Optimal decay rates of a nonlinear time-delayed viscoelastic wave equation. (English) Zbl 1463.35349

By use of the Galerkin method and potential well theory the authors prove global existence of weak solution of the initial boundary value problem to the viscoelastic wave equation with time-dependent delay \[u_{tt}(x,t)-\Delta u(x,t)+\int_0^tg(t-s)\Delta u(x,s)\, ds+\mu_1u_t(x,t)+\mu_2u_t(x,t-\tau(t))=|u(x,t)|^{\rho}u(x,t).\] Further, they study the optimal decay rates of energy to the problem and give the assumptions, under which the energy tends to zero as \(t\to\infty.\)


35L70 Second-order nonlinear hyperbolic equations
35R09 Integro-partial differential equations
45K05 Integro-partial differential equations