Blow Up for Some Semilinear Wave Equations in Multi-space Dimensions

Abstract

In this paper, we discuss a new nonlinear phenomenon. We find that in n ≥ 2 space dimensions, there exists two indexes p and q such that the Cauchy problems for the nonlinear wave equations and both have global existence for small initial data, while for the combined nonlinearity, the solutions to the Cauchy problem for the nonlinear wave equation with small initial data will blow up in finite time. In the two dimensional case, we also find that if q = 4, the Cauchy problem for the equation (0.1) has global existence, and the Cauchy problem for the equation has almost global existence, that is, the life span is at least exp (cϵ−2) for initial data of size ϵ. However, in the combined nonlinearity case, the Cauchy problem for the equation has a life span which is of the order of ϵ−18 for the initial data of size ϵ, this is considerably shorter in magnitude than that of the first two equations. This solves the final open optimality problem for general theory of fully nonlinear wave equations (see [7]). Furthermore, we consider the finite time blow-up of solutions to another natural generalization problem of (0.5).