Nonuniqueness for fractional parabolic equations with sublinear power-type nonlinearity

We show that the parabolic equation $u_t+(-\mathrm{\Delta }{)}^{s}u=q(x)\mathrm{\mid }u{\mathrm{\mid }}^{\alpha -1}u$ posed in a time-space cylinder $(0,T)\times {\mathbb{R}}^N$ and coupled with zero initial condition and zero nonlocal Dirichlet condition in $(0,T)\times ({\mathbb{R}}^N\setminus \mathrm{\Omega })$, where $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$. This fact contrasts with the (super)linear case $\alpha \ge 1$ in which the only bounded finite energy solution is identically zero.