$L^1$ contraction for bounded (non-integrable) solutions of degenerate parabolic equations
Abstract
We obtain new 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 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.
Keywords
Cite
@article{arxiv.1404.6418,
title = {$L^1$ contraction for bounded (non-integrable) solutions of degenerate parabolic equations},
author = {J. Endal and E. R. Jakobsen},
journal= {arXiv preprint arXiv:1404.6418},
year = {2014}
}