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April  2005, 13(1): 35-62. doi: 10.3934/dcds.2005.13.35

A Finite-dimensional attractor for a nonequilibrium Stefan problem with heat losses

Michael L. Frankel 1, and  Victor Roytburd 2,

1. 

Department of Mathematical Sciences, Indiana University – Purdue University Indianapolis, Indianapolis, IN 46202-3216, United States
2. 

Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, United States

Received  January 2004 Revised  October 2004 Published  March 2005

We study a two-phase modified Stefan problem modeling solid combustion and nonequilibrium phase transitions. The problem is known to exhibit a variety of non-trivial dynamical scenarios. In the presense of heat losses we develop a priori estimates and establish well-posedness of the problem in weighted spaces of continuous functions. The estimates secure sufficient decay of solutions that allows us to carry out analysis in Hilbert spaces. We demonstrate existence of a compact attractor in the weighted spaces and prove that the attractor consist of sufficiently regular functions. We show that the Hausdorff dimension of the attractor is finite.
Keywords: compact attractor, Free interface, Hausdorff dimension..
Mathematics Subject Classification: Primary: 35R35, 80A25; Secondary: 35K5.
Citation: Michael L. Frankel, Victor Roytburd. A Finite-dimensional attractor for a nonequilibrium Stefan problem with heat losses. Discrete & Continuous Dynamical Systems, 2005, 13 (1) : 35-62. doi: 10.3934/dcds.2005.13.35
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