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.


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