Global solutions and self-similar solutions of the coupled system of semilinear wave equations in three space dimensions

Abstract

In this paper, we treat the coupled system of wave equations whose nonlinearities are $|u|^{p_j}|v|^{q_j}$ and propagation speeds may be different from each other. We study the lower bounds of $p_j$ and $q_j$ to assure the global existence of a class of small amplitude solutions which includes self-similar solutions. The exponent of self-similar solutions plays crucial role to find the lower bounds. Moreover, we prove that the discrepancy of propagation speeds allow us to bring them down. Conversely, if such conditions for the global existence do not hold, then no self-similar solution exists even for small initial data.