Blowup for the defocusing septic complex-valued nonlinear wave equation in $\mathbb{R}^{4+1}$

In this paper, we prove blowup for the defocusing septic complex-valued nonlinear wave equation in $\mathbb{R}^{4+1}$. This work builds on the earlier results of Shao, Wei, and Zhang [SWZ2024a,SWZ2024b], reducing the order of the nonlinearity from $29$ to $7$ in $\mathbb{R}^{4+1}$. As in [SWZ2024a,SWZ2024b], 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 in [SWZ2024a] 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.