On semilinear Tricomi equations with critical exponents or in two space dimensions

Abstract

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.

Nonexistence of Global Solutions to A Semilinear Wave Equation with Scale Invariant Damping

“…Step2: The lower bound of R(u). The practices of reference [27,23], let ω ∈ R n be a unit vector. The Radon transform of u with respect to the space variables is defined a…”

On the strauss index of semilinear tricomi equation

“…However, there was still a gap for the range of p and the value of p crit was unkonwn in [34]. Recently, He, Witt and Yin have shown in [12,13,14] that for m ∈ N and n ≥ 2 there exists a critical exponent p crit (m, n) > 1 such that weak solutions u generally blow up when 1 < p ≤ p crit (m, n), while there exists a global small-data weak solution u when p > p crit (m, n). Here, p crit (m, n) is the positive root of the quadratic equation Thirdly, we consider the semilinear Tricomi equation…”

“…Now let us turn back to the 1-D Cauchy problem for Tricomi equation (1.1). Since the local existence of weak solution u to (1.1) under minimal regularity assumptions has been established in [26] and [34], without loss of generality, as in [13,14], we focus on the global small-data weak solution problem to (1.1) starting at a fixed positive time. Moreover, in order to establish the global existence result, we need to apply the Strichartz estimates with a characteristic weight ((φ(t) + M ) 2 − |x| 2 ) γ , however, the characteristic cone for Tricomi operator is φ 2 (t) = |x| 2 , which admits a cusp singularity at t = 0, due to this difficulty, we can only establish inhomogeneous Strichartz estimates for the case t is away from zero.…”

“…The authors of [2,24,25,26] have established the local existence as well as the singularity structure of low regularity solutions to the Cauchy problem for semilinear Tricomi equations in the degenerate hyperbolic region and the elliptic-hyperbolic mixed region, respectively. We have additionally given a complete study of the global existence versus blowup problem for small-data solutions u to the semilinear Tricomi equation ∂ 2 t u − t∆u = |u| p for n ≥ 2 (see [12,13,14]). In addition, Lin and Tu in [17] gave the upper bound of lifespan when 1 < p ≤ p c .…”

Blow-up and lifespan estimate for the generalized Tricomi equation with mixed nonlinearities

“…However, the question of global existence for the system (1.2) with small data in the supercritical case has been proved only for m = 1 and for dimensions N = 1, 2; see e.g. [8,9,10,11]. Hence, the critical exponent for (1.2), that we denote by q C (N, m), should be given by the greatest root of the following quadratic equation…”