article

Title: Blow-up for a degenerate diffusion problem not in divergence form

Authors: Arturo de Pablo, Raul Ferreira and Julio D. Rossi

Issue: Volume 55 (2006), Issue 3, 955-974

Abstract:

$We study the behaviour of solutions of the nonlinear diffusion equation in the half-line, $\mathbb{R}_{+} = (0,\infty)$, with a nonlinear boundary condition, \[ \begin{cases} u_t = uu_{xx}, \quad (x,t) \in \mathbb{R}_{+} \times (0,T),\\ -u_{x}(0,t) = u^{p}(0,t), \quad t \in (0,T),\\ u(x,0) = u_0(x), \quad x \in \mathbb{R}_{+},\\ \end{cases} \] with $p$ greater than $0$. We describe, in terms of $p$ and the initial datum, when the solution is global in time and when it blows up in finite time. For blowing up solutions we find the blow-up rate and the blow-up set and we describe the asymptotic behaviour close to the blow-up time in terms of a self-similar profile. The stationary character of the support is proved both for global solutions and blowing-up solutions. Also we obtain results for the problem in a bounded interval.$