The final problem on the optimality of the general theory for nonlinear wave equations

The general theory of the initial value problem for fully nonlinear wave equations is to clarify lower bounds of the lifespan, the maximal existence time, of classical solutions in terms of the amplitude of small initial data according to the order of smooth nonlinear terms and space dimensions. All the results had been obtained till 1995. So we have been interested in the optimality of the lower bounds. This can be obtained by blow-up results for model equations. Among such several results, only the case of the quadratic semilinear term in 4 space dimensions has been remained open for more than 20 years.This final problem on the optimality has been known to be the critical case of Strauss' conjecture on semilinear wave equations. The technical difficulty prevented us from proving even the blow-up of solutions in finite time. This was finally solved by Yordanov and Zhang [5] in 2006, or Zhou [6] in 2007 independently. But the upper bound of the lifespan was not clarified in both papers. Recently Takamura and Wakasa [4] have succeeded to obtain it including all the critical cases in higher dimensions than 4.In this note, we present the result of [4] in the most interesting case, 4 space dimensions. It is much simpler than higher dimensions.