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| We are interested in the fact that the lifespan $T(\varepsilon)$, the maximal existence time, of a classical solution of
$
\left\{
\begin{array}{l}
u_{tt}-\Delta u=u^2\quad\mbox{in}\quad{\bf R}^4\times[0,\infty),\\
u(x,0)=\varepsilon f(x),u_t(x,0)=\varepsilon g(x)
\end{array}
\right.
$
with a small parameter $\varepsilon>0$, compactly supported smooth functions $f$ and $g$, has an estimate
$
\exp\left(c\varepsilon^{-2}\right)\le T(\varepsilon)\le\exp\left(C\varepsilon^{-2}\right),
$
where $c$ and $C$ are positive constants depending only on $f$ and $g$.
This result is due to Li $\&$ Zhou in 1995 for the lower bound and our previous work, Takamura $\&$ Wakasa in 2011 for the upper bound. We note that its importance is quite huge as the problem is the final open part of Strauss' conjecture on semilinear wave equations as well as one of the last open optimality of the general theory for nonlinear wave equations.
In this talk, I would like to show you a special equation with $u^2$+some integral terms
admits a global solution. Such integral terms are deeply related to "derivative loss due to high dimensions". This fact may help us to make a criterion to get the global existence in four space dimensions for nonlinear term even with $u^2$. |
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