Movable singularity of semi linear Heun equation and application to blowup phenomenon

Abstract

In this paper we shall study the movable singularity of semi linear Heun equation and its application to the blowup of a semi linear wave equation. In fact, the semi linear Heun equation appears if we consider a radially symmetric self-similar solution of the semi linear wave equation. By the movable singularity we mean the singularity which does not appear in the coefficients of the equation and that depends on the respective solution. We focus on movable singularity when we construct a singular solution of the semi linear wave equation with singularities on the characteristic cone. In the proof of our theorem we reduce our equation to a simpler form by the method similar to the so-called Birkhoff reduction, then we analyze the reduced equation. The latter part is closely related with the parametrization of a solution in terms of the Jacobi elliptic function.