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Large time estimates for solutions to the porous medium equation with nonintegrable data via comparison

Published online by Cambridge University Press:  14 November 2011

Nicholas D. Alikakos  and
Rouben Rostamian
Nicholas D. Alikakos
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U.S.A.
Rouben Rostamian
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, U.S.A.
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Synopsis

We consider the Cauchy problem for the porous medium equation in one space dimension, with initial data which are locally integrable. We measure the asymptotic behaviour of the initial data near infinity in an integral sense and relate this to the pointwise rate of growth or decay of solution for large time. The emphasis is on a novel comparison method wherein the initial data are rearranged on the ×-axis to form a sequence of Dirac δ-masses. By using the explicit solution in the latter case, we derive upper and lower bounds for the solution to the original problem by comparisons.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 100 , Issue 1-2 , 1985 , pp. 1 - 10
Copyright
Copyright © Royal Society of Edinburgh 1985

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References

1Alikakos, N. and Rostamian, R.. On the uniformization of solutions of the porous medium equation in RN. Israel J. Math 74 (1984), 270–290.CrossRefGoogle Scholar
2Aronson, D. and Caffarelli, L.. The initial trace of a solution of the porous medium equation, manuscript.Google Scholar
3Barenblatt, G.. On some unsteady motions of a liquid or gas in a porous medium. Prikl. Mat. Meh. 16 (1952), 62–78 (Russian).Google Scholar
4Benilan, Ph., Crandall, M. and Pierre, M.. Solutions of the porous medium equation in RN under optimal conditions on initial values. Indiana Vniv. Math. J 33 (1984), 51–87.CrossRefGoogle Scholar
5Denisov, V.. Question of a necessary condition for the stabilization uniform on an arbitrary compact of a solution of Cauchy problem for the thermal-conduction equation. Differential 'nye Uravneniya 18 (1982), 7–16 (Russian).Google Scholar
6Denisov, V.. A necessary condition for the uniform stabilization of a solution of Cauchy's problem for the heat-conduction equation. Differential'nye Uravneniya 18 (1982), 290–305.Google Scholar
7Repnikov, V. and Eidelman, S.. A new proof of the theorem on the stabilization of the solutions of the Cauchy problem for the heat equation. Mat. Sb 73 (1967), 135–139.CrossRefGoogle Scholar
8Vazquez, J.. Asymptotic behaviour and propagation properties of the one dimensional flow of gas in a porous medium, manuscript.Google Scholar