Stability Results for a Weakly Dissipative Viscoelastic Equation with Variable-Exponent Nonlinearity: Theory and Numerics

Abstract

In this paper, we study the long-time behavior of a weakly dissipative viscoelastic equation with variable exponent nonlinearity of the form utt+Δ2u−∫0tg(t−s)Δu(s)ds+a|ut|n(·)−2ut−Δut=0, where n(.) is a continuous function satisfying some assumptions and g is a general relaxation function such that g′(t)≤−ξ(t)G(g(t)), where ξ and G are functions satisfying some specific properties that will be mentioned in the paper. Depending on the nature of the decay rate of g and the variable exponent n(.), we establish explicit and general decay results of the energy functional. We give some numerical illustrations to support our theoretical results. Our results improve some earlier works in the literature.

Keywords

• viscoelasticity
• relaxation function
• general decay
• convex functions
• variable exponent
• numerical computations