Fujita type results for quasilinear parabolic inequalities with nonlocal terms

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 $u_0\in L_{loc}^1({\mathbb{R}}^N)$, $L_{\mathcal{A}}$ 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.

References in corpus (2)

• Positive solutions for quasilinear elliptic inequalities and systems with nonlocal terms
• Quasilinear elliptic inequalities with Hardy potential and nonlocal terms