The initial-boundary value problem for the Kawahara equation on the half-line
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Nonlinear Differential Equations and Applications NoDEA
The initial-boundary value problem for the Kawahara equation on the half-line M´arcio Cavalcante and Chulkwang Kwak Abstract. This paper concerns the initial-boundary value problem of the Kawahara equation posed on the right and left half-lines. We prove the local well-posedness in the low regularity Sobolev space. We introduce the Duhamel boundary forcing operator, which is introduced by Colliander and Kenig (Commun Partial Differ Equ 27:2187–2266, 2002) in the context of Airy group operators, to construct solutions on the whole line. We also give the bilinear estimate in X s,b space for b < 12 , which is almost sharp compared to IVP of Kawahara equation (Chen et al. in J Anal Math 107:221–238, 2009; Jia and Huo in J Differ Equ 246:2448–2467, 2009). Mathematics Subject Classification. 35Q53, 35G31. Keywords. Kawahara equation, Initial-boundary value problem, Local well-posedness.
1. Introduction In this paper, we consider the following Kawahara equation1 : ∂t u − ∂x5 u + ∂x (u2 ) = 0.
(1.1)
The Kawahara equation was first proposed by Kawahara [23] describing solitarywave propagation in media. Also, the Kawahara equation can be described in the theory of magneto-acoustic sound waves in plasma and in theory of shallow water waves with surface tension. For another physical background of Kawahara equation or in view of perturbed equation of KdV equation, see [4,18,35] and references therein. 1 It
is well-known of the form ∂t u + au∂x u + b∂x3 u − ∂x5 u = 0 for arbitrary constants a, b ∈ R. We, however, use the form (1.1) for the simplicity, since the dominant dispersion term is the fifth-order term and constants do not affect our analysis. 0123456789().: V,-vol
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M. Cavalcante, C. Kwak
NoDEA
Kawahara equation (1.1), posed on the whole line R, admits following three conservation laws: M [u(t)] := u(t, x) dx = M [u(0)], 1 (1.2) E[u(t)] := u2 (t, x) dx = E[u(0)] 2 and 1 H[u(t)] := 2
(∂x2 u)2 (t, x)
1 dx − 3
u3 (t, x) dx = H[u(0)].
(1.3)
Moreover, (1.3) allow us to represent (1.1) as the Hamiltonian equation: ut = ∂x ∇u H (u (t)) , where ∇u is the variational derivative with respect to u, but not a completely integrable system in contrast to the KdV equation. 1.1. Well-posedness results on R The Cauchy problem for the Kawahara equation on R has been extensively studied, and here we only give some of previous works. The local well-posedness of Kawahara equation was first established by Cui and Tao [13]. They proved the Strichartz estimate for the fifth-order operator and obtained the local wellposedness in H s (R) s > 1/4 as its application, which implies, in addition to the energy conservation law (1.3), H 2 (R) global well-posedness. Later, Cui, Deng and Tao [9] improved the previous result to the negative regularity Sobolev space H s (R), s > −1, and Wang, Cui and Deng [42] further improved to the lower regularity s ≥ −7/5. In both paper, authors used Fourier restriction norm method, while more delicate analysis has been performed in t
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