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Weakly dissipative semilinear equations of viscoelasticity. (English) Zbl 1101.35016

The authors investigate global in time existence and qualitative behavior of solutions to a nonlinear partial integro-differential equation \[ u_{tt} +\alpha u_{t} - k(0) \Delta u - \int_ 0^ \infty k'(s) \Delta u (t-s,x)\,ds + g(u) = f \] \(x\in\Omega\subset \mathbb R^3\), \(t>0,\) subject to Dirichlet boundary conditions and initial conditions. Setting the equation in the history space framework they prove the existence of a regular global attractor for the semidynamical system generated by solutions to the above integro-differential equation.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
45K05 Integro-partial differential equations
74D99 Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials)
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