On the regularity of the blow-up set for semilinear heat equations. (English) Zbl 1012.35039

Zaag, Hatem

Ann. Inst. Henri Poincaré, Anal. Non Linéaire 19, No. 5, 505-542 (2002).

The blow-up phenomena arising in the following semilinear problem: \[ u_t=\Delta u+|u|^{p-1}u, \quad u(0)= u_0\in L^\infty (\mathbb{R}^N) \] is considered. There \(u:\mathbb{R}^N \times[0,T) \to\mathbb{R}\), \(p>1\), \((N-2) p<N+2\) and either \(u_0\geq 0\) or \((3N-4)p <3N+8\). A solution \(u(t)\) blows-up in finite time if its maximal existence time \(T\) is finite. In this case, \[ \lim_{t \to T}\bigl \|u(t)\bigr \|_{H^1(\mathbb{R}^N)} =\lim_{t\to T}\bigl \|u(t) \bigr\|_{L^\infty (\mathbb{R}^N)}= +\infty. \] A point \(a\in \mathbb{R}^N\) is called a blow-up point if \(|u(x,t) |\to+ \infty\) as \((x,t) \to(a,T)\). The blow-up set \(S\) is the set of all blow-up points.
The main result states that if some nondegeneracy condition is satisfied and \(S\) is continuous, then it is a \(C^1\) manifold.

Reviewer: A.Cichocka (Katowice)

MSC:

35K55	Nonlinear parabolic equations
35K15	Initial value problems for second-order parabolic equations
35K05	Heat equation

Keywords:

nondegeneracy condition

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