Long time behaviour of parabolic equations in time-dependent growing domains

Article type: Research Article

Authors: Guesmia, Senoussi^{a; *} | Harkat, Soumia^b

Affiliations: [a] College of Sciences, Mathematics Department, Qassim University, Saudi Arabia. E-mail: [email protected] | [b] Department of Mathematics and Computer Science, University of Oum El Bouaghi, Algeria. E-mail: [email protected]

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: In this paper, we study the large time behaviour of the solution to parabolic problems defined on noncylindrical domains becoming unbounded in many directions when t tends to infinity. That is to say, the state variable domains are also becoming unbounded when t→∞. Since the steady state problem is elliptic and defined on unbounded domains we need to define in which sense the solution is understood and to deal with its existence. The convergence and its rate are also investigated with respect to the growth rate of the domain when t→∞. As the convergence cannot be expected on the whole domain, correctors are built to describe the asymptotic behaviour in the distant regions.

Keywords: Parabolic problems, elliptic problems in unbounded domains, stability, noncylindrical domains, asymptotic behaviour, correctors

DOI: 10.3233/ASY-171461

Journal: Asymptotic Analysis, vol. 108, no. 4, pp. 187-219, 2018