Differential inclusion for the evolution p(x)-Laplacian with memory

Abstract

We consider the evolution differential inclusion for a nonlocal operator that involves p(x)-Laplacian, ut - ∆p(x)u - ∫t0 g(t - s) ∆p(x) u(x, s)ds ∈ F(u) in QT = Ω x (0, T), where Ω ⊂ ℝn, n ≥ 1, is a bounded domain with Lipschitz-continuous boundary. The exponent p(x) is a given measurable function, p⎺ ≤ p(x) ≤ p+ a.e. in Ω for some bounded constants p⎺ > max {1, 2n/n+2} and p+ < ∞. It is assumed that g, g′ ∈ L2(0, T), and that the multivalued function F(∙) is globally Lipschitz, has convex closed values and F(0) ≠ ∅. We prove that the homogeneous Dirichlet problem has a local in time weak solution. Also we show that when p⎺ > 2 and uF(u) ⊆ {v ∈ L2(Ω) : v ≤ εu2 a.e. in Ω} with a sufficiently small ε > 0 the weak solution possesses the property of finite speed of propagation of disturbances from the initial data and may exhibit the waiting time property. Estimates on the evolution of the null-set of the solution are presented.