On blow-up for the supercritical defocusing nonlinear scalar wave equation

Abstract

We study finite-time singularity formation for the scalar defocusing nonlinear wave equation $-{\partial }_t^{2}u+\Delta u=u^p$. First, for fixed $d\ge 4$ and odd $p\ge 3$, we establish a conditional blow-up result under the assumption that a smooth self-similar imploding profile exists for the leading-order relativistic Euler system, with associated normalized coefficients satisfying quantitative analytic radius conditions. Second, we verify both the profile and the analytic radius conditions in dimensions four and five for all sufficiently large odd $p$. Using finite speed of propagation, we obtain a dimension-independent threshold $p_0$ such that, for every $d\ge 4$ and every odd $p\ge p_0$, finite-time blow-up occurs for some smooth, compactly supported, real-valued initial data. The proof combines a normalized formulation that separates the leading periodic oscillation from the unknown modulation, growth of the spatial analytic radius induced by an outgoing property, and frequency-dependent matrix multipliers that control the cumulative growth caused by the periodic linearized potential.

134 pages, 1 figure