The blow-up and lifespan of solutions to systems of semilinear wave   equation with critical exponents in high dimensions
  In this paper we prove the blow-up theorem in the critical case for weakly coupled systems of semilinear wave equations in high dimensions. The upper bound of the lifespan of the solution is precisely clarified. 
Yuki Kurokawa, Hiroyuki Takamura, Kyouhei Wakasa
math.AP