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

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 $p\le 1$ 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 $p=1\ne m$.