Nonexistence of solutions for indefinite fractional parabolic equations

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.

References in corpus (1)

• Hopf's lemmas for parabolic fractional Laplacians and parabolic fractional $p$-Laplacians