Differential inclusion for the evolution p(x)-Laplacian with memory
Date
2019-02-13
Authors
Antontsev, Stanislav
Shmarev, Sergey
Simsen, Jacson
Mariza Stefanello, Simsen
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
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.
Description
Keywords
Evolution p(x)-Laplacian, Nonlocal equation, Differential inclusion, Finite speed of propagation, Waiting time
Citation
Antontsev, S., Shmarev, S., Simsen, J., & Stefanello Simsen, M. (2019). Differential inclusion for the evolution p(x)-Laplacian with memory. <i>Electronic Journal of Differential Equations, 2019</i>(26), pp. 1-28.
Rights
Attribution 4.0 International