Blow up for some semilinear wave equations in multi-space dimensions

In this paper, we discuss a new nonlinear phenomenon. We find that in $n\geq 2$ space dimensions, there exists two indexes $p$ and $q$ such that the cauchy problems for the nonlinear wave equations {equation} \label{0.1} \Box u(t,x) = |u(t,x)|^{q}, \ \ x\in R^{n}, {equation} and {equation} \label{0.2} \Box u(t,x) = |u_{t}(t,x)|^{p}, \ \ x\in R^{n} {equation} both have global existence for small initial data, while for the combined nonlinearity, the solutions to the Cauchy problem for the nonlinear wave equation {equation} \label{0.3} \Box u(t,x) = | u_{t}(t,x)|^{p} + |u(t,x)|^{q}, \ \ x\in R^{n}, {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 \eqref{0.1} has global existence, and the Cauchy problem for the equation {equation} \label{0.4} \Box u(t,x) = u (t,x)u_{t}(t,x)^{2}, \ \ x\in R^{2} {equation} has almost global existence, that is, the life span is at least $\exp (c\varepsilon^{-2})$ for initial data of size $\varepsilon$. However, in the combined nonlinearity case, the Cauchy problem for the equation {equation} \label{0.5} \Box u(t,x) = u(t,x) u_{t}(t,x)^{2} + u(t,x)^{4}, \ \ x\in R^{2} {equation} has a life span which is of the order of $\varepsilon^{-18}$ for the initial data of size $\varepsilon$, this is considerably shorter in magnitude than that of the first two equations. This solves an open optimality problem for general theory of fully nonlinear wave equations (see \cite{Katayama}).