Some results on the asymptotic behavior of the Caginalp system with singular potentials. (English) Zbl 1145.35042

Cherfils, Laurence; Miranville, Alain

Adv. Math. Sci. Appl. 17, No. 1, 107-129 (2007).

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.

Reviewer: Messoud A. Efendiev (Berlin)

MSC:

35B41	Attractors
35B40	Asymptotic behavior of solutions to PDEs
80A22	Stefan problems, phase changes, etc.

Keywords:

Łojasiewicz-Simon inequality; long-time dynamics