### Fujita type results for a class of degenerate parabolic operators

Andrea Pascucci

#### 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 ${\cal L}$.

#### Article information

Source
Adv. Differential Equations Volume 4, Number 5 (1999), 755-776.

Dates
First available in Project Euclid: 15 April 2013

Mathematical Reviews number (MathSciNet)
MR1696353

Zentralblatt MATH identifier
0978.35024

Subjects
Primary: 35B33: Critical exponents
Secondary: 35H10: Hypoelliptic equations 35K65: Degenerate parabolic equations

#### Citation

Pascucci, Andrea. Fujita type results for a class of degenerate parabolic operators. Adv. Differential Equations 4 (1999), no. 5, 755--776. https://projecteuclid.org/euclid.ade/1366030979.