2025/12/18 by Glogić, Irfan, Hilditch, David, Wallauch, David
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2512.16623
We study finite-time blowup for a nonlinear wave equation for maps from the Minkowski space ℝ1+d into the 1-sphere \mathbbS1, whose nonlinearity exhibits a null-form structure. We construct, for every dimension d ≥ 1, a countable family of discretely self-similar blowup solutions, which are even for d=1 and radial for d ≥ 2. The main contribution of the paper is a detailed nonlinear stability analysis of this family of solutions. For d ≥ 2, we consider radial data, while in d=1 we allow for general perturbations. After linearizing around the self-similar profiles in similarity variables, we construct resolvents of the resulting highly non-self-adjoint operators through Liouville-Green transformations and precise Volterra-type asymptotics. The construction itself, which occupies most of the paper, is technically challenging, as it is performed in arbitrary dimensions and for a countable family of operators in each. Combined with a detailed spectral analysis of the linearized operators, this yields sharp semigroup bounds and allows us to establish nonlinear stability of all discretely self-similar profiles in all dimensions, with precise co-dimension determined by the unstable spectrum. To our knowledge, this is the first result on the existence and stability of discretely self-similar blowup for a geometric wave equation.