Existence And Exponential Decay Of Solutions To A Nonlinear Viscoelastic Equation

LetΩbe a bounded open domain of Rⁿ with a smooth boundary. For a uniformly bounded function a ( x)≥0onΩ, and a constantρ≥0, we consider the nonlinear viscoelastic equationWe will allow a(x) to vanish on any part. Firstly, based on Faedo-Galerkin approximation method, the existence of global weak solutions is proved.Secondly, according to the estimation of F (t)= E(t)+ε₁φ(t)+ε₂χ(t),an exponential decay result is established.