The fractional p-Laplacian evolution equation in $\mathbb{R}^N$ in the sublinear case

Abstract

We consider the natural time-dependent fractional $p$-Laplacian equation posed in the whole Euclidean space, with parameter $1<p<2$ and fractional exponent $s\in (0,1)$. Rather standard theory shows that the Cauchy Problem for data in the Lebesgue $L^q$ spaces is well posed, and the solutions form a family of non-expansive semigroups with regularity and other interesting properties. The superlinear case $p>2$ has been dealt with in a recent paper. We study here the "fast" regime $1<p<2$ which is more complex. As main results, we construct the self-similar fundamental solution for every mass value $M$ and any $p$ in the subrange $p_c=2N/(N+s)<p<2$, and we show that this is the precise range where they can exist. We also prove that general finite-mass solutions converge towards the fundamental solution having the same mass, and convergence holds in all $L^q$ spaces. Fine bounds in the form of global Harnack inequalities are obtained. Another main topic of the paper is the study of solutions having strong singularities. We find a type of singular solution called Very Singular Solution that exists for $p_c<p<p_1$, where $p_1$ is a new critical number that we introduce, $p_1\in (p_c,2)$. We extend this type of singular solutions to the "very fast range" $1<p<p_c$. They represent examples of weak solutions having finite-time extinction in that lower $p$ range. We briefly examine the situation in the limit case $p=p_c$. Finally, very singular solutions are related to fractional elliptic problems of nonlinear eigenvalue form.in the limit case $p=p_c$. Finally, very singular solutions are related to fractional elliptic problems of nonlinear eigenvalue form.