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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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