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Stability of the blow-up profile of nonlinear heat equations from the dynamical system point of view. (English) Zbl 0971.35038
The following nonlinear heat equation: ${\partial u\over\partial t}=\Delta u+|u|^{p- 1}u,\quad u(0)= u_0\tag{1}$ is considered, where $$u:\mathbb{R}^N\times [0,T)\to \mathbb{R}$$, $$u(t)\in L^\infty$$ and $$1< p$$, $$(N-2)p< N+ 2$$. The authors are interested in blow-up solutions $$u(t)$$ of (1) for which exists $$T$$ such that $$\lim_{t\to T} \|u(t)\|_{L^\infty}= \infty$$. The existence of such solutions is known. A solution $$u(t)$$ of (1) is called a blow-up solution of type I if there exists $$C>0$$ such that $\|u(t)\|_{L^\infty}\leq C(T- t)^{-{1\over p-1}},\;t\in[0,T].$ The main result of this paper concerns a stability of the blow-up profile with respect to the initial data and states: Theorem. Let $$\widetilde u(t)$$ be a type I blow-up solution of (1) with initial data $$\widetilde u_0$$ which blows up at time $$T$$ at only one point $$\widetilde a=0$$. Assume that for some $$R> 0$$ and $$M>0$$ $\text{for all }|x|\geq R\text{ and }t\in [0,\widetilde T):|\widetilde u(x,t)|\leq M,$
$\text{for all }x\in \mathbb{R}^N\setminus\{0\}:\widetilde u(x,t)\underset{t\to\widetilde T}\rightarrow \widetilde u^*(x)\text{ where }u^*(x)\underset{x\sim 0}\sim U_1(x),$ where $$U_1(x)= ({8p|\log|x||\over (p-1)^2|x|^2})^{{1\over p- 1}}$$. Then, there is a neighborhood $${\mathcal V}$$ in $$L^\infty$$ of $$\widetilde u_0$$ such that for all $$u_0\in{\mathcal V}$$ the solution of (1) with initial data $$u_0$$ blows up at time $$T= T(u_0)$$ at a unique point $$a= a(u_0)$$ and $\text{for all }x\in\mathbb{R}^N\setminus \{a\}: u(x,t)\underset{t\to T} \rightarrow u^*(x)\text{ where }u^*(x)\underset{x\sim a} \sim U_1(x- a).$ Moreover, $$(a(u_0), T(u_0))$$ goes to $$(0,\widetilde T)$$ as $$u_0$$ goes to $$\widetilde u_0$$.
It is known that under conditions $$u(0)\geq 0$$ or $$(3N- 4)p< 3N+ 8$$ all blow-up solutions are of type I. It is still an open problem for $$p\geq {3N+8\over 3N-4}$$ to know if there are blow-up solutions which are not of type I. The proof of the theorem is closely related to some uniform estimates with respect to initial data following from a Liouville Theorem and a solution of a dynamical system of ordinary differential equations.

##### MSC:
 35K55 Nonlinear parabolic equations 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 35B33 Critical exponents in context of PDEs 35A20 Analyticity in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs
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