Existence and asymptotic behaviour of solutions for a class of quasi-linear evolution equations with non-linear damping and source terms.

*(English)*Zbl 1011.35121The paper is devoted to the study of a class of quasilinear evolution equations arising from models of nonlinear viscoelasticity which have the following form:

$${u}_{tt}-{\Delta}{u}_{t}-\sum _{i=1}^{N}{\partial}_{{x}_{i}}{\sigma}_{i}\left({u}_{{x}_{i}}\right)+f\left({u}_{t}\right)=g\left(u\right)$$

considered in a bounded domain and supplemented with the Dirichlet boundary conditions and initial data at $t=0$. By a Galerkin approximation scheme combined with energy estimates, it is proved that this initial boundary value problem admits global weak solutions which decay to zero as $t\to \infty $. Here, it is assumed that $m<p$, where $m$ and $p$ are, respectively, the growth orders of the non-linear strain terms and the source term.

Reviewer: Grzegorz Karch (Wrocław)