Blow Up And Lifespan Estimation Of Solutions To Semi-linear Wave Equation

In this paper,we focus on the rupture mechanism and life span estimation of solutions for a class of semilinear wave equation utt-?u=(1+|x|2)?|u|p initial value problems.In the first part,we prove that the problem satisfies its initial condition in high-dimensional case,and there is no global solution at the critical p = pc(n).Firstly,the semilinear wave equation is transformed into a constant differential inequality of a functional of the solution,and 2 functions are introduced by using the test function.Secondly,in the high dimensional case,n?5 is used to establish an improved lower bound estimate of a nonlinear term by using the method of Randon transformation to the radial function,and to determine the appropriate a and q to prove that the problem is broken in a limited time.Finally,the range of parameters ? is given.The second part,the lifespan estimate to the Cauchy problem of the semi-linear wave equationl utt-?u=(1+|x|2)?|u|p in Rn is studied.The upper bound of the lifespan is improved for the case n= 2,1<p ?2 and n = 1,p>1,by using the improved Kato's type lemma.