The local strong and weak solutions to a generalized Novikov equation
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RESEARCH
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The local strong and weak solutions to a generalized Novikov equation Shaoyong Lai* and Meng Wu *
Correspondence: [email protected] Department of Mathematics, Southwestern University of Finance and Economics, Chengdu, 610074, China
Abstract A nonlinear partial differential equation, which includes the Novikov equation as a special case, is investigated. The well-posedness of local strong solutions for the equation in the Sobolev space Hs (R) with s > 32 is established. Although the H1 -norm of the solutions to the nonlinear model does not remain constant, the existence of its local weak solutions in the lower order Sobolev space Hs (R) with 1 ≤ s ≤ 32 is established under the assumptions u0 ∈ Hs and u0x L∞ < ∞. MSC: 35Q35; 35Q51 Keywords: local strong solution; local weak solution; generalized Novikov equation
1 Introduction Novikov [] derived the integrable equation with cubic nonlinearities ut – utxx + u ux = uux uxx + u uxxx ,
()
which has been investigated by many scholars. Grayshan [] studied both the periodic and the non-periodic Cauchy problem for Eq. () and discussed continuity results for the data-to-solution map in the Sobolev spaces. A Galerkin-type approximation method was used in Himonas and Holliman’s paper [] to establish the well-posedness of Eq. () in the Sobolev space H s (R) with s > on both the line and the circle. Hone et al. [] applied the scattering theory to find non-smooth explicit soliton solutions with multiple peaks for Eq. (). This multiple peak property is common with the Camassa-Holm and Degasperis-Procesi equations (see [–]). A matrix Lax pair for Eq. () was acquired in [, ] and was shown to be related to a negative flow in the Sawada-Kotera hierarchy. Sufficient conditions on the initial data to guarantee the formation of singularities in finite time for Eq. () were given in Jiang and Li []. Mi and Mu [] obtained many dynamic results for a modified Novikov equation with a peak solution. It is shown in Ni and Zhou [] that the Novikov equation associated with the initial value is locally well-posed in Sobolev space H s with s > by using the abstract Kato theorem. Two results about the persistence properties of the strong solution for Eq. () are established in []. Tiglay [] proved the local well-posedness for the periodic Cauchy problem of the Novikov equation in Sobolev space H s (R) with s > . The orbit invariants are used to show the existence of a periodic global strong solution if the Sobolev index s ≥ and a sign condition holds. For analytic initial data, the existence and uniqueness of analytic solutions for Eq. () are obtained in []. Using the Littlewood-Paley decomposition and nonhomogeneous Besov © 2013 Lai and Wu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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