Nonuniqueness for fractional parabolic equations with sublinear power-type nonlinearity

Abstract

We show that the parabolic equation $u_t+(-\Delta {)}^{s}u=q(x)|u{|}^{\alpha -1}u$ posed in a time-space cylinder $(0,T)\times {\mathbb{R}}^N$ and coupled with zero initial condition and zero nonlocal Dirichlet condition in $(0,T)\times ({\mathbb{R}}^N\setminus \Omega )$, where $\Omega$ is a bounded domain, has at least one nontrivial nonnegative finite energy solution provided $\alpha \in (0,1)$ and the nonnegative bounded weight function $q$ is separated from zero on an open subset of $\Omega$. This fact contrasts with the (super)linear case $\alpha \ge 1$ in which the only bounded finite energy solution is identically zero.

16 pages