Nonexistence of solutions for indefinite fractional parabolic equations

Abstract

We study fractional parabolic equations with indefinite nonlinearities∂u∂t(x,t)+(−Δ)su(x,t)=x1up(x,t),(x,t)∈Rn×R, where 0<s<1 and 1<p<∞. We first prove that all positive bounded solutions are monotone increasing along the x1 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.