Project Euclid

In this paper, we study the class of non-linear viscoelastic equation with lower order perturbation of $\overrightarrow{p}(x,t)-$ Laplacian type and a memory term \[ u_{tt}+\Delta^{2}u-\Delta_{\overrightarrow{p}(x,t)}u+\int_{0}^{t}g(t-s)\Delta u(s)ds -\epsilon\Delta u_{t}+f(u) =0\text{, } \] \[ (x,t)\in Q_{T}=\Omega\times(0,T),\quad\Omega\in\mathbb{R}^{n}. \] We prove local and global existence and uniqueness of weak solutions. These results are obtained assuming a strong damping $\epsilon\Delta u_{t}$ $(\epsilon>0)$ acting in the domain and provided the memory term decays exponentially and $f(u)$ is a nonlinear perturbation.