The critical case for a semilinear weakly hyperbolic equation

We prove a global existence result for the Cauchy problem, in the three-dimensional space, associated with the equation $$ u_{tt}-a_lambda(t) Delta_x u=-u|u|^{p(lambda)-1} $$ where $a_lambda(t)ge 0$ and behaves as $(t-t_0)^lambda$ close to some $t_0$ greater than zero with $a(t_0)=0$, and $p(lambda)=(3lambda+10)/(3lambda+2)$ with $3le p(lambda)le 5$. This means that we deal with the superconformal, critical nonlinear case. Moreover we assume a small initial energy.