Convergence to equilibrium of solutions to a nonautonomous semilinear viscoelastic equation with finite or infinite memory

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In this paper we consider the nonautonomous semilinear viscoelastic equation u t t − Δ u + ∫ 0 τ k ( s ) Δ u ( t − s ) d s + 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 R n 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 .