Abstract
A nonlinear generalization of the Degasperis-Procesi equation is investigated. The well-posedness of entropy weak solutions for the Cauchy problem of the equation is established in the space L 1 (R)∩ L ∞ (R).MSC:35G25, 35L05.
Highlights
1 Introduction The objective of this work is to study the well-posedness in the space L (R) ∩ L∞(R) for the generalized Degasperis-Procesi equation ut – utxx + muux = uxuxx + uuxxx, (t, x) ∈ R+ × R, ( )
Dullin et al [ ] proved that the Degasperis-Procesi equation can be obtained from the shallow water elevation equation by an appropriate Kodama transformation
Coclite and Karlsen [ ] established the existence, uniqueness and L (R) stability of entropy weak solutions belonging to the class L (R) ∩ BV (R) for Eq ( )
Summary
Coclite and Karlsen [ ] established the existence, uniqueness and L (R) stability of entropy weak solutions belonging to the class L (R) ∩ BV (R) for Eq ( ). They obtained the existence of at least one weak solution satisfying a restricted set of entropy inequalities in the space L (R) ∩ L (R). In Coclite and Karlsen [ ], the well-posedness of entropy weak solution is investigated in the space L (R) ∩ L∞(R). Motivated by the desire to extend the weak solution results presented in Coclite and Karlsen [ ], we consider Eq ( ) with its Cauchy problem in the form
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