Nonexistence of solutions for indefinite fractional parabolic equations

Wenxiong Chen; Leyun Wu; Pengyan Wang

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arXiv · 2108.02345

Nonexistence of solutions for indefinite fractional parabolic equations

Wenxiong Chen·Leyun Wu·Pengyan Wang

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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.

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Wenxiong Chen, Leyun Wu, Pengyan Wang. 2021-08-05. Nonexistence of solutions for indefinite fractional parabolic equations. https://arxiv.org/abs/2108.02345

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