On similarity in the evolution of semilinear wave and Klein-Gordon equations: Numerical surveys

Abstract

The aim of this paper is threefold. The first and also main purpose is to provide numerical evidence for the conjecture proposed by Bizoń et al. [“Dynamics near the threshold for blowup in the one-dimensional focusing nonlinear Klein-Gordon equation,” J. Math. Phys. 52, 103703 (2011)]10.1063/1.3645363 that the blowup evolution of spherically symmetric semilinear Klein-Gordon equations is similar to the evolution of spherically symmetric semilinear wave equations, i.e., the mass term can be neglected when the amplitude of a solution grows. The second aim is to describe the relationship between different types of blowup for energy critical semilinear wave equations. The third goal is to present numerical evidence for the fact that the special class of self-similar profiles of semilinear wave equations found by Kycia [“On self-similar solutions of semilinear wave equations in higher space dimensions,” Appl. Math Comput. 217, 9451–9466 (2011)]10.1016/j.amc.2011.04.039 play the same role in the evolution of semilinear wave and Klein-Gordon equations as the previously known ordinary profiles. All the results are presented in spherical symmetry.