Small data blow-up for the weakly coupled system of the generalized Tricomi equations with multiple propagation speeds

Abstract

In the present paper, we study the Cauchy problem for the weakly coupled system of the generalized Tricomi equations with multiple propagation speed. 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 test function method developed in the paper [16]. One of our new contributions is to construct two families of special solutions to the free equation (see (2.16) or (2.18)) and prove their several properties. We also give an improvement of the previous results [10, Theorem 1.1], [12, Theorem 1.1] and [25, Theorems 1.2 and 1.3] about the single case, that is, remove the point-wise positivity of the initial data assumed previously.