Exsistence Of A Global Solution To A Semi-linear Wave Equations

Consider the following Cauchy problem for semi - Linear wave equations in RnWhere = is a positive parameter. T. C. Sideris[3] has obtained the global existenceof a classical solution to (0. 1) with spherical symmetry in the case -where G = | ut |p(p > 2) and n = 3 under the assumption that f and g have compact support.Hiroyuki Takamura has obtained the existence of the global C2 - solution to (0.1) with spherical symmetry in the case G - | ut | p(p > 2, by removing the Compactness assumption of the support of intial data. Kunio Hidano also conjectured that the cauchy problem(0. 1)î–´the case G = | ut |p, p > admits a global C2 - solution for small data -without sphericalsymmetry, and that if 1 < p < the solution blows up at finite time.This paper established the global and local C2 - solution for the semilinear -wave equations without spherical symmetry in three space dinensions, the problem put forward by HiroyukiTakamura [2] is partially answered.