TY - GEN

T1 - On the local and global existence of solutions to the nonlocal Whitham equation on half-line

AU - Hayashi, Nakao

AU - Kaikina, Elena I.

PY - 1999/1/1

Y1 - 1999/1/1

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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U2 - 10.1109/DD.1999.816186

DO - 10.1109/DD.1999.816186

M3 - Conference contribution

AN - SCOPUS:85041477023

T3 - International Seminar: Day on Diffraction - Proceedings

SP - 77

EP - 86

BT - International Seminar

A2 - Grikuro, V.E.

A2 - Babic, V.M.

A2 - Androno, I.V.

A2 - Buldyrev, V.S.

A2 - Kiselev, A.P.

PB - Institute of Electrical and Electronics Engineers Inc.

T2 - International Seminar: Day on Diffraction, IS-DoD 1999

Y2 - 1 June 1999 through 3 June 1999

ER -