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.
For 1-D semilinear Tricomi equation ∂ 2 t u − t∂ 2 x u = |u| p with initial data (u(0, x), ∂ t u(0, x)) = (u 0 (x), u 1 (x)), where t ≥ 0, x ∈ R, p > 1, and u i ∈ C ∞ 0 (R) (i = 0, 1), we shall prove that there exists a critical exponent p crit = 5 such that the small data weak solution u exists globally when p > p crit ; on the other hand, the weak solution u, in general, blows up in finite time when 1 < p < p crit . We specially point out that for 1-D semilinear wave equation, the weak solution v will generally blow up in finite time for any p > 1. By this paper and [9]-[11], we have given a systematic study on the blowup or global existence of small data solution u to the equation ∂ 2 t u − t∆u = |u| p for all space dimensions. One of the main ingredients in the paper is to establish a crucial weighted Strichartz-type inequality for 1-D linear degenerate equation ∂ 2 t w − t∂ 2 x w = F (t, x) with (w(0, x), ∂ t w(0, x)) = (0, 0), i.e., an inequality with the weight ( 4 9 t 3 − |x| 2 ) α between the solution w and the function F is derived for some real numbers α.
In our previous papers, we have given a systematic study on the global existence versus blowup problem for the small-data solution u of the multi-dimensional semilinear Tricomi equation ∂ 2 t u − t ∆u = |u| p , u(0, •), ∂tu(0, •) = (u 0 , u 1), where t > 0, x ∈ R n , n ≥ 2, p > 1, and u i ∈ C ∞ 0 (R n) (i = 0, 1). In this article, we deal with the remaining 1-D problem, for which the stationary phase method for multi-dimensional case fails to work and the large time decay rate of u(t, •) L ∞ x (R) is not enough. The main ingredient of the proof in this paper is to use the structure of the linear equation to get the suitable decay rate of u in t, then the crucial weighted Strichartz estimates are established and the global existence of solution u is proved when p > 5.
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