Abstract
The Kato theorem for abstract differential equations is applied to establish the local well‐posedness of the strong solution for a nonlinear generalized Novikov equation in space C([0, T), Hs(R))∩C1([0, T), Hs−1(R)) with s > (3/2). The existence of weak solutions for the equation in lower‐order Sobolev space Hs(R) with 1 ≤ s ≤ (3/2) is acquired.
Highlights
The Novikov equation with cubic nonlinearities takes the form ut − utxx 4u2ux 3uuxuxx u2uxxx, 1.1 which was derived by Vladimir Novikov in a symmetry classification of nonlocal partial differential equations 1
The ideas of proving the second result come from those presented in Li and Olver 8
For any real number s, Hs Hs R denotes the Sobolev space with the norm defined by h Hs
Summary
The Novikov equation with cubic nonlinearities takes the form ut − utxx 4u2ux 3uuxuxx u2uxxx, 1.1 which was derived by Vladimir Novikov in a symmetry classification of nonlocal partial differential equations 1. Ni and Zhou 11 proved that the Novikov equation associated with initial value is locally well-posedness in Sobolev space Hs with s > 3/2 by using the abstract Kato theorem. Two results about the persistence properties of the strong solution for 1.1 were established It is shown in 12 that the local well-posedness for the periodic Cauchy problem of the Novikov equation in Sobolev space Hs with s > 5/2. By using the Kato theorem for abstract differential equations, we establish the local existence and uniqueness of solutions for the 1.2 with any β and arbitrary positive integer N in space C 0, T , Hs R C1 0, T , Hs−1 R with s > 3/2. The ideas of proving the second result come from those presented in Li and Olver 8
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