On semilinear Tricomi equations in one space dimension

Abstract

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.

Blow-up results for semilinear damped wave equations in Einstein-de Sitter spacetime

“…On the other hand, in the undamped case µ = 0 (that is, for the semilinear wave equation with speed of propagation t −k ) the exponent max{p 0 (k, n), p 1 (k, n)} is consistent with the result for the generalized semilinear Tricomi equation (i.e., the semilinear wave equation with speed of propagation t ℓ , where ℓ > 0) obtained in the recent works [13,14,15,21].…”

Lifespan of semilinear generalized Tricomi equation with Strauss type exponent

“…They have shown that the solution to problem (1) would blow up in finite time in the sub-critical case 1 < p < p crit (m, n) in [7], and the small data solution would exist globally in the super-critical case p > p crit (m, n) in [8]. The results of low dimension case n = 1, 2 were given in [10,9]. In particular, they also showed the blow-up result for the critical case for m = 1 in [9], by applying the test function method and the Riccati-type ordinary differential inequality.…”

“…Proof. (i) Since z = −2λφ(t) < 0, we shall apply (10) for M (α, γ; z) when |z| is large. For small |z|, we only need to require the integral is convergent around 0, i.e, q − α + 1 > 0 and q + α > 0.…”

Blow-up and lifespan estimate for generalized Tricomi equations related to Glassey conjecture

“…Yagdjian [37] obtained some partial results, in the sense that there was still a gap between the blow-up and global existence ranges. The critical power is finally established in a recent series of works by He, Witt and Yin [9][10][11][12] (see also the Doctoral dissertation by He [8]). For k ≥ 1, p c (k, n) admits the following form:…”