Stable Type I blow-up for the one-dimensional wave equation with time-derivative nonlinearity

We study finite-time blow-up for the one-dimensional nonlinear wave equation with a quadratic time-derivative nonlinearity, $$u_{tt}-u_{xx}=(u_t)^2,\qquad (x,t)\in\mathbb R\times[0,T).$$ Building on the work of Ghoul, Liu, and Masmoudi \cite{ghoul2025blow} on the spatial-derivative analogue, we establish the non-existence of smooth, exact self-similar blow-up profiles. Instead we construct an explicit family of \emph{generalised self-similar} solutions, bifurcating from the ODE blow-up, that are smooth within the past light cone and exhibit type-I blow-up at a prescribed point $(x_0,T)$. We further prove asymptotic stability of these profiles under small perturbations in the energy topology.