Life-Span of Solutions to Critical Semilinear Wave Equations

The final open part of the famous Strauss conjecture on semilinear wave equations of the form u = |u|(p), i.e., blow-up theorem for the critical case in high dimensions was solved by Yordanov and Zhang [21], or Zhou [25] independently. But the estimate for the lifespan, the maximal existence time, of solutions was not clarified in both papers. Recently, Takamura and Wakasa [18] have obtained the sharp upper bound of the lifespan of the solution to the critical semilinear wave equations, and their method is based on the method in Yordanov and Zhang [21]. In this paper, we give a much simple proof of the result of Takamura and Wakasa [18] by using the method in Zhou [25] for space dimensions n >= 2.