Abstract
In this paper, we are concerned with the decay rate of the solution of a viscoelastic plate equation with infinite memory and logarithmic nonlinearity. We establish an explicit and general decay rate results with imposing a minimal condition on the relaxation function. In fact, we assume that the relaxation function h satisfies h′(t)≤−ξ(t)H(h(t)),t≥0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ h^{\\prime}(t)\\le-\\xi(t) H\\bigl(h(t)\\bigr),\\quad t\\geq0, $$\\end{document} where the functions ξ and H satisfy some conditions. Our proof is based on the multiplier method, convex properties, logarithmic inequalities, and some properties of integro-differential equations. Moreover, we drop the boundedness assumption on the history data, usually made in the literature. In fact, our results generalize, extend, and improve earlier results in the literature.
Highlights
In this work, we consider the following viscoelastic plate problem with velocity-dependent material density and logarithmic nonlinearity: +∞|ut|ρ utt + 2u + 2utt – h(s) 2u(t – s) ds = αu ln |u| in Ω × (0, ∞), (1)equipped with initial and boundary conditions∂u u(x, t) = (x, t) = 0 in ∂Ω × (0, ∞), ∂n (2)u(x, –t) = u0(x, t), ut(x, 0) = u1(x) in Ω, where Ω is a bounded domain of R2 with smooth boundary ∂Ω, n is the unit outer normal to ∂Ω, and ρ and α are positive constants
(b) To extend some general decay results, known for the case of finite history, to the case of infinite history where the relaxation function satisfies a wider class of relaxation functions instead of those considered in [7, 8, 12, 19, 21, 29, 31]
5 Conclusion As far as we know, there are no decay results in the literature known for logarithmic plate equation with infinite memory and a wider class of relaxation functions
Summary
The relaxation function h satisfies the following Let us recall some results regarding problems with logarithmic nonlinearity. Proved existence and decay results of the solutions under the following condition on the relaxation function: h (t) ≤ –ξ (t)hp(t),
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