$L^1$ contraction for bounded (non-integrable) solutions of degenerate parabolic equations

J. Endal; E. R. Jakobsen

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arXiv · 1404.6418

$L^1$ contraction for bounded (non-integrable) solutions of degenerate parabolic equations

J. Endal·E. R. Jakobsen

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Abstract

We obtain new $L^1$ contraction results for bounded entropy solutions of Cauchy problems for degenerate parabolic equations. The equations we consider have possibly strongly degenerate local or non-local diffusion terms. As opposed to previous results, our results apply without any integrability assumption on the %(the positive part of the difference of) solutions. They take the form of partial Duhamel formulas and can be seen as quantitative extensions of finite speed of propagation local $L^1$ contraction results for scalar conservation laws. A key ingredient in the proofs is a new and non-trivial construction of a subsolution of a fully non-linear (dual) equation. Consequences of our results are maximum and comparison principles, new a priori estimates, and in the non-local case, new existence and uniqueness results.

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BibTeX RIS

J. Endal, E. R. Jakobsen. 2014-10-03. $L^1$ contraction for bounded (non-integrable) solutions of degenerate parabolic equations. https://arxiv.org/abs/1404.6418

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