On the local and global existence of solutions to the nonlocal Whitham equation on half-line

Nakao Hayashi, Elena I. Kaikina

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We study the following initial-boundary value problem for the nonlocal Whitham equation ut+N(u)+Ku=0, (x,t) ∈R+×R+, u(x,0)=u~(x), x∈R+, where the nonlinearity is N(u)=uxu and K is the pseudodifferential operator on the half-line of order α satisfying 1<α<2 and some dissipative conditions. We prove that if the initial data are such that xδu~∈L1, with δ∈(0, 1/2 ) and the norm par/u~par/X+par/xδu~par/(L1) is sufficiently small, where X={psi/∈(L1), psi/'∈L1;par/psi/par/x=par/psi/par/(L1)+par/psi/xpar/(L1)<∞}, then there exists a unique solution u∈C ([0, +∞); L2)cap/C(R+, H1) of the initial-value problem (1), where Hk is the Sobolev space with norm par/φpar/(Hk)=par/(1-part/2x)k/2φpar/(L2). We also study large time asymptotics of the solutions.

Original languageEnglish
Title of host publicationInternational Seminar
Subtitle of host publicationDay on Diffraction - Proceedings
EditorsV.E. Grikuro, V.M. Babic, I.V. Androno, V.S. Buldyrev, A.P. Kiselev
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages77-86
Number of pages10
ISBN (Electronic)5799701569, 9785799701567
DOIs
Publication statusPublished - 1999 Jan 1
Externally publishedYes
EventInternational Seminar: Day on Diffraction, IS-DoD 1999 - St. Petersburg, Russian Federation
Duration: 1999 Jun 11999 Jun 3

Publication series

NameInternational Seminar: Day on Diffraction - Proceedings

Conference

ConferenceInternational Seminar: Day on Diffraction, IS-DoD 1999
CountryRussian Federation
CitySt. Petersburg
Period99/6/199/6/3

ASJC Scopus subject areas

  • Radiation
  • Acoustics and Ultrasonics
  • Atomic and Molecular Physics, and Optics

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