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April  2010, 26(2): 713-736. doi: 10.3934/dcds.2010.26.713

Boundary dynamics of a two-dimensional diffusive free boundary problem

Micah Webster 1, and  Patrick Guidotti 2,

1. 

Mathematics and Computer Science Department, Goucher College, 1021 Dulaney Valley Road, Baltimore, MD, 21204, United States
2. 

Department of Mathematics, University of California, Irvine, CA 92697-3875

Received  February 2009 Revised  July 2009 Published  October 2009

Numerous models of industrial processes such as diffusion in glassy polymers or solidification phenomena, lead to general one-phase free boundary value problems with phase onset. In this paper we develop a framework viable to prove global existence and stability of planar solutions to one such multi-dimensional model whose application is in controlled-release pharmaceuticals. We utilize a boundary integral reformulation to allow for the use of maximal regularity. To this effect, we view the operators as pseudo-differential and exploit knowledge of the relevant symbols. Within this framework, we give a local existence and continuous dependence result necessary to prove planar solutions are locally exponentially stable with respect to two-dimensional perturbations.
Keywords: Free Boundary Problem, Well-Posedness, Non-Regular Pseudodifferential Operators., Maximal Regularity.
Mathematics Subject Classification: Primary: 35A07, 35C15, 35M1.
Citation: Micah Webster, Patrick Guidotti. Boundary dynamics of a two-dimensional diffusive free boundary problem. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 713-736. doi: 10.3934/dcds.2010.26.713
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