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We consider the semilinear wave equation with time varying delay and linear internal feedback u t t ( x , t ) − Δ u ( x , t ) + μ 1 u t ( x , t ) + μ 2 u t ( x , t − τ ( t ) ) + f ( x , u ) = g , in a regular bounded domain of R n with Dirichlet boundary conditions. Under suitable relations between μ 1 and μ 2 and suitable conditions on f and τ we derive first a boundedness result and we show that any global bounded solution of the above equation has a relative compact range in the natural energy space. More importantly, when the nonlinear term f is analytic in the second variable, we show the convergence to equilibrium as well as estimates for the rate of convergence for any global bounded solution. We also discuss existence of global solutions.

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.

In this article we present a unified approach to study the asymptotic behavior and the decay rate to a steady state of bounded weak solutions of nonlinear, gradient-like evolution equations of mixed first and second order. The proof of convergence is based on the Lojasiewicz–Simon inequality, the construction of an appropriate Lyapunov functional, and some differential inequalities. Applications are given to nonautonomous semilinear wave and heat equations with dissipative, dynamical boundary conditions, a nonlinear hyperbolic–parabolic partial differential equation, a damped wave equation and some coupled system.