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January  1995, 1(1): 59-76. doi: 10.3934/dcds.1995.1.59

Semilinear degenerate parabolic systems and distributed capacitance models

Brooke L. Hollingsworth 1, and  R.E. Showalter 2,

1. 

Department of Mathematics, The Universit of Texas at Austin, Austin, TX 78712, United States
2. 

Texas Institute for Computational and Applied Mathematics, Department of Mathematics, University of Texas at Austin, Austin, TX 78712

Received  October 1994 Published  December 1994

A two-scale microstructure model of current flow in a medium with continuously distributed capacitance is extended to include nonlinearities in the conductance across the interface between the local capacitors and the global conducting medium. The resulting degenerate system of partial differential equations is shown to be in the form of a semilinear parabolic evolution equation in Hilbert space. It is shown directly that such an equation is equivalent to a subgradient flow and, hence, displays the appropriate parabolic regularizing effects. Various limiting cases are identified and the corresponding convergence results obtained by letting selected parameters tend to infinity.
Keywords: semilinear parabolic evolution equation in Hilbert space., subgradient flow, microstructure model, degenerate system, the layered medium equation.
Mathematics Subject Classification: 35K9.
Citation: Brooke L. Hollingsworth, R.E. Showalter. Semilinear degenerate parabolic systems and distributed capacitance models. Discrete and Continuous Dynamical Systems, 1995, 1 (1) : 59-76. doi: 10.3934/dcds.1995.1.59
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