Asymptotic Behavior Of Several Degenerate Evolution Equations With Variable Density

In this paper we consider the solution of nonlinear viscoelastic equation with variable density. Under some assumption on initial data and memory kernel, we can find that the decay rate of the energy is related to the memory kernel.There are three parts in this paper.Part one, we study the equation with nonlinear internal dissipation and variable density second-order terms. |ut(t)|putt(t)+|ut(t)|put(t)+Au(t)-∫0tg(t-s)Au(s)ds=-|u(t)|pu(t). We develops a method to construct energy functional and other functionals, then by explicit estimate to get the general decay rate of the energy function.Part two, we study the equation with force term but without nonlinear internal dis-sipation term. |ut(t)|putt(t)+Au(t)-∫0tg(t-s)Au(s)ds=|u(t)|pu(t). We develops a method to construct energy functional and other functionals, then by explicit estimate to get the general decay rate of the energy function.Part three, we study the wave equation with resistant term but without nonlinear internal dissipation term. |ut(t)|putt(t)+Au(t)-∫0tg(t-s)Au(s)ds=-|u(t)|pu(t). We develops a method to construct energy functional and other functionals, then by explicit estimate to get the general decay rate of the energy functional.The way to get the result is different when the force term and the memory kernel are different, and the corresponding result is different too. In this paper, we prove that the energy decays polynomially or exponentially to zero as the time tends to infinity.