New general decay results for a viscoelastic plate equation with a logarithmic nonlinearity

Boundary Value Problems

Mohammad M. Al-Gharabli¹ 

[1] The Preparatory Year Math Program, King Fahd University of Petroleum and Minerals;

关键词: Optimal decay;    Plate equations;    Viscoelastic;    Convexity;   

DOI  :  10.1186/s13661-019-01308-0

来源: DOAJ

【 摘 要 】

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 condition g ′ ( t ) ≤ − ξ ( t ) G ( g ( t ) ) , $$ g^{\prime }(t)\le -\xi (t) G\bigl(g(t)\bigr), $$ 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 $G(s) = s^{p}$ and p covers the full admissible range [ 1 , 2 ) $[1, 2)$ . 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).

【 授权许可】

Unknown