Restriction theorems and semilinear Klein-Gordon equations in \((1+3)\)-dimensions. (English) Zbl 0865.35077

Lindblad, Hans; Sogge, Christopher D.

Duke Math. J. 85, No. 1, 227-252 (1996).

The authors consider the semilinear Klein-Gordon equation with right-hand side \(F_p(u)\), \[ |\partial^jF_p(u)|\leq C|u|^{p-j},\quad 0\leq j\leq [p], \qquad |F_p'(u)-F_p'(v)|\leq C|u-v|^{p-j}, \] if \(1<p<2\), and prove global and semiglobal existece results for the Cauchy problem. More precisely, in the case \(1\leq n\leq 3\), \(p>1+{2\over n}\) it is shown that for each compactly supported Cauchy data \((f,g)\in C^{1+[n/2]}\times C^{[n/2]}\cap H^{(n+2)/2}\times H^{n/2}\) one can find a global \(C^1\) solution if \(|f|_{H^{(n+2)/2}}+|g|_{H^{n/2}}\) is sufficiently small. A semiglobal existence result, i.e. \(u\in C^1([0,T_\varepsilon]\times\mathbb{R}^n)\), \(T_\varepsilon=\exp(c/\varepsilon^{p-1})\), \(c=\text{const}>0\), \(\varepsilon>0\) sufficiently small, is proved under the weaker condition \(p=1+{2/n}\).

Reviewer: P.Popivanov (Sofia)

MSC:

35L15	Initial value problems for second-order hyperbolic equations
35L70	Second-order nonlinear hyperbolic equations

Keywords:

semiglobal existence result

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