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

TL;DR

This paper establishes new $L^1$ contraction results for bounded solutions of degenerate parabolic equations, including non-local cases, without requiring integrability assumptions, thereby extending the scope of previous local $L^1$ contraction results.

Contribution

It introduces $L^1$ contraction results for bounded solutions without integrability assumptions, applicable to both local and non-local degenerate parabolic equations.

Findings

Derived $L^1$ contraction inequalities for bounded solutions

Established maximum and comparison principles

Proved new existence and uniqueness results in non-local cases

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.