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Convergence towards equilibria for a hyperbolic system arising in ferroelectricity. (English) Zbl 1172.35328

Summary: We analyze the asymptotic behavior of a hyperbolic system describing the evolution of the electromagnetic field inside a ferroelectric material according with a model proposed by Greenberg, MacCamy and Coffman. The system consists of a linear damped wave equation for a field \(u\) coupled with a semilinear damped wave equation for the polarization field \(p\). Assuming homogeneous Dirichlet and Neumann boundary conditions for \(u\) and \(p\) and that the nonlinearity \(\phi\) is analytic, we prove that any weak solution converges to a stationary state. This is done by adapting an argument already used by Haraux-Jendoubi and based on the Simon-Lojasiewicz inequality. Furthermore, another goal of the paper is to provide the estimate of the decay rate to equilibrium. We give two results in this direction, according to different assumptions on \(\phi\).

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35B41 Attractors
35L70 Second-order nonlinear hyperbolic equations
35Q60 PDEs in connection with optics and electromagnetic theory
35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables)
35R25 Ill-posed problems for PDEs
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
35L55 Higher-order hyperbolic systems
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