Differential and Integral Equations

Asymptotic self-similar global blow-up for a quasilinear heat equation

Victor A. Galaktionov

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study the asymptotic behavior near finite blow-up time $t = T$ of the solutions to the one-dimensional degenerate quasilinear parabolic equation $$ u_t = (u^\sigma u_x)_x + u^\beta \quad \text{in} \quad \mathbb{R} \times (0,T) ; \quad \sigma>0, \,\, 1<\beta<\sigma+1, $$ with bounded, nonnegative, compactly supported initial data. This parameter range corresponds to global blow-up where $ u(x,t) \to \infty$ as $t \to T^-$ for any $x \in \mathbb{R}$. We prove that the rescaled function $$ f(\xi,t) = (T-t)^{1/(\beta-1)} u(\xi(T-t)^m,t), \quad m = \frac {\beta-(\sigma+1)}{2(\beta-1)} < 0, $$ converges uniformly as $t \to T$ to a unique compactly supported, symmetric self-similar profile $\theta(\xi) \ge 0$ satisfying a nonlinear ordinary differential equation. The proof is based on the intersections comparison (the Sturmian argument) with a two-parametric family of self-similar solutions.

Article information

Source
Differential Integral Equations, Volume 10, Number 3 (1997), 487-497.

Dates
First available in Project Euclid: 2 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1367525664

Mathematical Reviews number (MathSciNet)
MR1744858

Zentralblatt MATH identifier
0890.35072

Subjects
Primary: 35K65: Degenerate parabolic equations
Secondary: 35K55: Nonlinear parabolic equations

Citation

Galaktionov, Victor A. Asymptotic self-similar global blow-up for a quasilinear heat equation. Differential Integral Equations 10 (1997), no. 3, 487--497. https://projecteuclid.org/euclid.die/1367525664


Export citation