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Universality in blow-up for nonlinear heat equations. (English) Zbl 0857.35018

The authors study blow-up profiles of solutions of the nonlinear heat equation \(u_t= u_{xx}+u^p\), where \(p>1\) and \(x\in \mathbb{R}\). They show that there exist infinitely many profiles around the blow-up point, and, for each integer \(k\), they construct a set \({\mathcal M}_k\) of codimension \(2k\) in the space \(C^0(\mathbb{R})\) of initial data for which the solutions blow up according to the given profile. The ideas of the proofs are close to the ones of the renormalization group where the long-time behavior of the solutions is related to the existence of fixed points of the renormalization group transformation. The set \({\mathcal M}_k\) consists of initial data which are close to the corresponding fixed point and which lie in the basin of attraction of this point.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
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