SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

The local strong and weak solutions to a generalized Novikov equation

Shaoyong Lai* and Meng Wu

Author Affiliations

Department of Mathematics, Southwestern University of Finance and Economics, Chengdu, 610074, China

For all author emails, please log on.

Boundary Value Problems 2013, 2013:134  doi:10.1186/1687-2770-2013-134

Published: 20 May 2013

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M1">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M2">View MathML</a> is established. Although the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M3">View MathML</a>-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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M1">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M5">View MathML</a> is established under the assumptions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/134/mathml/M7">View MathML</a>.

MSC: 35Q35, 35Q51.

Keywords:
local strong solution; local weak solution; generalized Novikov equation