## Advances in Differential Equations

### Self-similar blow-up for a quasilinear parabolic equation with gradient diffusion and exponential source

#### Abstract

We study the asymptotic behaviour near a finite blow-up time $t=T$ of the solutions to the initial-boundary value problem for the quasilinear equation $$u_t = \nabla \cdot (|\nabla u|^\sigma \nabla u) + e^u \mbox{ in } \{ | x| < R\} \times (0,T), \ \ \sigma >0,$$ with zero Dirichlet boundary condition and a radial symmetric initial function $u_0 (|x|) >0$ in $\{ |x| <R\}, \ u'_0 (r)<0$ in $(0,R)$. We prove single point blow-up and different sharp lower and upper estimates of the solution as $t \rightarrow T^-$ and of the final-time profile. For the one-dimensional problem the asymptotic behaviour is proved to be described by nonconstant self-similar solutions of the form $u_* (x,t) =-\log (T-t) + \theta (\xi), \ \xi = x/(T-t)^{1/(\sigma+2)}$, where $\theta (\xi) \sim - (\sigma+2) \log \xi$ as $\xi \rightarrow\infty$.

#### Article information

Source
Adv. Differential Equations, Volume 2, Number 1 (1997), 85-124.

Dates
First available in Project Euclid: 24 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1366809229

Mathematical Reviews number (MathSciNet)
MR1424764

Zentralblatt MATH identifier
1023.35516

#### Citation

Budd, Chris J.; Dold, James W.; Galaktionov, Victor A. Self-similar blow-up for a quasilinear parabolic equation with gradient diffusion and exponential source. Adv. Differential Equations 2 (1997), no. 1, 85--124. https://projecteuclid.org/euclid.ade/1366809229