Article ID Journal Published Year Pages File Type
8899142 Journal of Differential Equations 2017 30 Pages PDF
Abstract
In this paper we consider the nonautonomous semilinear viscoelastic equationutt−Δu+∫0τk(s)Δu(t−s)ds+f(x,u)=g,τ∈{t,∞}, in R+×Ω, with Dirichlet boundary conditions and finite (τ=t) or infinite (τ=∞) memory. Here Ω is a bounded domain in Rn with smooth boundary and the nonlinearity f:Ω×R+→R is analytic in the second variable, uniformly with respect to the first one. For this equation, we derive an appropriate Lyapunov function and we use the Łojasiewicz-Simon inequality to show that the dissipation given by the memory term is strong enough to prove the convergence to a steady state for any global bounded solution. In addition, we discuss the rate of convergence to equilibrium which is polynomial or exponential, depending on the Łojasiewicz exponent and the decay of the time-dependent right-hand side g.
Related Topics
Physical Sciences and Engineering Mathematics Analysis
Authors
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