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, we show that very singular solutions are related to fractional elliptic problems of the nonlinear eigenvalue type, of interest in their own right.