Sharp lower bound for the lifespan of systems of semilinear wave equations with multiple speeds

Abstract

We consider the Cauchy problem for a system of semilinear wave equations with multiple propagation speeds in three space dimensions. We assume that quadratic terms in its nonlinearity consist of products of unknowns whose speeds are different from each other. It is known that some classical solution to the above type of system may blow up in finite time. In this article, we get the sharp lower bound for the lifespan of classical solutions to this kind of system.