Lower Bound Lifespan Estimates For Semi-linear Klein-Gordon/Wave Equations Associated To The Harmonic Oscillator

In this paper,we mainly consider and discuss the semi-linear Klein Gordon equation and the semi-linear wave equation((?)t2-?+|x|2+m2)v=vp+1 related to the harmonic oscillator with small initial data in dimension d(d?1),where m?0,p ? N*.We prove that its solution exists in a longer time interval than that obtained by the local existence theory,give the lower bound estimate of the lifespan of the solution,and obtain the uniform boundedness of the Sobolev norm of the solution in the existence interval.The structure of this paper is as follows:In the introduction,the author introduces the research background and current research status of the semi-linear Klein Gordon equation and the semi-linear wave equation,and gives the problems and main conclusions of this paper.In the second chapter,for the semi-linear Klein Gordon equation related to the harmonic oscillator with small initial data,by using an LP-Lq estimate for eigenfunctions of the harmonic oscillator,and by analysis on the nonlinearity,we control the Sobolev energy of the solution in a certain time interval by using the normal form method.And the lower bound of the time interval is obtained,which is better than the one given by the local existence theorem.This conclusion extends the results in one dimension to higher dimensions.In the third chapter,for the semi-linear wave equation related to the harmonic oscillator with small initial data,based on the same method as in the second chapter,the author uses different high and low frequency decompositions when determining the perturbation,and finally obtains the lower bound estimate of the lifespan of the solution to the problem.