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).