In this paper, we are concerned with the global Cauchy problem for the semilinear generalized Tricomi equationWe show that there exists a critical exponent p crit (m, n) > 1 such that the solution u, in general, blows up in finite time when 1 < p < p crit (m, n). We further show that there exists a conformal exponent p conf (m, n) > p crit (m, n) such that the solution u exists globally when p > p conf (m, n) provided that the initial data is small enough. In case p crit (m, n) < p ≤ p conf (m, n), we will establish global existence of small data solutions u in a subsequent paper [13].

This paper is a complement of our recent works on the semilinear Tricomi equations in [8] and [9]. For the semilinear Tricomi equation ∂ 2 t u−t∆u = |u| p with initial data (u(0, ·), ∂ t u(0, ·)) = (u 0 , u 1 ), where t ≥ 0, x ∈ R n (n ≥ 3), p > 1, and u i ∈ C ∞ 0 (R n ) (i = 0, 1), we have shown in [8] and [9] that there exists a critical exponent p crit (n) > 1 such that the solution u, in general, blows up in finite time when 1 < p < p crit (n), and there is a global small solution for p > p crit (n). In the present paper, firstly, we prove that the solution of ∂ 2 t u − t∆u = |u| p will generally blow up for the critical exponent p = p crit (n) and n ≥ 2, secondly, we establish the global existence of small data solution to ∂ 2 t u − t∆u = |u| p for p > p crit (n) and n = 2. Thus, we have given a systematic study on the blowup or global existence of small data solution u to the equationRemark 1.1. For the semilinear wave equation ∂ 2 t u − ∆u = |u| p (p > 1), the critical exponent p 0 (n) in Strauss' conjecture (see [26]) is determined by the algebraic equation (n−1)p 2 0 (n)−(n+1)p 0 (n)− 2 = 0 (so far the global existence of small data solution u for p > p 0 (n) or the blowup of solution u for 1 < p < p 0 (n) have been proved in [4]-[6], [12]-[13], [23] and the references therein). The finite time blowup for the critical wave equations ∂ 2 t u − ∆u = |u| p 0 (n) has been established in [4], [12], [22], and [31]-[32], respectively. Motivated by the techniques in [31] and [8], we prove the blowup result for the critical semilinear Tricomi equation in (1.1). Remark 1.2. For brevity, we only study the semilinear Tricomi equation instead of the generalized semilinear Tricomi equation ∂ 2 t u − t m ∆u = |u| p (m ∈ N) in problem (1.1). In fact, by the methods in Theorem 1.1 and Theorem 1.2, one can establish the analogous results to Theorem 1.1-Theorem 1.2 for the generalized semilinear Tricomi equation. Remark 1.3. It follows from a direct computation that p crit (2) = 3+ √ 33 4and p conf (2) = 3 in Theorem 1.2.

In this paper, we are concerned with the local existence and singularity structure of low regularity solutions to the semilinear generalized Tricomi equation ∂ 2 t u − t m ∆u = f (t, x, u) with typical discontinuous initial data (u(0, x), ∂ t u(0, x)) = (0, ϕ(x)); here m ∈ N, x = (x 1 , ..., x n ), n ≥ 2, and f (t, x, u) is C ∞ smooth in its arguments. When the initial data ϕ(x) is a homogeneous function of degree zero or a piecewise smooth function singular along the hyperplane {t = x 1 = 0}, it is shown that the local solutionand is C ∞ away from the forward cuspidal cone Γ 0 = (t, x): t > 0, |x| 2 = 4t m + 2 , respectively. On the other hand, for n = 2 and piecewise smooth initial data ϕ(x) singular along the two straight lines {t = x 1 = 0} and {t = x 2 = 0}, we establish the local existence of a solution2 )) in general due to the degenerate character of the equation under study; here Γ. This is an essential difference to the well-known result for

SummaryEdge Sobolev spaces are proposed as a main new tool for the investigation of weakly hyperbolic equations. The well-posedness of the linear and the semilinear Cauchy problem in the class of such edge Sobolev spaces is proved. Applications to the propagation of singularities for solutions to semilinear problems are considered.

We prove the local existence and uniqueness of minimal regularity solutions u of the semilinear generalized Tricomi equationunder the assumption that |F (u)| |u| κ and |F ′ (u)| |u| κ−1 for some κ > 1. Our results improve previous results of M. Beals [2] and of ourselves [15][16][17]. We establish Strichartz-type estimates for the linear generalized Tricomi operator ∂ 2 t − t m ∆ from which the semilinear results are derived.