Global existence and lifespan for semilinear wave equations with mixed nonlinear terms

Abstract

Firstly, we study the equation □u=|u|qc+|∂u|p with small data, where qc is the critical power of Strauss conjecture and p≥qc. We obtain the optimal estimate of the lifespan ln⁡(Tε)≈ε−qc(qc−1) in n=3, and improve the lower bound of Tε from exp⁡(cε−(qc−1)) to exp⁡(cε−(qc−1)2/2) in n=2. Then, we study the Cauchy problem with small initial data for a system of semilinear wave equations □u=|v|q, □v=|∂tu|p in 3-dimensional space with q<2. We obtain that this system admits a global solution above a p−q curve for spherically symmetric data. On the contrary, we get a new region where the solution will blow up.