Blow-up directions for quasilinear parabolic equations

Abstract

We consider the Cauchy problem for quasilinear parabolic equations $u_t=\Delta\phi(u)+f(u)$, with the bounded non-negative initial data $u_0(x)$ ($u_0(x)\not\equiv0$), where $f(\xi)$ is a positive function in $\xi>0$ satisfying a blow-up condition $\int_1^{\infty}1/f(\xi)\,\mathrm{d}\xi<\infty$. We study blow-up of non-negative solutions with the least blow-up time, i.e. the time coinciding with the blow-up time of a solution of the corresponding ordinary differential equation $\mathrm{d} v/\mathrm{d} t=f(v)$ with the initial data $\|u_0\|_{L^{\infty}(\mathbb{R}^N)}>0$. Such a blow-up solution blows up at space infinity in some direction (directional blow-up) and this direction is called a blow-up direction. We give a sufficient condition on $u_0$ for directional blow-up. Moreover, we completely characterize blow-up directions by the profile of the initial data, which gives a sufficient and necessary condition on $u_0$ for blow-up with the least blow-up time, provided that $f(\xi)$ grows more rapidly than $\phi(\xi)$.