## Advances in Differential Equations

- Adv. Differential Equations
- Volume 9, Number 5-6 (2004), 625-643.

### Instantaneous shrinking of the support in degenerate parabolic equations with strong absorption

#### Abstract

We investigate the phenomenon of instantaneous shrinking of the support of nonnegative solutions to the Cauchy problem in $\\mathbb R^n$ for $$ u_t=f(u) \Delta u - g(u), \quad \mbox{where } f(0)=0. $$ Among other results, it is shown by means of comparison and integral techniques that under some structural assumptions on $f$ and $g$, a necessary {\em and} sufficient condition on the growth of $g$ near zero for instantaneous shrinking to occur is $\int_0^1 \frac{ds}{g(s)} <\infty$.

#### Article information

**Source**

Adv. Differential Equations, Volume 9, Number 5-6 (2004), 625-643.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867938

**Mathematical Reviews number (MathSciNet)**

MR2099974

**Zentralblatt MATH identifier**

1101.35010

**Subjects**

Primary: 35K55: Nonlinear parabolic equations

Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35B40: Asymptotic behavior of solutions 35K65: Degenerate parabolic equations

#### Citation

Winkler, Michael. Instantaneous shrinking of the support in degenerate parabolic equations with strong absorption. Adv. Differential Equations 9 (2004), no. 5-6, 625--643. https://projecteuclid.org/euclid.ade/1355867938