Abstract
We study the existence of global weak solutions for a hyperbolic differential inclusion with a source term, and then investigate the asymptotic stability of the solutions by using Nakao lemma.
Highlights
We are concerned with the global existence and the asymptotic stability of weak solutions for a hyperbolic differential inclusion with nonlinear damping and source terms: ytt − Δyt − div |∇y|p−2∇y + Ξ = λ|y|m−2 y in Ω × (0, ∞), Ξ(x, t) ∈ φ yt(x, t) a.e. (x, t) ∈ Ω × (0, ∞), y = 0 on ∂Ω × (0, ∞), y(x, 0) = y0(x), yt(x, 0) = y1(x) in x ∈ Ω, (1.1)
(1.1) with λ = 0 by making use of the Faedo-Galerkin approximation, and considered asymptotic stability of the solution by using Nakao lemma [8]. The background of these variational problems are in physics, especially in solid mechanics, where nonconvex, nonmonotone, and multivalued constitutive laws lead to differential inclusions
In this paper, we will deal with the existence and the asymptotic behavior of a global weak solution for the hyperbolic differential inclusion (1.1) involving p-Laplacian, a nonlinear, discontinuous, and multivalued damping term and a nonlinear source term
Summary
We are concerned with the global existence and the asymptotic stability of weak solutions for a hyperbolic differential inclusion with nonlinear damping and source terms: ytt − Δyt − div |∇y|p−2∇y + Ξ = λ|y|m−2 y in Ω × (0, ∞), Ξ(x, t) ∈ φ yt(x, t) a.e. In this paper, we will deal with the existence and the asymptotic behavior of a global weak solution for the hyperbolic differential inclusion (1.1) involving p-Laplacian, a nonlinear, discontinuous, and multivalued damping term and a nonlinear source term. As far as we are concerned, there is a little literature dealing with asymptotic behavior of solutions for differential inclusions with source terms. For every q ∈ (1, ∞), we denote the dual of W01,q by W−1,q with q = q/(q − 1).
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