Abstract: We consider the evolution differential inclusion for a nonlocal operator that involvesZ p(x)-Laplacian, (Formula presented) where Ω ⊂ R n , 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, n+2 2n } and p + < ∞. It is assumed that g, g ′ ∈ L 2 (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 ∈ L 2 (Ω): v ≤ ɛu 2 a.e. in Ω} with a sufficiently small ɛ > 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 -set of the solution are presented.

Cite: Antonsev S. , Shmarev S. , Simsen J. , Simsen M.
Differential inclusion for the evolution p(x)-laplacianwith memory
Electronic Journal of Differential Equations. 2019. V.2019. N26. P.1-28. РИНЦ

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38694386

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