General Decay of Solutions in a Viscoelastic Equation with Nonlinear Localized Damping

Abstract:

      In this paper, we consider the following viscoelastic equation $$u_{tt} -\Delta u+\int_0^t {g(t-s)\Delta u(s)\d s} +a(x)u_t +u\left| u\right|^r=0$$ with initial condition and Dirichlet boundary condition.\,The decay property of the energy function closely depends on the properties of the relaxation function $g(t)$ at infinity. In the previous works of [3,7,11], it was required that the relaxation function $g(t)$ decay exponentially or polynomially as $t\rightarrow +\infty$. In the recent work of Messaoudi [12,13], it was shown that the energy decays at a similar rate of decay of the relaxation function, which is not necessarily dacaying in a polynomial or exponential fashion. Motivated by [12,13], under some assumptions on $g(x)$, $a(x)$ and $r$, and by introducing a new perturbed energy, we also prove the similar results for the above equation.