### The global existence and finite time extinction of a quasilinear parabolic equation

#### Abstract

We consider the quasilinear differential equation $u_t = \mbox{ div} (|\bigtriangledown u|^{m-1} \bigtriangledown u )$ in $R^n \times (0, \infty)$, where $0 <m < 1$, $n \geq 2$. We show that there exist various global self-similar solutions with different decaying rate when $|x| \rightarrow \infty$. In particular, when $(n-1)/(n+1) < m < 1$, for every $c > 0$, there exists a unique self-similar solution which has the property that $u(x,t) \rightarrow c \delta (x)$ as $t \rightarrow 0$, where $\delta (x)$ is the Dirac delta function (Theorem 1). On the other hand, if $0 < m <(n-1)/(n+1)$, we show that there exist global self-similar solutions $u_{gs}$ which satisfy $u_{gs}(\cdot, t) \in L^p(R^n)$ for any $p > s \equiv n(1-m)/(1+m)$ but they are not in $L^s(R^n)$ (Theorem 1). At the same time, we demonstrate the existence of finite time extinction self-similar solutions $u_{ls}$ which satisfy the same integrability conditions (Theorem 2).

#### Article information

Source
Adv. Differential Equations, Volume 4, Number 5 (1999), 731-753.

Dates
First available in Project Euclid: 15 April 2013

Mathematical Reviews number (MathSciNet)
MR1696352

Zentralblatt MATH identifier
0959.35104

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

#### Citation

Qi, Yuan-Wei; Wang, Mingxin. The global existence and finite time extinction of a quasilinear parabolic equation. Adv. Differential Equations 4 (1999), no. 5, 731--753. https://projecteuclid.org/euclid.ade/1366030978