On classification of blow-up patterns for a quasilinear heat equation
The authors study the asymptotic blow-up behavior of nonnegative solutions to the quasilinear heat equation \[ (u)_t= (u^2)_{xx}+ u^2\quad \text{for } x\in \mathbb{R},\;t> 0,\tag{1} \] with nonnegative, bounded, continuous initial data \(u_0(x)\) in \(\mathbb{R}\). Theu give a complete classification of all possible types of blow-up behavior for compactly supported initial data. The main theorems are: Theorem 1. Let \(u_0\) be continuous and compactly supported. Assume that \(u_0\) intersects the level \(T^{- 1}\) exactly at two points. Then there exists a constant \(a_0= a_0(u_0)\in \mathbb{R}\) such that as \(t\to T^{- 1}\) \((T- t) u(x, t)\to f_*(x- a_0)\) uniformly in \(x\in \mathbb{R}\), where \(f_*(y)= \{(4/3 \cos^2(y/4)\) for \(|y|< 2\pi\), \(0\) for \(|y|\geq 2\pi\}\). Let \(I_f(t)\) be the number of intersections (in \(x\)) of the solutions \(u(x, t)\) and the flat solution \(u_f(t)= (T- t)^{- 1}\) with the same blow-up time \(T= T(u_0)\) for a fixed \(t\in (0, T)\). They classify the blow-up patterns in the following theorem. Theorem 2. Let \(u_0\) be continuous, compactly supported and suppose \[ \lim_{t\to T^{-1}} I_f(t)= 2k \] holds. Then there exist \(k\) numbers, \(a_1< a_2<\cdots< a_k\), with \(a_{i+ 1}- a_i\geq 4\pi\), such that as \(t\to T\) for \(i= 1, 2,\dots, k\) \((T- t) u(x, t)\to f_*(x- a_i)\) uniformly on \(|x- a_i|\leq 2\pi\) and \((T- t) u(x, t)\to 0\) uniformly on \(\mathbb{R}\backslash \bigcup_{(i)} \{|x- a_i|\leq 2\pi\}\). They also study the asymptotic blow-up behavior of nonnegative solutions of (1) when the initial data \(u_0(x)\) is neither flat nor bell-shaped.
scientific article; zbMATH DE number 883498