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A necessary and sufficient condition for finite speed of propagation in the theory of doubly nonlinear degenerate parabolic equations

Published online by Cambridge University Press:  14 November 2011

B. H. Gilding  and
R. Kersner
B. H. Gilding
Affiliation:
Faculty of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
R. Kersner
Affiliation:
Computer and Automation Research Institute, Hungarian Academy of Sciences, P.O. Box 63, H-1518 Budapest, Hungary
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Abstract

A degenerate parabolic partial differential equation with a time derivative and first- and second-order derivatives with respect to one spatial variable is studied. The coefficients in the equation depend nonlinearly on both the unknown and the first spatial derivative of a function of the unknown. The equation is said to display finite speed of propagation if a non-negative weak solution which has bounded support with respect to the spatial variable at some initial time, also possesses this property at later times. A criterion on the coefficients in the equation which is both necessary and sufficient for the occurrence of this phenomenon is established. According to whether or not the criterion holds, weak travelling-wave solutions or weak travelling-wave strict subsolutions of the equation are constructed and used to prove the main theorem via a comparison principle. Applications to special cases are provided.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 4 , 1996 , pp. 739 - 767
Copyright
Copyright © Royal Society of Edinburgh 1996

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