We consider the global regularity problem for defocusing nonlinear wave systems
on Minkowski spacetime with d’Alembertian , where the field is vector-valued, and is a smooth potential which is positive and homogeneous of order outside of the unit ball for some . This generalises the scalar defocusing nonlinear wave (NLW) equation, in which and . It is well known that in the energy-subcritical and energy-critical cases when or and , one has global existence of smooth solutions from arbitrary smooth initial data , at least for dimensions . We study the supercritical case where and . We show that in this case, there exists a smooth potential for some sufficiently large (in fact we can take ), positive and homogeneous of order outside of the unit ball, and a smooth choice of initial data for which the solution develops a finite-time singularity. In fact the solution is discretely self-similar in a backwards light cone. The basic strategy is to first select the mass and energy densities of , then itself, and then finally design the potential in order to solve the required equation. The Nash embedding theorem is used in the second step, explaining the need to take relatively large.
"Finite-time blowup for a supercritical defocusing nonlinear wave system." Anal. PDE 9 (8) 1999 - 2030, 2016. https://doi.org/10.2140/apde.2016.9.1999