Decay estimates of a higher-order viscoelastic wave equation with general strong damping and source terms

Abstract

In this paper, we consider the initial-boundary value problem for a higher-order viscoelastic wave equation with general strong damping and general strong source terms: u_tt + (−Δ)^mu − ∫₀^t h(t−s)(−Δ)^mu(s) ds − μ(t) g₁(Δu_t) − g₂(Δu) = 0 in a bounded domain. Under appropriate conditions on μ, g₁, g₂ and on the kernel h, we establish the existence of global solutions by using the energy method combined with Faedo-Galarkin procedure. Furthermore, we utilize a direct method based on a the multipliers technique and Martinez's inequality to show a decay estimate for the energy of our problem, which depends on the initial energy and μ.