## Advances in Differential Equations

- Adv. Differential Equations
- Volume 7, Number 2 (2002), 155-176.

### On a class of parabolic equations with variable density and absorption

Robert Kersner, Guillermo Reyes, and Alberto Tesei

#### Abstract

We investigate qualitative properties of solutions to the Cauchy problem for the equation $\rho(x)u_t=(u^m)_{xx}-c_0 u^p$, where $m>1$ and $c_0, p >0$; the initial data are nonnegative with compact support and the density $\rho(x)>0$ satisfies suitable decay conditions as $|x|\to\infty$. If $p \ge m$ and $\rho(x)$ decays not faster than $|x|^{-k}$, where $0 <k \le k^*:=2(p-1)/(p-m)$, the interfaces exist globally in time. On the contrary, if $\rho(x)$ decays faster than $|x|^{-k}$ with $k>k^*$, the interfaces can disappear in finite time. It is also proved that solutions go to zero uniformly as $t \to\infty $, at variance from the case $c_0=0$.

#### Article information

**Source**

Adv. Differential Equations Volume 7, Number 2 (2002), 155-176.

**Dates**

First available in Project Euclid: 27 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1356651849

**Mathematical Reviews number (MathSciNet)**

MR1869559

**Zentralblatt MATH identifier**

1223.35209

**Subjects**

Primary: 35K65: Degenerate parabolic equations

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

#### Citation

Kersner, Robert; Reyes, Guillermo; Tesei, Alberto. On a class of parabolic equations with variable density and absorption. Adv. Differential Equations 7 (2002), no. 2, 155--176. https://projecteuclid.org/euclid.ade/1356651849