An evolutionary double-well problem

Tang Qi; Zhang Kewei

doi:10.1016/j.anihpc.2006.11.002

JournalsaihpcVol. 24, No. 3pp. 341–359

An evolutionary double-well problem

Tang Qi
Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9RF, United Kingdom
Zhang Kewei
Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9RF, United Kingdom

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Abstract

We establish the existence theorem and study the long time behaviour of the following PDE problem:

⎩ ⎨ ⎧ u_{t} - div \nabla W (\nabla u) - f (x) = 0 \nabla W (\nabla u) \cdot n ∣_{\partial Ω \times (0, \infty)} = 0, u (x, 0) = u_{0} (x) in Ω \times (0, - \infty), in Ω

where $W$ is a specially given quasiconvex double-well function and $f \in L^{2} (Ω)$ is a given function independent of time $t$ . In particular, the existence theorem is established for general given source term $f$ , the long time behaviour is analyzed under the assumption that $\int_{Ω} f (x) d x = 0$ .

The system is an evolutionary quasimonotone system. We believe that the existence of solutions established here is stronger than the usual Young Measure solution and is the first of its kind. The existence of a compact $ω$ -limit set as $t \to \infty$ is also established under some non-restrictive conditions.

Cite this article

Tang Qi, Zhang Kewei, An evolutionary double-well problem. Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. 3, pp. 341–359

DOI 10.1016/J.ANIHPC.2006.11.002