Abstracts Mathematics

On the Stability of Self-Similar Blow-Up in Nonlinear Wave Equations

by Michael Liam McNulty

In this dissertation, we study the stability of self-similar blow-up for two nonlinear wave equations: the equation of motion of the strong-field Skyrme model and the quadratic wave equation.We begin by studying the stability of an explicitly known, self-similar solution of the equation of motion of the strong-field Skyrme model in the lowest energy supercritical dimension. This model originates from the Skyrme model which is itself a quasilinear modification of wave maps into a sphere. The strong-field Skyrme model restores the scaling invariance not present in the Skyrme model which allows for the existence of self-similar solutions. This equation is a semilinear wave equation that is, in particular, nonlinear in the derivatives of the unknown. As a consequence, standard techniques for establishing linear stability of this solution do not apply. Via application to a toy model, we present techniques that will be used to establish linear stability of this solution in a forthcoming paper.Next, we study the stability of an explicitly known, self-similar solution of the quadratic wave equation in the lowest energy supercritical dimension. For radial data close to this self-similar blowup solution with blowup time $T=1$ adjusted along a one-dimensional subspace, we are able to prove convergence of the corresponding solution to another self-similar blowup solution with blowup time close to $T=1$. In particular, this result holds true in a region of spacetime that can be made arbitrarily close to the Cauchy horizon of the singularity. This will be achieved via hyperboloidal similarity coordinates.