Semilinear degenerate parabolic systems and distributed capacitance models

Brooke L. Hollingsworth; R.E. Showalter

doi:10.3934/dcds.1995.1.59

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1995, Volume 1, Issue 1: 59-76. Doi: 10.3934/dcds.1995.1.59

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Semilinear degenerate parabolic systems and distributed capacitance models

Brooke L. Hollingsworth¹ and
R.E. Showalter^2,

1.
Department of Mathematics, The Universit of Texas at Austin, Austin, TX 78712
2.
Texas Institute for Computational and Applied Mathematics, Department of Mathematics, University of Texas at Austin, Austin, TX 78712

Received: September 30, 1994

Published: December 1994

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Abstract

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:

microstructure model,
degenerate system,
subgradient ﬂow,
the layered medium equation,
semilinear parabolic evolution equation in Hilbert space.

Mathematics Subject Classification: 35K90.

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