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Local well-posedness and persistence properties for a model containing both Camassa-Holm and Novikov equation

Shouming Zhou1*, Chun Wu1 and Baoshuai Zhang2

Author Affiliations

1 College of Mathematics Science, Chongqing Normal University, Chongqing, 401331, P.R. China

2 School of Economics Management, Chongqing Normal University, Chongqing, 401331, P.R. China

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Boundary Value Problems 2014, 2014:9  doi:10.1186/1687-2770-2014-9

Published: 8 January 2014

Abstract

This paper deals with the Cauchy problem for a generalized Camassa-Holm equation with high-order nonlinearities,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M3">View MathML</a>. This equation is a generalization of the famous equation of Camassa-Holm and the Novikov equation. The local well-posedness of strong solutions for this equation in Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M4">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M5">View MathML</a> is obtained, and persistence properties of the strong solutions are studied. Furthermore, under appropriate hypotheses, the existence of its weak solutions in low order Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M4">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M7">View MathML</a> is established.

Keywords:
persistence properties; local well-posedness; weak solution