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

Nakao Hayashi; Elena I. Kaikina

doi:10.1109/DD.1999.816186

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

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Abstract

We study the following initial-boundary value problem for the nonlocal Whitham equation u_t+N(u)+Ku=0, (x,t) ∈R⁺×R⁺, u(x,0)=u~(x), x∈R⁺, where the nonlinearity is N(u)=u_xu 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 x^δu~∈L¹, with δ∈(0, 1/2 ) and the norm par/u~par/X⁺par/x^δu~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 H^k is the Sobolev space with norm par/φpar/(H^k)=par/(1-part/₂^x)k/2φpar/(L²). We also study large time asymptotics of the solutions.

Original language	English
Title of host publication	International Seminar
Subtitle of host publication	Day on Diffraction - Proceedings
Editors	V.E. Grikuro, V.M. Babic, I.V. Androno, V.S. Buldyrev, A.P. Kiselev
Publisher	Institute of Electrical and Electronics Engineers Inc.
Pages	77-86
Number of pages	10
ISBN (Electronic)	5799701569, 9785799701567
DOIs	https://doi.org/10.1109/DD.1999.816186
Publication status	Published - 1999 Jan 1
Externally published	Yes
Event	International Seminar: Day on Diffraction, IS-DoD 1999 - St. Petersburg, Russian Federation Duration: 1999 Jun 1 → 1999 Jun 3

Publication series

Name	International Seminar: Day on Diffraction - Proceedings

Conference

Conference	International Seminar: Day on Diffraction, IS-DoD 1999
Country/Territory	Russian Federation
City	St. Petersburg
Period	99/6/1 → 99/6/3

ASJC Scopus subject areas

Radiation
Acoustics and Ultrasonics
Atomic and Molecular Physics, and Optics

Access to Document

10.1109/DD.1999.816186

Cite this

Hayashi, N., & Kaikina, E. I. (1999). On the local and global existence of solutions to the nonlocal Whitham equation on half-line. In V. E. Grikuro, V. M. Babic, I. V. Androno, V. S. Buldyrev, & A. P. Kiselev (Eds.), International Seminar: Day on Diffraction - Proceedings (pp. 77-86). [816186] (International Seminar: Day on Diffraction - Proceedings). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/DD.1999.816186

On the local and global existence of solutions to the nonlocal Whitham equation on half-line. / Hayashi, Nakao; Kaikina, Elena I.

International Seminar: Day on Diffraction - Proceedings. ed. / V.E. Grikuro; V.M. Babic; I.V. Androno; V.S. Buldyrev; A.P. Kiselev. Institute of Electrical and Electronics Engineers Inc., 1999. p. 77-86 816186 (International Seminar: Day on Diffraction - Proceedings).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Hayashi, N & Kaikina, EI 1999, On the local and global existence of solutions to the nonlocal Whitham equation on half-line. in VE Grikuro, VM Babic, IV Androno, VS Buldyrev & AP Kiselev (eds), International Seminar: Day on Diffraction - Proceedings., 816186, International Seminar: Day on Diffraction - Proceedings, Institute of Electrical and Electronics Engineers Inc., pp. 77-86, International Seminar: Day on Diffraction, IS-DoD 1999, St. Petersburg, Russian Federation, 99/6/1. https://doi.org/10.1109/DD.1999.816186

Hayashi N, Kaikina EI. On the local and global existence of solutions to the nonlocal Whitham equation on half-line. In Grikuro VE, Babic VM, Androno IV, Buldyrev VS, Kiselev AP, editors, International Seminar: Day on Diffraction - Proceedings. Institute of Electrical and Electronics Engineers Inc. 1999. p. 77-86. 816186. (International Seminar: Day on Diffraction - Proceedings). doi: 10.1109/DD.1999.816186

Hayashi, Nakao ; Kaikina, Elena I. / On the local and global existence of solutions to the nonlocal Whitham equation on half-line. International Seminar: Day on Diffraction - Proceedings. editor / V.E. Grikuro ; V.M. Babic ; I.V. Androno ; V.S. Buldyrev ; A.P. Kiselev. Institute of Electrical and Electronics Engineers Inc., 1999. pp. 77-86 (International Seminar: Day on Diffraction - Proceedings).

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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.",

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N2 - 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.

AB - 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.

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

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