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A necessary and sufficient condition of existence of global solutions for some nonlinear hyperbolic equations. (English) Zbl 0836.35098

Summary: The author considers the Klein-Gordon equations \(u_{tt} - \Delta u + \mu u = f(u)\) \((\mu > 0\), \(|f(u) |\leq c |u |^{\alpha + 1})\). The necessary and sufficient condition of existence of global solutions is obtained for \(E(0) = {1 \over 2} (|u_1 |^2_{L^2} + |\nabla u_0 |^2_{L^2} + \mu |u_0 |^2_{L^2}) - \int_{\mathbb{R}^n} \int_0^{u_0} f(s) dsdx < d\) \((d\) is a given constant).

MSC:

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