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




Antontsev, Stanislav
Shmarev, Sergey
Simsen, Jacson
Mariza Stefanello, Simsen

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Texas State University, Department of Mathematics


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.



Evolution p(x)-Laplacian, Nonlocal equation, Differential inclusion, Finite speed of propagation, Waiting time


Antontsev, S., Shmarev, S., Simsen, J., & Stefanello Simsen, M. (2019). Differential inclusion for the evolution p(x)-Laplacian with memory. Electronic Journal of Differential Equations, 2019(26), pp. 1-28.


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