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Existence and asymptotic behaviour of solutions for a class of quasi-linear evolution equations with non-linear damping and source terms. (English) Zbl 1011.35121

The paper is devoted to the study of a class of quasilinear evolution equations arising from models of nonlinear viscoelasticity which have the following form: \[ u_{tt}-\Delta u_t -\sum_{i=1}^N\partial_{x_i} \sigma_i(u_{x_i})+f(u_t)=g(u) \] 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.

MSC:

35Q72 Other PDE from mechanics (MSC2000)
74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity)
74B20 Nonlinear elasticity
35G25 Initial value problems for nonlinear higher-order PDEs
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