Global attractors for nonlinear      viscoelastic equations with memory

Monica Conti; Elsa M. Marchini; V. Pata

doi:10.3934/cpaa.2016021

Article Contents

2016, Volume 15, Issue 5: 1893-1913. Doi: 10.3934/cpaa.2016021

This issue Previous Article Improved higher order poincaré inequalities on the hyperbolic space via Hardy-type remainder terms Next Article Distributionally chaotic families of operators on Fréchet spaces

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

Published: August 2016

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Abstract

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.

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Mathematics Subject Classification: Primary: 35B41, 45G10; Secondary: 35L72.

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