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Some results on the asymptotic behavior of the Caginalp system with singular potentials. (English) Zbl 1145.35042
The authors consider the following system of partial differential equations in a bounded smooth domain $$\Omega$$ of $$\mathbb{R}^3$$ $\delta\partial_t\varphi- \Delta_x\varphi+ f(\varphi)- u= g,$
$\varepsilon\partial_t u+\partial_t\varphi- \Delta_x u= 0,$
${\partial u\over\partial n}\Biggl|_{\partial\Omega}= {\partial\varphi\over\partial n}\Biggl|_{\partial\Omega}= 0,\quad \varphi\Biggl|_{t= 0}= \varphi_0,\;u\Biggl|_{t= 0}= u_0,$ where $$0<\varepsilon< 1$$, $$\delta> 0$$. The authors are mainly interested in the study of finite-dimensional attractors and the convergence of solutions to steady states.

##### MSC:
 35B41 Attractors 35B40 Asymptotic behavior of solutions to PDEs 80A22 Stefan problems, phase changes, etc.
##### Keywords:
Łojasiewicz-Simon inequality; long-time dynamics