We study fractional parabolic equations with indefinite nonlinearities$\frac{\partial u}{\partial t}(x,t)+{(-\mathrm{\Delta })}^{s}u(x,t)=x_1{u}^p(x,t),(x,t)\in {\mathbb{R}}^n\times \mathbb{R},$ 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.

我们研究具有不定非线性的分数式抛物线方程$\frac{\partial 你}{\partial 吨}(X,吨)+{(-\mathrm{\Delta })}^{秒}你(X,吨)=X_1{你}^{磷}(X,吨),(X,吨)\in {\mathbb{电阻}}^n\times \mathbb{电阻},$ 在哪里 $0<秒<1$ 和 $1<磷<\infty$. 我们首先证明所有正有界解都是单调递增的$X_1$方向。基于此，我们得出一个矛盾，从而得出不存在解。这些单调性和不存在性结果是有界域中分数抛物线方程的先验估计和完整爆炸的关键工具。为此，我们引入了几个新想法并开发了一种系统方法，该方法也可用于研究许多其他分数抛物线问题的解决方案的定性属性。