# American Institute of Mathematical Sciences

September  2016, 15(5): 1893-1913. doi: 10.3934/cpaa.2016021

## Global attractors for nonlinear viscoelastic equations with memory

 1 Dipartimento di Matematica "F.Brioschi", Politecnico di Milano, I-20133 Milano 2 Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy

Published  July 2016

We study the asymptotic properties of the semigroup $S(t)$ arising from the nonlinear viscoelastic equation with hereditary memory on a bounded three-dimensional domain \begin{eqnarray} |\partial_t u|^\rho \partial_{t t} u-\Delta \partial_{t t} u-\Delta \partial_t u\\ -\Big(1+\int_0^\infty \mu(s)\Delta s \Big)\Delta u +\int_0^\infty \mu(s)\Delta u(t-s)\Delta s +f(u)=h \end{eqnarray} written in the past history framework of Dafermos [10]. We establish the existence of the global attractor of optimal regularity for $S(t)$ when $\rho\in [0,4)$ and $f$ has polynomial growth of (at most) critical order 5.
Citation: Monica Conti, Elsa M. Marchini, V. Pata. Global attractors for nonlinear viscoelastic equations with memory. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1893-1913. doi: 10.3934/cpaa.2016021
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