Nonuniqueness for fractional parabolic equations with sublinear power-type nonlinearity

TL;DR

This paper demonstrates the existence of nontrivial solutions for a fractional parabolic equation with sublinear nonlinearity, contrasting with the linear or superlinear cases where solutions are trivial.

Contribution

It establishes the existence of nontrivial solutions for fractional parabolic equations with sublinear power nonlinearities, a phenomenon not present in the linear or superlinear cases.

Findings

Existence of at least one nontrivial nonnegative finite energy solution for sublinear case

Contrasts with the superlinear case where only trivial solutions exist

Solution existence depends on the weight function being separated from zero

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 $\alpha \geq 1$ in which the only bounded finite energy solution is identically zero.