Abstract ：In this talk, we will talk about our recent work on the long time existence of small solutions, for a class of semi-linear wave equations with nonlinearities of the form |u|p+|ut|q, with p,q>1, which is in relation with both the Strauss conjecture and the Glassey conjecture. We determine the full region of (p, q) to admit global existence of small solutions, at least for spatial dimensions n≧3. Moreover, for many (p, q) when there is no global existence, we obtain sharp lower bound of the lifespan, which is of the same order as the upper bound of the lifespan. This is joint work with Kunio Hidano and Kazuyoshi Yokoyama.
