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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 lithium battery research, when we need to establish equivalent circuit based on impedance measurement, we can find several suitable circuit models. We usually use the chi-squared to determine the validity of equivalent models; when the chi-squared is less than 0.01, it indicates that those models have good fitting effect. In this case, we propose to verify the Kramers–Kronig transformation by extrapolating the data with equivalent circuit model and compare the residual errors of the Kramers–Kronig transform to determine the most suitable equivalent circuit model.

This paper develops a unified method to derive compactness properties and convergence to a steady state as well as decay estimates of global bounded solutions of the abstract semilinear second order integro-differential equation $$\begin{aligned} u_{tt}(t)+ Au(t) -\int ^\tau _0 k(s) Au(t-s)ds+ f(u)=g(t), \ \tau \in \{t, \infty \}, \end{aligned}$$ with finite ( $$\tau =t$$ ) or infinite ( $$\tau =\infty $$ ) memory. Here, A is a continuous linear operator, k is a positive nonincreasing function, f is a nonlinear function such that $$A+f$$ is the gradient of a potential satisfying the Łojasiewicz–Simon inequality, and the time-dependent right-hand side g decays to 0 at infinity. The general results are applied to particular semilinear evolution viscoelastic problems at the end of the paper.

By a small perturbation of the usual Lyapunov energy and by using the Łojasiewicz–Simon inequality, we show that the dissipation given by the memory term is strong enough to prove the convergence to equilibrium as well as estimates for the rate of convergence for any global bounded solution to the semilinear viscoelastic equation $$\begin{aligned} |u_t |^\rho u_{tt} -\Delta u_{tt}-\Delta u +\int ^\tau _0 k(s) \Delta u(t-s)\mathrm{d}s+ f(x,u)=g, \ \tau \in \{t, \infty \}, \end{aligned}$$ in a bounded regular domain of \({\mathbb {R}}^n\) with Dirichlet boundary conditions. Here, the kernel function \(k > 0\) is assumed to decay exponentially at infinity, the nonlinearity f is analytic in the second variable, and the forcing term g is supposed to decay polynomially or exponentially at infinity. The present work extends the previous result where the given equation was studied in the presence of an additional strong linear damping \(-\,\Delta u_t\).

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.

We study the long-time behavior as time goes to infinity of global bounded solutions to the following nonautonomous semilinear viscoelastic equation: $$\begin{aligned} |u_t |^\rho u_{tt} -\Delta u_{tt}-\Delta u_{t}-\Delta u +\int ^\tau _0 k(s) \Delta u(t-s)ds+ f(x,u)=g, \ \tau \in \{t, \infty \}, \end{aligned}$$|ut|?utt-Δutt-Δut-Δu+?0?k(s)Δu(t-s)ds+f(x,u)=g,??{t,?},in $${\mathbb {R}}^+\times \Omega $$R+×Ω, with Dirichlet boundary conditions, where $$\Omega $$Ω is a bounded domain in $${\mathbb {R}}^n$$Rn and the nonlinearity f is analytic. Based on an appropriate (perturbed) new Lyapunov function and the ?ojasiewicz---Simon inequality we prove that any global bounded solution converges to a steady state. We discuss also the rate of convergence which is polynomial or exponential, depending on the ?ojasiewicz exponent and the decay of the term g.

Our aim in this article is to study a nonautonomous semilinear wave equation with nonlinear damping and dynamical boundary condition. First we prove the existence and uniqueness of global bounded solutions having relatively compact range in the natural energy space. Then, by deriving an appropriate Lyapunov energy, we show that if the exponent in the Łojasiewicz-Simon inequality is large enough (depending on the damping), then weak solutions converge to equilibrium.

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.

We consider a nonautonomous, semilinear, hyperbolic-parabolic equation subject to a dynamical boundary condition of memory type. First we prove the existence and uniqueness of global bounded solutions having relatively compact range in the natural energy space. Under the assumption that the nonlinear term $f$ is real analytic, we then derive an appropriate Lyapunov energy and we use the {\L}ojasiewicz-Simon inequality to show the convergence of global weak solutions to single steady states as time tends to infinity. Finally, we provide an estimate for the convergence rate.