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On classification of blow-up patterns for a quasilinear heat equation. (English) Zbl 0851.35057
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.
Reviewer: A.Tsutsumi (Osaka)

##### MSC:
 35K55 Nonlinear parabolic equations 35K65 Degenerate parabolic equations 35B40 Asymptotic behavior of solutions to PDEs
##### Keywords:
quasilinear heat equation; blow-up behavior