Global Existence of Solutions for the Kawahara Equation in Sobolev Spaces of Negative Indices

The Cauchy problem for Kawahara equation in Sobolev spaces with low regularity

“…By using (39), Lemma 3.6 in [11], and To attain the desired result, we choose N such that L 5 T. As L ∼ CN The proof of Theorem 1.1 is completed.…”

“…In [9], the authors showed that (2), (3) have a local solution for u 0 in H s (R) with s>−1 and a global solution when u 0 ∈ L 2 . In [11], the authors proved that (2), (3) are locally solvable for u 0 in H s (R) with s − 7 5 and globally solvable for u 0 in H s (R) with s>− 1 2 . In [12], the authors considered (1), (3).…”

“…In [16], the authors established the local well-posedness result of (1), (3) for u 0 in H s (R) with s>− 7 4 . In this paper, based on the local well-posedness result of [16] and [11], by using I-method introduce in [1] and developed in [2] as well as the homogeneous Bourgain spaces that are used to prove some crucial bilinear estimates, we prove that (1), (3) are globally well-posed for u 0 in H s (R) with s>− 63 58 . Thus, we improve the result of [11], where the authors proved that the fifth-order shallow water wave equation is globally well-posed for the initial data in the Sobolev spaces H s (R) with s>− 1 2 .…”

Local well-posedness for the periodic higher order KdV type equations

“…LWP of these equations are studied for non-periodic case. For the known results of the non-periodic Kawahara equation, see [3,6,12,25] and the Eq. (1.3), see [20,23].…”

Odd-order quasilinear evolution equations posed on a bounded interval

“…Therefore, precise mathematical analysis of boundary value problems in bounded domains for general dispersive equations is welcome and attracts attention of specialists in the area of dispersive equations, especially KdV and BBM equations, [3,5,6,7,8,9,12,15,16,17,18,21,25,26,27,28,31,37,38]. Cauchy problem for dispersive equations of high orders was successfully explored by various authors, [2,10,11,15,23,33,36]. On the other hand, we know few published results on initialboundary value problems posed on a finite interval for general nonlinear odd-order dispersive equations, such as the Kawahara equation, see [13,14,29,20].…”