Fujita type results for quasilinear parabolic inequalities with nonlocal terms

TL;DR

This paper establishes conditions under which nonnegative solutions do not exist for certain nonlocal quasilinear parabolic inequalities, extending Fujita-type results to broader operators and convolution terms without relying on comparison principles.

Contribution

It introduces a novel nonlocal capacity estimate approach to determine nonexistence of solutions for quasilinear parabolic inequalities with convolution terms, including operators like the m-Laplacian.

Findings

Fujita-type critical exponent identified for problem (P^-)

No critical exponent exists for problem (P^+)

Nonexistence results hold without comparison principles

Abstract

In this paper we investigate the nonexistence of nonnegative solutions of parabolic inequalities of the form $$\begin{cases} &u_t \pm L_\mathcal A u\geq (K\ast u^p)u^q \quad\mbox{ in } \mathbb R^N \times \mathbb (0,\infty),\, N\geq 1,\\ &u(x,0) = u_0(x)\ge0 \,\, \text{ in } \mathbb R^N,\end{cases} \qquad (P^{\pm}) $$ where $u_0\in L^1_{loc}({\mathbb R}^N)$, $L_{\mathcal{A}}$ denotes a weakly $m$-coercive operator, which includes as prototype the $m$-Laplacian or the generalized mean curvature operator, $p,\,q>0$, while $K\ast u^p$ stands for the standard convolution operator between a weight $K>0$ satisfying suitable conditions at infinity and $u^p$. For problem $(P^-)$ we obtain a Fujita type exponent while for $(P^+)$ we show that no such critical exponent exists. Our approach relies on nonlinear capacity estimates adapted to the nonlocal setting of our problems. No comparison results or maximum principles are required.