科研之友
微信
新浪微博
Facebook
分享链接
摘要
In this paper, we investigate the initial value problem (IVP henceforth) associated with the generalized Kawahara equation [Z.Y. Zhang, J.H. Huang, Z.H. Liu and M.B. Sun, On the unique continuation property for the modified Kawahara equation, Adv Math (China).45(2016),pp.80-88) as follows: @@@ {partial derivative(t)u + alpha partial derivative(5)(x)u +beta partial derivative(3)(x)u + gamma partial derivative(x)u + mu partial derivative(x)(u(k)) = 0, x is an element of R, t >= 0, @@@ u(x,0) = u(0) (x) @@@ with initial data in the Sobolev space H-s (R). Benefited from ideas of [Z.Y. Zhang and J.H. Huang, Well-posedness and unique continuation property for the generalized Ostrovsky equation with low regularity, Math Meth Appl Sci. 39(2016),pp.2488-2513; Z.Y. Zhang, J.H. Huang, Z.H. Liu and M.B. Sun, Almost conservation laws and global rough solutions of the defocusing nonlinear wave equation on R-2; Acta Math Sci.37(2017),pp.385C39], first, we show that the local well-posedness is established for the initial data u(0) is an element of H-s(R) with s >= -7/4(k = 2) and s >= -1/4(k = 3) respectively. Then,using these results and conservation laws, we also prove that the IVP is globally well-posed for the initial data u(0) is an element of H-s(R) with s = 0(k = 2, 3). Finally, benefited from ideas of [Z.Y. Zhang and J.H. Huang, Well-posedness and unique continuation property for the generalized Ostrovsky equation with low regularity, Math Meth Appl Sci. 39(2016),pp.2488-2513; Z.Y. Zhang, J.H. Huang, Z.H. Liu and M.B. Sun, On the unique continuation property for the modified Kawahara equation,Adv Math (China).45(2016),pp.80-88], i.e. using complex variables technique and Paley-Wiener theorem, we prove the unique continuation property (UCP henceforth) for the IVP.
