Well-posedness and asymptotic stability of solutions to the semilinear wave equation with analytic nonlinearity and time varying delay

Abstract

We consider the semilinear wave equation with time varying delay and linear internal feedbackutt(x,t)−Δu(x,t)+μ1ut(x,t)+μ2ut(x,t−τ(t))+f(x,u)=g, in a regular bounded domain of Rn with Dirichlet boundary conditions. Under suitable relations between μ1 and μ2 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.