Global dynamics of the generalized fifth-order KdV equation with quintic nonlinearity
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Journal of Evolution Equations
Global dynamics of the generalized fifth-order KdV equation with quintic nonlinearity Yuexun Wang
Abstract. We prove global existence and scattering for the solutions of the generalized fifth-order KdV equation with quintic nonlinearity for small and localized initial data. The proof uses the space-time resonance method and the stationary phase argument.
1. Introduction We consider the following generalized fifth-order KdV equation with quintic nonlinearity: u t = ∂x5 u + αu 4 u x ,
(1.1)
where u is a real function which maps R × R to R, and α = 1 (defocusing case) or α = −1 (focusing case). Equation (1.1) is a member of the general fifth-order KdV equations: u t = α1 ∂x5 u + α2 ∂x3 u + ∂x g(u, ∂x u, ∂x2 u), α1 , α2 ∈ R, α1 = 0, which contains in particular some models of plasma waves and capillary–gravity waves and the fifth order equation in the KdV integrable hierarchy. (See the introduction of [2,12,13] for a useful survey.) It is standard to show that the Cauchy problem of (1.1) is well posed in C [−T, T ]; H s (R) (s > 23 ) for a short time T > 0, for instance, one may refer to [1,19]. Our aim in the present work is to study the global existence and scattering for the solutions of (1.1) with small and localized initial data, in the framework of the space-time resonance method [4,11] and the stationary phase argument [11]. The main ingredient is to study the evolutionary equation of the profile of the solutions in Fourier space which is unfolded by a careful stationary phase analysis based on an adaptation of the argument of [5]. Since we only focus on small solutions, the sign of α will not matter and will be taken to be ‘−1’ in the rest of the paper. Our main result can be stated precisely as follows: Mathematics Subject Classification: 76B15, 76B03, 35S30, 35A20 Keywords: Global existence, Scattering, Quintic nonlinearity.
J. Evol. Equ.
Y. Wang
Theorem 1.1. Given the initial data u 0 as u(x, 0) = u 0 (x).
(1.2)
u 0 H 2 (R) + xu 0 L 2 (R) ≤ ε0 ≤ ε,
(1.3)
Assume that u 0 satisfies
for some constant ε sufficiently small. Then, the Cauchy problem (1.1)–(1.2) admits 2 a unique global solution u ∈ C R; H (R) satisfying the decay estimates for t ≥ 1 and x ∈ R β |∂x |β u(x, t) ε0 t −(β+1)/5 x/t 1/5 − 38 + 4 , β ∈ [0, 3]. (1.4) Moreover, the solution has the following asymptotics as t → +∞: (Decaying region) When x ≥ t 1/5 , we have the decay estimate |u(x, t)| ε0 t −1/5 (x/t 1/5 )−7/8 . 1
(Self-similar region) When |x| ≤ t 5 +4γ , with γ = approximately self-similar :
1 1 5 10
1
(1.5) 4 − Cε05 , the solution is
7γ
|u(x, t) − t −1/5 Q(x/t 1/5 )| ε0 t − 5 − 2 ,
(1.6)
where Q is a bounded solution of the nonlinear ordinary differential equation Q (4) − 5−1 x Q − Q 5 = 0
(1.7)
Q L ∞ (R) ε0 .
(1.8)
with 1 5 +4γ
(Oscillatory region) When x ≤ −t , the leading order asymptotic behavior is linear: There exists f ∞ ∈ L ∞ (R) with f ∞ L ∞ (R) ε0 such that u(x, t) − 1 exp −4itξ 5 + iπ + i | f ∞ (ξ0 )|4 f ∞ (ξ0
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