Blow up of the Solutions of Nonlinear Wave Equation

We construct for every fixed[inline-graphic not available: see fulltext] the metric[inline-graphic not available: see fulltext], where[inline-graphic not available: see fulltext],[inline-graphic not available: see fulltext],[inline-graphic not available: see fulltext],[inline-graphic not available: see fulltext], are continuous functions,[inline-graphic not available: see fulltext], for which we consider the Cauchy problem[inline-graphic not available: see fulltext], where[inline-graphic not available: see fulltext],[inline-graphic not available: see fulltext];[inline-graphic not available: see fulltext],[inline-graphic not available: see fulltext], where[inline-graphic not available: see fulltext],[inline-graphic not available: see fulltext],[inline-graphic not available: see fulltext],[inline-graphic not available: see fulltext],[inline-graphic not available: see fulltext],[inline-graphic not available: see fulltext],[inline-graphic not available: see fulltext] and[inline-graphic not available: see fulltext] are positive constants. When[inline-graphic not available: see fulltext], we prove that the above Cauchy problem has a nontrivial solution[inline-graphic not available: see fulltext] in the form[inline-graphic not available: see fulltext] for which[inline-graphic not available: see fulltext]. When[inline-graphic not available: see fulltext], we prove that the above Cauchy problem has a nontrivial solution[inline-graphic not available: see fulltext] in the form[inline-graphic not available: see fulltext] for which[inline-graphic not available: see fulltext].