Abstract
In this paper, we investigate the stability of the solutions of a viscoelastic plate equation with a logarithmic nonlinearity. We assume that the relaxation function g satisfies the minimal conditiong′(t)≤−ξ(t)G(g(t)),\\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}$$ g^{\\prime }(t)\\le -\\xi (t) G\\bigl(g(t)\\bigr), $$\\end{document} where ξ and G satisfy some properties. With this very general assumption on the behavior of g, we establish explicit and general energy decay results from which we can recover the exponential and polynomial rates when G(s) = s^{p} and p covers the full admissible range [1, 2). Our new results substantially improve and generalize several earlier related results in the literature such as Gorka (Acta Phys. Pol. 40:59–66, 2009), Hiramatsu et al. (J. Cosmol. Astropart. Phys. 2010(06):008, 2010), Han and Wang (Acta Appl. Math. 110(1):195–207, 2010), Messaoudi and Al-Khulaifi (Appl. Math. Lett. 66:16–22, 2017), Mustafa (Math. Methods Appl. Sci. 41(1):192–204, 2018), and Al-Gharabli et al. (Commun. Pure Appl. Anal. 18(1):159–180, 2019).
Highlights
In the present paper, we consider the following viscoelastic plate problem with logarithmic nonlinearity: ⎧ ⎪⎪⎨utt +2u + u – t 0 g(t s)2u(s) ds = ku ln |u|, in Ω × (0, ∞), ⎪⎪⎩uu(=x, ∂u ∂ν 0) = =0, u0(x), ut(x, 0) = u1(x), in ∂Ω × (0, ∞), in Ω, (1)
We investigate the stability of the solutions of a viscoelastic plate equation with a logarithmic nonlinearity
Remark 1.1 Let us note here that though the logarithmic nonlinearity is somehow weaker than the polynomial nonlinearity, both the existence and stability result are not obtained by straightforward application of the method used for polynomial nonlinearity
Summary
In [15], Han considered the initial boundary value problem (4) in Ω ⊂ R3 and obtained global existence of weak solutions for all (u0, u1) ∈ H01(Ω) × L2(Ω). Messaoudi [23] established an existence result of the following problem: Established various existence results and proved, for smooth monotone decreasing relaxation functions, that the solutions go to zero as t goes to infinity.
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