Equações parabólicas do tipo Kirchhoff com fontes de expoentes variáveis

Abstract

In this work, we are interested in the existence and non-existence of global weak solutions for parabolic problems of the type p(x)-Kirchhoff with variable exponent sources of the type    ut − M Z Ω |∇u| p(x) p(x) dx! ∆p(x)u = |u| m(x)−2u, (x, t) ∈ Ω × (0, T) u(x, t) = 0, (x, t) ∈ ∂Ω × (0, T) u(x, 0) = u0(x), x ∈ Ω, (2) where Ω is a bounded domain of R N with smooth boundary ∂Ω, initial condition u0 ∈ W 1,p(·) 0 (Ω), the Kirchhoff function M : [0,∞) → [0,∞) and the functions m, p : Ω → R. We use the Galerkin method and potential well method to study the main objectives of this work.