Critical Curve for p-q Systems of Nonlinear Wave Equations in Three Space Dimensions

Abstract

The existence of the critical curve for p-q systems for nonlinear wave equations was already established by D. Del Santo, V. Georgiev, and E. Mitidieri [1997, Global existence of the solutions and formation of singularities for a class of hyperbolic systems, in``Geometric Optics and Related Topics'' (F. Colombini and N. Lerner, Eds.), Progress in Nonlinear Differential Equations and Their Applications, Vol. 32, pp. 117 139, Birkha user, Basel] except for the critical case. Our main purpose is to prove a blow-up theorem for which the nonlinearity ( p, q) is just on the critical curve in three space dimensions. Moreover, the lower and upper bounds of the lifespan of solutions are precisely estimated, including the sub-critical case. 2000Academic Press

Critical Blowup for Systems of Semilinear Wave Equations in Low Space Dimensions

“…This paper gives an affirmative answer for the question. After we proved Theorem 1.1 in this paper, we learned from Professor Takamura that Agemi, Kurokawa, and Takamura [18] obtained a similar result to Theorem 1.1 by another approach (see Remark 1.6 in [18]). They also obtained the lower bounds of the life span of classical solutions for (1.1)-(1.4) for the case when n = 3 and 2 I P I q, which show the sharpness of the upper bounds (1.10) and (1.11) in Theorem 1.1.…”

Nonexistence of Global Solutions for a Weakly Coupled System of Semilinear Damped Wave Equations of Derivative Type in the Scattering Case

“…the cases G 1 (v, ∂ t v) = |v| p , G 2 (u, ∂ t u) = |u| q and G 1 (v, ∂ t v) = |∂ t v| p , G 2 (u, ∂ t u) = |∂ t u| q have been studied in [5,3,4,2,23,22,8,24] and in [6,52,21,16], respectively. While in the case of power nonlinearities (that is, for G 1 (v, ∂ t v) = |v| p , G 2 (u, ∂ t u) = |u| q ) the critical curve is given by…”

Existence and Asymptotic Behavior of Radially Symmetric Solutions to a Semilinear Hyperbolic System in Odd Space Dimensions*

“…The aim here is to show the global existence of a small solution to the Cauchy problem by assuming only the condition (3.2), i.e., Γ > 0, where Γ is defined by (1.3) with p * = p − 2 and q * = q − 2. This condition is optimal for the global existence, since if Γ ≤ 0, then solutions of the Cauchy problem generically blow up in finite time even though the initial data are small (see [1,[3][4][5][6]15] for the case c 1 = c 2 and [16] for the case c 1 = c 2 ).…”

“…Since solutions of the Cauchy problem generically blow up in finite time if Γ < 0 even though the initial data are small enough (see [3,4,6,7]), and the blow-up occurs also when Γ = 0 and either n = 2 or n = 3 (see [1,5,15,16]), we need to assume Γ > 0 for the global existence. While, it is an open problem whether (1.6) is an optimal condition to prove the existence result or not.…”