It is well-known that there are global small data smooth solutions for the
3-D semilinear Klein-Gordon equations $\square u + u = F(u,{\partial u})$ with
cubic nonlinearities. Jedoch, for the short pulse initial data $(u,
\partial_tu)(0, x)=({\delta^{\nu+1}}{u_0}({\frac{x}{\Delta}}),{\delta^\nu
}{u_1}({\frac{x}{\Delta}}))$ with $\nu\in\Bbb R$ and $(u_0, u_1)\in
C_0^{\infty}(\Bbb R)$, which are a class of large initial data, we establish
that when $\nu\le -\frac{1}{2}$, the solution $u$ can blow up in finite time
for some suitable choices of $(u_0, u_1)$ and cubic nonlinearity $F(u,{\partial
u})$; when $\nu>-\frac{1}{2}$, the smooth solution $u$ exists globally.
Therefore, $\nu=-\frac{1}{2}$ is just the critical power corresponding to the
global existence or blowup of smooth
short pulse solutions for the cubic semilinear Klein-Gordon equations.

Dieser Artikel untersucht Zeitreisen und deren Auswirkungen.

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