Nonexistence of solutions for indefinite fractional parabolic equations

TL;DR

This paper proves the nonexistence of solutions for a class of indefinite fractional parabolic equations by establishing monotonicity properties and deriving contradictions, providing new tools for analyzing fractional PDEs.

Contribution

The paper introduces a systematic approach to prove nonexistence of solutions and monotonicity properties for fractional parabolic equations with indefinite nonlinearities.

Findings

All positive bounded solutions are monotone increasing in the $x_1$ direction.

Nonexistence of solutions for the studied fractional parabolic equations.

New ideas and systematic methods applicable to other fractional PDEs.

Abstract

We study fractional parabolic equations with indefinite nonlinearities $$ \frac{\partial u} {\partial t}(x,t) +(-\Delta)^s u(x,t)= x_1 u^p(x, t),\,\, (x, t) \in \mathbb{R}^n \times \mathbb{R}, $$ where $0<s<1$ and $1<p<\infty$. We first prove that all positive bounded solutions are monotone increasing along the $x_1$ direction. Based on this we derive a contradiction and hence obtain non-existence of solutions. These monotonicity and nonexistence results are crucial tools in a priori estimates and complete blow-up for fractional parabolic equations in bounded domains. To this end, we introduce several new ideas and developed a systematic approach which may also be applied to investigate qualitative properties of solutions for many other fractional parabolic problems.