Nonexistence of solutions to fractional parabolic problem with general nonlinearities

Abstract

In this content, we investigate a class of fractional parabolic equation with general nonlinearities $$\begin{aligned} \frac{\partial z(x,t)}{\partial t}-(\Delta +\lambda )^{\frac{\beta }{2}}z(x,t)=a(x_{1})f(z), \end{aligned}$$ ∂ z ( x , t ) ∂ t - ( Δ + λ ) β 2 z ( x , t ) = a ( x 1 ) f ( z ) , where a and f are nondecreasing functions. We first prove that the monotone increasing property of the positive solutions in $$x_{1}$$ x 1 direction. Based on this, nonexistence of the solutions are obtained by using a contradiction argument. We believe these new ideas we introduced will be applied to solve more fractional parabolic problems.