Fujita type results for a class of degenerate parabolic operators

Abstract

In this paper we study the global existence of non-negative solutions to the Cauchy problem for $Lu= - u^{p}$ where $L$ belongs to a class ${\cal L}$ of hypoelliptic operators of degenerate parabolic type and $p>1$. Extending some old results by Fujita, we prove the existence and we determine explicitly a critical exponent $p^{*}$ for the problem. Namely, we prove that if $p>p^{*}$ then there are global positive solutions to the problem, while if $1 <p<p^{*}$ then all non-trivial solutions blow up in finite time. We also study the critical case $p=p^{*}$ for a remarkable subclass of ${\cal L}$.