Movable Singularity of Some Hamiltonian Systems Associated with Blowup Phenomena

Abstract

The motivation for this paper comes from a blowup problem for a semi-linear wave equation and a heat equation. Following an idea of J. Leray, we study a radially symmetric self-similar solution which has singularities on the characteristic cone. This naturally leads to the study of a Hamiltonian system called a profile equation. The novelty of this paper is that we focus on the movable singularity of the Hamiltonian system and we use Borel summability in constructing a singular solution. By “movable singularity” we mean that the singularity does not appear in the coefficients of the equation and depends on the respective solution. In the proof of our theorem we reduce the Hamiltonian system to a simpler form by a method similar to the so-called Birkhoff reduction. We obtain the parametrization of a singular solution by an elementary function. We also give applications to a semi-linear wave equation and a heat equation.