## Differential and Integral Equations

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

Victor A. Galaktionov

#### 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

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