标题： 不定分数抛物方程解的存在性不存在 显示英文标题

标题： Nonexistence of solutions for indefinite fractional parabolic equations

摘要： 我们研究带有不定非线性的分数抛物方程$$ \frac{\partial u} {\partial t}(x,t) +(-\Delta)^s u(x,t)= x_1 u^p(x, t),\,\, (x, t) \in \mathbb{R}^n \times \mathbb{R}, $$，其中$0<s<1$且$1<p<\infty$。 我们首先证明所有正有界解在$x_1$方向上都是单调递增的。 基于此，我们推导出一个矛盾，从而得到解的不存在性。 这些单调性和不存在性结果是关于有界域中分数抛物方程先验估计和完全爆破的关键工具。 为此，我们引入了几种新思想并发展了一种系统方法，该方法也可用于研究许多其他分数抛物问题解的定性性质。 显示英文摘要

摘要： 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.