction-diffusion equations posed in a time-dependent family of domains which collapses to a lower dimensional set as the parameter ϵ goes to 0.
Evolutionary PDEs on time-varying domains have been the subject of intense research over the past few years; we can cite, for example, [1–6] and the references given therein. Fundamental questions such as existence, regularity, and the asymptotic behavior of solutions are frequent topics of study and, since such problems are intrinsically non-autonomous, they need to be considered in a non-autonomous setting. These problems generate evolution process, and we can indicate the references [7, 8] which include a substantial overview on this topic.
Inspired by the recent works [4, 5], both of them related to reaction-diffusion equations on time-dependent domains, in this paper we are concerned with reaction-diffusion equations in a time-dependent thin domain. To the best of our knowledge, this is an untouched topic in the literature. We are going to prove that the evolution process generated by a family of singularly perturbed reaction-diffusion equations, which is equivalent in the autonomous case to the flow generated by such a family, converges to the evolution process generated by a limiting equation posed in a lower dimensional domain. This is the first step in order to consider the continuity of asymptotic dynamics (attractors), which will be considered in a further work.
One of the pioneering works on (autonomous) nonlinear reaction-diffusion equations on thin domains was , where the authors had considered one reaction-diffusion equation posed in the ordinate set under a smooth function satisfying (uniformly in x). They found a limiting equation (), which was defined in a lower dimensional domain, and showed that the flow generated by such an equation behaves continuously (with respect to ϵ in -norms). Our aim in this paper is to prove a similar result for the evolution process generated by equations on time-dependent domains.
In order to set up the problem, let ω be a smooth bounded domain in , , and satisfying
where denotes the partial derivative of g with respect to t, denotes the partial derivative of g with respect to , , and is the Euclidian norm in of the vector .
Recalling that , we see that .
where for all .
where denotes the unit outward normal vector field to , denotes the outward normal derivative and is a -function with bounded derivatives up second order.
We indicate to the interested reader the monograph [, Section 4.3].
for all .
where ν denotes the unit outward normal vector field to ∂ω.
The paper is organized as follows. In Section 2, we introduce the abstract framework for perturbed problem (1.6) as well as for the limiting one (1.9) and we prove the existence of the associated evolution process. In Section 3 we prove the continuity of such an evolution process uniformly in compact subsets of the real line.
2 Functional setting
In this section we recall the definitions of suitable spaces and operators as well as some of their properties. We start recalling that varies in accordance with a positive parameter ϵ, collapsing itself to the lower dimensional domain ω as ϵ goes to 0. Therefore, in order to preserve the ‘relative capacity’ of a mensurable subset , we rescale the Lebesgue measure of E, , by a factor and we are led to consider the singular measure . This measure has been widely considered in studies involving thin domains, e.g., [9, 11–15], and it allows us to introduce the Lebesgue and the Sobolev spaces.
for some positive constant c independent of ϵ and t.
for all and .
where is the Nemitskii operator (composition operator) associated to f.
This allows us to consider, for each value of the parameter ϵ, each initial time and each initial data , the solution of (2.5). This gives rise to a linear process defined by . We notice that (2.5) is the abstract Cauchy problem associated to equation (1.6) in the case .
there exist time and a unique solution of (2.6). Under assumption (1.3) on the nonlinearity f, one can show that actually . Further details can be found in [, Section 4.3] and [, Theorem 3.2].
where is the linear evolution process associated to homogeneous problem (2.5).
For the reader’s convenience, we recall the definition of an evolution process in a Banach space.
for any ,
is continuous for all and .
2.1 Limiting consideration
for all .
where , .
for all and .
where is the evolution process associated to the linear homogeneous counterpart.
Now we have the elements to state our main result.
uniformly for , , in bounded subsets of , where the extension operator E is defined in (3.2).
3 Convergence results
Due to the nature of this specific kind of singular perturbations, it is natural to introduce the following operators in order to compare functions defined in the different domains Ω and ω.
Notice that the extension operator E maps the family of spaces into .
Lemma 3.1 If is a bounded family in , then is a bounded family in .
which proves the result. □
The following lemma describes the behavior of the operators as .
uniformly in t in bounded subsets of ℝ.
Taking in the last inequality, we derive the result from (2.2) recalling Lemma 3.1. □
in compact subsets of ℝ.
for some constant independent of ϵ.
for t in compact subsets of ℝ. □
This paper is dedicated in memoriam to José Carlos Arcuri da Silva. RPS is partially supported by FAPESP #2012/06753-8, FUNDUNESP # 0135812 and FUNDUNESP–PROPe #0019/008/13, Brazil. The authors thank the referees for their careful reading and suggestions which led to an improvement of the work.
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