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
Full Text: