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

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 $$p=1+{2/n}$$.

### MSC:

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

### Keywords:

semiglobal existence result
Full Text:

### References:

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