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On the long-time behavior of ferroelectric systems. (English) Zbl 0980.78008

The authors present a new model for the long-time behavior of ferroelectric material. The material is supposed to occupy a cylinder parallel to the \(z\)-axis with a simply connected cross-section \(\Omega\). They consider only electric fields \(E\) (resp. electric polarization \(P\) and magnetic fields \(H\)) which are independent of \(z\) and have the special form \(E= aee_3\), \(P= ape_3\), \(H= b(h_1e_1+ h_2e_2)\), where \(a\) and \(b\) are suitable constants. The governing equations for \(e\), \(p\), \(h_1\), \(h_2\) are Maxwell’s equations with one or the other of the boundary conditions \(e=0\) or \(-n_2h_1+ n_1h_2= 0\) on \(\partial\Omega\). The long time behavior of these functions and of the energy is studied. It is shown that the system tends to a steady state which depends on the boundary condition on \(\partial\Omega\). Their method is based on a series of convenient a priori estimates and properties of suitable Sobolev spaces. The limit \(p^\infty\) of \(p\), as \(t\to \infty\), is a solution of a nonlinear elliptic problem; this equilibrium problem has multiple solutions and is investigated in the last section of the paper.

MSC:

78A99 General topics in optics and electromagnetic theory
35Q60 PDEs in connection with optics and electromagnetic theory
35B45 A priori estimates in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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