Smooth continuation criterion for semilinear wave equations

For a smooth semilinear wave equation whose nonlinearity is a smooth function of the solution and its first derivatives, local Sobolev space theory gives a continuation criterion: a finite maximal existence time requires divergence of a sufficiently high solution norm. A bounded base norm $H^{s+1}\times H^s$, with $s>n/2$, allows restarting local existence. Higher regularity persists by differentiated energy estimates. Precise norms can be weakened for particular nonlinearities.