Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity and a scale-invariant damping

Abstract

In this article, we investigate the blow-up behavior of solutions to the one-dimensional damped nonlinear wave equation, namely $$ \partial_t^2 u - \partial_x^2 u + \frac{\mu}{1 + t} \partial_t u = |\partial_t u|^p \quad (p > 1). $$ Under the assumption of sufficiently large and smooth initial data, we establish that the blow-up curve is continuously differentiable ($\mathcal{C}^1$). A key step in our analysis involves the characterization of the blow-up profile of the solution. The proof relies on transforming the equation into a first-order system and adapting the techniques of Sasaki in \cite{Sasaki2018,Sasaki2019} which have elegantly extended the method of Caffarelli and Friedman \cite{Caffarelli1986} to nonlinear wave equations with time derivative nonlinearity, but without the scale-invariant term ($\mu =0$).