Long-term dynamical behavior of the wave model with locally distributed frictional and viscoelastic damping

•Our wave system (1.1) models the vibrations in a flexible body made of materials consisting of an elastic part and a viscoelastic part. It has both viscoelastic and frictional dampings acting in a subset ω0 of Ω.•We give an answer to the problem “when the memory kernel g(·) just satisfies some basic assumptions, how do the memory damping and frictional damping, both active just in a subset, in uence the energy decay rate of the whole wave system (1.1)?”.•We show that the solutions of (1.3) decay uniformly with a unified decay rate, which is determined by the two dampings and described by the solution of a differential equation. When the viscoelastic damping acts in the whole subset ω0, the viscoelastic damping dominates the decay rates.•Because the dampings just act in a subset of Ω, we retrieve the negative kinetic energy term by constructing some complex auxiliary functions.•We present numerical simulations for some cases, which illustrate the theoretical result well. We investigate the long-term dynamical behavior of the partially viscoelastic wave equation subject to a localized frictional damping, given byutt−Δu+∫0tg(t−s)div(a(x)∇u(s))ds+b(x)g1(ut)=0,inΩ×R+,where g denotes the memory kernel, a(x)∈C1(Ω¯),b(x)∈L∞(Ω) are nonnegative functions satisfying the assumptiona(x)+b(x)≥2δ>0,∀x∈ω0,ω0 is a subset of Ω, and b(x)g1(ut) denotes the frictional damping. Under as less as possible restrictions imposed on memory kernel g(·) and some geometric condition on the subset ω0, we show that there does not exist bifurcation and chaos for this physical model and actually the energy of the solution for the equation above decays definitely to zero with uniform decay rate as the time goes to infinity. Moreover, such a uniform decay rate is determined by the solution of an ordinary differential equation.