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Nonexistence of solutions for indefinite fractional parabolic equations. (English) Zbl 1476.35073

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

MSC:

35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs
35R11 Fractional partial differential equations
30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
35K15 Initial value problems for second-order parabolic equations
35K58 Semilinear parabolic equations
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