  In the present paper, we study the Cauchy problem for the weakly coupled
system of the generalized Tricomi equations with multiple propagation speeds.
Our aim of this paper is to prove a small data blow-up result and an upper
estimate of lifespan of the problem for a suitable compactly supported initial
data in the subcritical and critical cases of the Strauss type. The proof is
based on the framework of the argument in the paper [17]. One of our new
contributions is to construct two families of special solutions to the free
equation (see (2.16) or (2.18) as the test functions and prove their several
properties. We emphasize that the system with two different propagation speeds
is treated in this paper and the assumption on the initial data is improved
from the point-wise positivity to the integral positivity.
