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The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions

The final open part of Strauss' conjecture on semilinear wave equations was the blow-up theorem for the critical case in high dimensions. This problem was solved by Yordanov and Zhang in 2006, or Zhou in 2007 independently. But the estimate for the lifespan, the maximal existence time, of solutions was not clarified in both papers. In this paper, we refine their theorems and introduce a new iteration argument to get the sharp upper bound of the lifespan. As a result, with the sharp lower bound by Li and Zhou in 1995, the lifespan $T(\e)$ of solutions of $u_{tt}-Δu=u^2$ in $\R^4\times[0,\infty)$ with the initial data $u(x,0)=\e f(x),u_t(x,0)=\e g(x)$ of a small parameter $\e>0$, compactly supported smooth functions $f$ and $g$, has an estimate \[ \exp(c\e^{-2})\le T(\e)\le\exp(C\e^{-2}), \] where $c$ and $C$ are positive constants depending only on $f$ and $g$. This upper bound has been known to be the last open optimality of the general theory for fully nonlinear wave equations.