The Cauchy problem for a nonhomogeneous heat equation with reaction

Abstract

We study the behaviour of the solutions to the Cauchy problem$$\left\{\begin{array}{ll}\rho(x)u_t=\Delta u+u^p,&\quad x\in\mathbb{R}^N ,\;t\in(0,T),\\ u(x,\, 0)=u_0(x),&\quad x\in\mathbb{R}^N ,\end{array}\right.$$with $p>0$ and a positive density $\rho$ satisfying$\rho(x)\sim|x|^{-\sigma}$ for large $|x|$, $0 We show that instantaneous blow-up at space infinity takes placewhen $p\le1$. We also briefly discuss the case $2<\sigma< N$: we prove that theFujita exponent in this case does not depend on $\sigma$,$\tilde{p}_c=1+\frac2{N-2}$, and for initial values not too smallat infinity a phenomenon of instantaneous complete blow-up occurs inthe range $1< p < \tilde{p}_c$