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

In this article, we investigate the blow-up behavior of solutions to the one-dimensional damped nonlinear wave equation, namely $${\mathrm{\partial }}_t^{2}u-{\mathrm{\partial }}_x^{2}u+\frac{\mu }{1+t}{\mathrm{\partial }}_{t}u=\mathrm{\mid }{\mathrm{\partial }}_{t}u{\mathrm{\mid }}^p(p>1)\mathrm{.}$$ 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 ($\mu =0$).