Blowup behaviour for the nonlinear Klein--Gordon equation

We analyze the blowup behaviour of solutions to the focusing nonlinear Klein–Gordon equation in spatial dimensions $d\ge 2$. We obtain upper bounds on the blowup rate, both globally in space and in light cones. The results are sharp in the conformal and sub-conformal cases. The argument relies on Lyapunov functionals derived from the dilation identity. We also prove that the critical Sobolev norm diverges near the blowup time.