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:

{utdivW(u)f(x)=0in Ω×(0,),W(u)nΩ×(0,)=0,u(x,0)=u0(x)in Ω\left\{\begin{matrix} u_{t}−\mathrm{div}\mathrm{∇}W(\mathrm{∇}u)−f(x) = 0\:\text{in }\Omega \times (0,−\infty ), \\ \mathrm{∇}W(\mathrm{∇}u) \cdot \mathbf{n}|_{\partial \Omega \times (0,\infty )} = 0, \\ u(x,0) = u_{0}(x)\:\text{in }\Omega \\ \end{matrix}\right.

where W is a specially given quasiconvex double-well function and fL2(Ω)f \in L^{2}(\Omega ) 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 Ωf(x)dx=0\int _{\Omega }f(x)\:\mathrm{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 tt\rightarrow \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