Existence of maximal solutions for some very singular nonlinear fractional diffusion equations in 1D

TL;DR

This paper proves the existence of maximal solutions for certain very singular nonlinear fractional diffusion equations in one dimension, including power and logarithmic nonlinearities, and studies their long-term behavior.

Contribution

It solves an open problem on the existence of solutions for very singular nonlinearities in fractional diffusion equations in 1D, introducing new methods and a comparison principle.

Findings

Existence of unique, mass-conserving maximal solutions.

Construction of Barenblatt-type self-similar solutions.

Asymptotic convergence of solutions to self-similar profiles.

Abstract

We consider nonlinear parabolic equations involving fractional diffusion of the form $\partial_t u + (-\Delta)^s \Phi(u)= 0,$ with $0<s<1$, and solve an open problem concerning the existence of solutions for very singular nonlinearities $\Phi$ in power form, precisely $\Phi'(u)=c\,u^{-(n+1)}$ for some $0< n<1$. We also include the logarithmic diffusion equation $\partial_t u + (-\Delta)^s \log(u)= 0$, which appears as the case $n=0$. We consider the Cauchy problem with nonnegative and integrable data $u_0(x)$ in one space dimension, since the same problem in higher dimensions admits no nontrivial solutions according to recent results of the author and collaborators. The {\sl limit solutions} we construct are unique, conserve mass, and are in fact maximal solutions of the problem. We also construct self-similar solutions of Barenblatt type, that are used as a cornerstone in the existence theory, and we prove that they are asymptotic attractors (as $t\to\infty$) of the solutions with general integrable data. A new comparison principle is introduced.