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On the Cauchy problem and initial traces for a degenerate parabolic equation

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Di Benedetto, E. and Herrero, Miguel A. (1989) On the Cauchy problem and initial traces for a degenerate parabolic equation. Transactions of the American Mathematical Society, 314 (1). pp. 187-224. ISSN 0002-9947

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Official URL: http://www.ams.org/journals/tran/1989-314-01/S0002-9947-1989-0962278-5/S0002-9947-1989-0962278-5.pdf


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Abstract

The authors study the Cauchy problem for the degenerate parabolic equation ut = div(|Du| p−2 Du)(p<2), and find sufficient conditions on the initial trace u0 (and in particular on its behaviour as |x|→∞) for existence of a solution in some strip RN × (0,T). Using a Harnack type inequality they show that these conditions are optimal in the case of nonnegative solutions. Uniqueness of solutions is shown if u0 belongs to L1loc(RN), but is left open in the case that u0 is merely a locally bounded measure. The results are closely related to papers by Aronson-Caffarelli, Benilan-Crandall-Pierre, and Dahlberg-Kenig about the porous medium equation ut = Δum. The proofs are different and allow one to generalize some of the above results to equations with variable coefficients.


Item Type:Article
Uncontrolled Keywords:Cauchy problem; porous medium equation; existence; Harnack-type inequality; Uniqueness
Subjects:Sciences > Mathematics > Differential equations
ID Code:17482
Deposited On:18 Dec 2012 09:32
Last Modified:12 Dec 2018 15:08

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