Grow-up for a quasilinear heat equation with a localized reaction in higher dimensions

TL;DR

This paper investigates the long-term behavior of solutions to a quasilinear heat equation with a localized reaction in higher dimensions, revealing how the size of the reaction region influences boundedness and growth rates.

Contribution

It provides new insights into the critical role of the reaction region size in the growth and boundedness of solutions, especially in the case where reaction and diffusion are balanced.

Findings

The size of the reaction region determines whether solutions are bounded or unbounded.

The critical length L influences the growth rate in the case p=m for dimensions N≥3.

Unbounded solutions exhibit different grow-up rates depending on the size of the reaction region.

Abstract

We study the behaviour of nonnegative solutions to the quasilinear heat equation with a reaction localized in a ball $$ u_t=\Delta u^m+a(x)u^p, $$ for $m>0$, $0<p\le\max\{1,m\}$, $a(x)=\mathds{1}_{B_L}(x)$, $0<L<\infty$ and $N\ge2$. We study when solutions, which are global in time, are bounded or unbounded. In particular we show that the precise value of the length $L$ plays a crucial role in the critical case $p=m$ for $N\ge3$. We also obtain the asymptotic behaviour of unbounded solutions and prove that the grow-up rate is different in most of the cases to the one obtained when $L=\infty$.