Finite degrees of freedom for the refined blow-up profile of the semilinear heat equation

TL;DR

This paper refines the understanding of blow-up behavior in the semilinear heat equation by constructing a finite-parameter family of non-explicit profiles, advancing the theoretical description of singularity formation.

Contribution

It introduces a novel approach that abandons explicit profiles, constructing special solutions with finite degrees of freedom to better describe blow-up behavior.

Findings

Constructed a family of special solutions with finite degrees of freedom.

Refined the asymptotic description of blow-up solutions.

Used topological and index theory methods for the construction.

Abstract

We refine the asymptotic behavior of solutions to the semilinear heat equation with Sobolev subcritical power nonlinearity which blow up in some finite time at a blow-up point where the (supposed to be generic) profile holds. In order to obtain this refinement, we have to abandon the explicit profile function as a first order approximation, and take a non explicit function as a first order description of the singular behavior. This non explicit function is in fact a special solution which we construct, obeying some refined prescribed behavior. The construction relies on the reduction of the problem to a finite dimensional one and the use of a topological argument based on index theory to conclude. Surprisingly, the new non explicit profiles which we construct make a family with finite degrees of freedom, namely $\frac{(N+1)N}{2}$ if $N$ is the dimension of the space.