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Small data blow-up for semilinear Klein-Gordon equations. (English) Zbl 0931.35105
The global existence of the solution of the Cauchy problem for the semilinear Klein-Gordon equations \[ \square u-u=F(u, \partial_tu, \nabla_x u),\;\square= -\partial^2_t +\Delta_x,\;x\in \mathbb{R}^n\tag{1} \] is studied in this paper. The nonlinearity \(F\) satisfies \[ \bigl|\partial^\infty F(v) \bigr |\leq C|v|^{p- |\alpha |},\;0\leq|\alpha |\leq [p] \quad \text{and} \]
\[ \bigl|\partial^\alpha F(v)- \partial^\alpha F(w) \bigr |\leq C|v-w|^{p -|\alpha |},\;p-1\leq |\alpha |\leq p\tag{2} \] for some \(p>1\) and \(C>0\). Here \(v\) and \(w\) stand for the arguments of \(F\). In the paper of H. Lindblad and Ch. D. Sogge [Duke Math. J. 85, No. 1, 227-252, (1996; Zbl 0865.35077)] was proved that the Cauchy problem for such equations has always a global solution if \(p>1+2/n\) and if the initial data are compactly supported and sufficiently small. M. Keel and T. Tao prove now that if \(p<1+2/n\), then there exist functions \(F\) which satisfy the conditions from above and such that the solution of the Cauchy problem for (1) blows up, even if the initial data are small. More precisely, the following theorems are proved:
Theorem 1.1. If \(n\geq 0\) and \(1<p\leq 1+ 2/n\) then there exists a scalar equations, (1) with the nonlinearity \(F(u, \partial_t u)\) which does not have a global finite-energy solution in the forward light cone for any Cauchy data \((\varepsilon f,\varepsilon g)\) with \(f\) and \(g\) compactly supported and \(\varepsilon>0\).
Theorem 1.2. If \(n=0,1,2,3\) and \(1<p< 1+2/n\) then the Cauchy problem for the Klein-Gordon equations \[ \begin{aligned} \square u-u & =|u |^{p-1}u\\ \square w-w & =|w|^p +|u |^p \end{aligned} \] does not have a global distributional solution supported in the forward light cone for any nontrivial compactly supported smooth initial data.
Estimates of the lifespan of the solution are obtained as a corollary of the proof of Theorem 1.2.

35L70 Second-order nonlinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
35B45 A priori estimates in context of PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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