Blow-up profile for the complex-valued semilinear wave equation

TL;DR

This paper characterizes the blow-up behavior of solutions to the complex-valued semilinear wave equation in one dimension, identifying the blow-up profile and addressing additional neutral directions through modulation techniques.

Contribution

It extends the analysis of blow-up profiles from real to complex-valued wave equations, introducing new methods to handle the extra neutral direction.

Findings

Characterization of stationary solutions as a two-parameter family

Identification of the blow-up profile in the non-characteristic case

Development of a modulation technique to control neutral directions

Abstract

In this paper, we consider a blow-up solution for the complex-valued semilinear wave equation with power nonlinearity in one space dimension. We first characterize all the solutions of the associated stationary problem as a two-parameter family. Then, we use a dynamical system formulation to show that the solution in self-similar variables approaches some particular stationary one in the energy norm, in the non-characteristic case. This gives the blow-up profile for the original equation in the non-characteristic case. Our analysis is not just a simple adaptation of the already handled real case. In particular, there is one more neutral-direction in our problem, which we control thanks to a modulation technique.