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Long-time behaviour of strong solutions of retarded nonlinear P. D. E. s. (English) Zbl 0891.35159
The paper deals with the following retarded PDE: \[ \mu\ddot u+ \gamma\dot u+ \Delta^2u- f\Biggl(\int_\Omega|\nabla u(x,t)|^2dx\Biggr) \Delta u+ \rho{\partial u\over\partial x_1}- q(u_t)= p_0(x),\quad x\in\Omega,\quad t>0, \] with boundary conditions \[ u\bigl|_{t=0+}= u_0,\quad \dot u\bigl|_{t=0+}= u_1,\quad u\bigl|_{t\in(- t_*,0)}= \varphi(x, t) \] and initial conditions \[ u\bigl|_{\partial\Omega}= \Delta u\bigl|_{\partial\Omega}= 0, \] where \(\Omega\subset\mathbb{R}^n\) is a bounded domain, \(u_t= u_t(\theta)= u(t+\theta)\), \(\theta\in (-t_*,0)\) is the retarded function. The authors prove existence of strong solutions and investigate the long-time behaviour of these solutions. The existence of a finite-dimensional attractor is proved as well as its continuous dependence on the parameters of the system.
Reviewer: D.Bainov (Sofia)

35R10 Functional partial differential equations
35L75 Higher-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
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