Blowup for the defocusing septic complex-valued nonlinear wave equation in ℝ⁴⁺¹

In this paper, we prove blowup for the defocusing septic complex-valued nonlinear wave equation in R 4 + 1 R⁴⁺¹ . This work builds on the earlier results of Shao, Wei, and Zhang [Forum Math. Pi 13 (2025); Self-similar imploding solutions of the relativistic Euler equations , arXiv:2403.11471, 2024], reducing the order of the nonlinearity from 29 29 to 7 7 in R 4 + 1 R⁴⁺¹ . As by Shao, Wei, and Zhang [Forum Math. Pi 13 (2025); Self-similar imploding solutions of the relativistic Euler equations , arXiv:2403.11471, 2024], the proof hinges on a connection between solutions to the nonlinear wave equation and the relativistic Euler equations via a front compression blowup mechanism. More specifically, the problem is reduced to constructing smooth, radially symmetric, self-similar imploding profiles for the relativistic Euler equations. As with implosion for the compressible Euler equations, the relativistic analogue admits a countable family of smooth imploding profiles. The result of Shao, Wei, and Zhang [ Self-similar imploding solutions of the relativistic Euler equations , arXiv:2403. 11471, 2024] represents the construction of the first profile in this family. In this paper, we construct a sequence of solutions corresponding to the higher-order profiles in the family. This allows us to saturate the inequalities necessary to show blowup for the defocusing complex-valued nonlinear wave equation with an integer order of nonlinearity and radial symmetry via this mechanism.