Grow-up for a quasilinear heat equation with a localized reaction

TL;DR

This paper analyzes the long-term behavior of solutions to a quasilinear heat equation with a localized reaction term, identifying conditions for boundedness and growth rates depending on parameters.

Contribution

It establishes the critical exponent for boundedness and characterizes the growth rates of solutions in the presence of a localized reaction.

Findings

Solutions are bounded if p > p_0, where p_0 = max{1, (m+1)/2}.

For p ≤ p_0, solutions are global and unbounded, with different growth rates depending on p and m.

Inside the reaction support, growth rates match the homogeneous case; outside, they are slower.

Abstract

We study the behaviour of global solutions to the quasilinear heat equation with a reaction localized $$ u_t=(u^m)_{xx}+a(x) u^p, $$ $m, p>0$ and $a(x)$ being the characteristic function of an interval. we prove that there exists $p_0=\max\{1,\frac{m+1}2\}$ such that all global solution are bounded if $p>p_0$, while for $p\le p_0$ all the solution are global and unbounded. In the last case, we prove that if $p<m$ the grow-up rate is different to the one obtained when $a(x)\equiv1$, while if $p>m$ the grow-up rate coincides with that rate, but only inside the support of $a$; outside the interval the rate is smaller.