A note on blow-up of a nonlinear integral equation

Abstract

Let us deal with the positive solutions of \begin{equation*} \frac{\partial u(t)}{\partial t}=k(t)\Delta _{\alpha }u(t)+h(t)u^{1+\beta }(t),\text{ \ }u(0,x)=\varphi (x)\geq 0,\text{ }x\in \mathbb{R}^{d}, \end{equation*} where $\Delta _{\alpha }$ is the fractional Laplacian, $0<\alpha \leq 2$, and $\beta >0$ is a constant. We prove that under certain regularity condition on $\varphi $, $h$ and $k$ any non-trivial positive solution blows up in finite time. In this way we answer, in particular, the question raised in [4] for the critical case.