Stable blow-up on a sphere for a quadratic-derivative nonlinear wave equation
We study finite-time blow-up for the nonlinear wave equation \begin{equation*} v_{tt}-\Delta v=|\nabla_x v|^2 \end{equation*} in dimensions $n\geq2$, under radial symmetry. For every prescribed radius $r_0>0$, we construct solutions which blow up in finite time $T>0$ on the sphere $\{|x|=r_0\}$ with logarithmic Type-I rate. The leading singular dynamics are governed by ``generalised self-similar'' profiles of the associated one-dimensional equation, while the radial geometry generates a curvature correction of size $\mathcal{O}((\frac{T}{r_0})^2)$. A key simplification in our approach is a logarithmic radial correction which removes the first-order radial drift and reduces the geometry to a decaying inverse-square forcing. We further prove asymptotic stability of the resulting family under radial perturbations. A new feature compared with the one-dimensional theory is that the stable blow-up family is not fully explicit. To overcome this, we develop spectral and semigroup estimates on an extended light cone, together with Lipschitz dependence on the modulation parameters for the spectral projections, the stable flow, and the non-explicit correction.