## Advances in Differential Equations

### Universal bounds at the blow-up time for nonlinear parabolic equations

#### Abstract

We prove a priori supremum bounds for solutions to \begin{equation*} u_{t} - {\text{\rm div}} \big(u^{m-1} | {Du}| ^{\lambda -1} Du \big) = f(x) u^{p}\,, \end{equation*} as $t$ approaches the time when $u$ becomes unbounded. Such bounds are universal in the sense that they do not depend on $u$. Here $f$ may become unbounded, or vanish, as $x\to 0$. When $f\equiv1$, we also prove a bound below, as well as uniform localization of the support, for subsolutions to the corresponding Cauchy problem.

#### Article information

Source
Adv. Differential Equations, Volume 10, Number 1 (2005), 89-120.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867897

Mathematical Reviews number (MathSciNet)
MR2106122

Zentralblatt MATH identifier
1122.35042

Subjects
Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B40: Asymptotic behavior of solutions

#### Citation

Andreucci, Daniele; Tedeev, Anatoli F. Universal bounds at the blow-up time for nonlinear parabolic equations. Adv. Differential Equations 10 (2005), no. 1, 89--120. https://projecteuclid.org/euclid.ade/1355867897