Asymptotic behavior for a class of parabolic equations in weighted variable Sobolev spaces

Article type: Research Article

Authors: Shmarev, Sergey^{a; *} | Simsen, Jacson^b | Stefanello Simsen, Mariza^b | Primo, Marcos Roberto T.^c

Affiliations: [a] Departamento de Matemáticas, Universidad de Oviedo, c/Calvo Sotelo, s/n, 33007, Oviedo, Spain. E-mail: shmarev@uniovi.es | [b] Instituto de Matemática e Computação, Universidade Federal de Itajubá, Av. BPS n. 1303, Bairro Pinheirinho, 37500-903, Itajubá, MG, Brasil. E-mails: jacson@unifei.edu.br, mariza@unifei.edu.br | [c] Departamento de Matemática, Universidade Estadual de Maringá, 87020-900, Maringá, Paraná, Brasil. E-mail: mrtprimo@uem.br

Correspondence: [*] Corresponding author. E-mail: shmarev@uniovi.es.

Abstract: We study the homogeneous Dirichlet problem for the class of nonlinear parabolic equations with variable nonlinearity ut−div(D(x)|∇u|p(x)−2∇u)=f(x,t,u)−A(x)|u|q(x)−2u in the cylinder Ω×(0,T) with given nonnegative weights D(x), A(x), measurable bounded exponents p(x)∈[p−,p+], q(x)∈[q−,q+] and a globally Lipschitz function f(x,t,u). Sufficient conditions of existence and uniqueness of weak and strong solutions are derived. We find conditions on the exponents p(x), q(x) which guarantee that the associated semigroup has a compact global attractor in L2(Ω). It is shown that in case the exponents p(x) and q(x) do not meet the sufficient conditions of existence of a nontrivial global attractor and ‖u(0)‖L2(Ω) is sufficiently small, then every solution with bounded ‖u(t)‖L2(Ω)2 either vanishes in a finite time, or decays exponentially as t→∞.

Keywords: Nonlinear parabolic equation, variable nonlinearity, global attractors

DOI: 10.3233/ASY-181486

Journal: Asymptotic Analysis, vol. 111, no. 1, pp. 43-68, 2019