## Advances in Differential Equations

- Adv. Differential Equations
- Volume 13, Number 11-12 (2008), 1001-1030.

### The evolution of dead cores in strongly degenerate diffusion problems

#### Abstract

This paper studies the large time behavior of nonnegative solutions of $$ u_t = u^p \Delta u - u^{-q} \chi_{\{u>0\}} \qquad \mbox{in } \Omega \times (0,\infty), \qquad p \ge 1, q>-1, $$ with prescribed boundary values $u|_{{\partial\Omega}}=1$ in smoothly bounded domains $\Omega \subset \mathbb R^n$. The particular interest here is in the question of how the interplay of strongly degenerate diffusion and strong absorption influences the evolution of the dead core set $\{u(t)=0\}$ (which is enforced to be nontrivial for all time by the assumption that the initial data $u_0$ vanish in some ball). The main results state that there exist four parameter regimes in which, widely independent of $u_0$, all solutions undergo

(i)a complete extinction at time $t=0$ ($q>p-1$)

(ii) a total extinction in finite but positive time ($0 <q\le p-1$),

(iii) an extinction in infinite time ($1-p <q <0$), or

(iv) no extinction ($q <1-p$),

respectively. Specifically, the first two types of behavior are in sharp contrast to the non-degenerate and weakly degenerate cases $p=0$ and $p \in (0,1)$, respectively, where only (iv) occurs.

#### Article information

**Source**

Adv. Differential Equations, Volume 13, Number 11-12 (2008), 1001-1030.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867284

**Mathematical Reviews number (MathSciNet)**

MR2483128

**Zentralblatt MATH identifier**

1194.35068

**Subjects**

Primary: 35K55: Nonlinear parabolic equations

Secondary: 35B33: Critical exponents 35B40: Asymptotic behavior of solutions 35K57: Reaction-diffusion equations 35K65: Degenerate parabolic equations

#### Citation

Winkler, Michael. The evolution of dead cores in strongly degenerate diffusion problems. Adv. Differential Equations 13 (2008), no. 11-12, 1001--1030. https://projecteuclid.org/euclid.ade/1355867284