The Well‐Posedness of Solutions for a Generalized Shallow Water Wave Equation

Shaoyong Lai and Aiyin Wang

Abstract and Applied Analysis, 2012, vol. 2012, issue 1

Abstract: A nonlinear partial differential equation containing the famous Camassa‐Holm and Degasperis‐Procesi equations as special cases is investigated. The Kato theorem for abstract differential equations is applied to establish the local well‐posedness of solutions for the equation in the Sobolev space Hs(R) with s > 3/2. Although the H1‐norm of the solutions to the nonlinear model does not remain constant, the existence of its weak solutions in the lower‐order Sobolev space Hs with 1 ≤ s ≤ 3/2 is proved under the assumptions u0 ∈ Hs and ∥u0x∥L∞

Date: 2012
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https://doi.org/10.1155/2012/872187

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:2012:y:2012:i:1:n:872187

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