A nonlinear diffusion equation with reaction localized to the half-line

TL;DR

This paper investigates the behavior of solutions to a nonlinear heat equation with localized reaction on the half-line, identifying critical exponents for global existence and analyzing growth and blow-up rates.

Contribution

It characterizes the global existence and Fujita exponents for the equation and analyzes the growth and blow-up rates depending on parameters.

Findings

Global existence exponent p_0=1

Fujita exponent p_c=m+2

Different grow-up rates for p>m and p=1

Abstract

We study the behaviour of the solutions to the quasilinear heat equation with a reaction restricted to a half-line $$ u_t=(u^m)_{xx}+a(x) u^p, $$ $m, p>0$ and $a(x)=1$ for $x>0$, $a(x)=0$ for $x<0$. We first characterize the global existence exponent $p_0=1$ and the Fujita exponent $p_c=m+2$. Then we pass to study the grow-up rate in the case $p\le1$ and the blow-up rate for $p>1$. In particular we show that the grow-up rate is different as for global reaction if $p>m$ or $p=1\neq m$.