Nonlinear nonlocal Whitham equation on a segment

✍ Scribed by Elena I. Kaikina

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 301 KB
• Volume: 59
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2004.07.002

✦ Synopsis

We study global existence and large time asymptotic behavior of solutions to the initial-boundary value problem for the nonlinear nonlocal equation on a segment

where the pseudodifferential operator Ku on a segment [0, a] is defined by Ku=

C is chosen such that ReK(p) > 0 for Rep = 0. We prove that if the initial data u 0 ∈ L ∞ and u 0 L ∞ < ε, then there exists a unique solution u ∈ C([0, ∞); L 2 (0, a)) of the initial-boundary value problem (0.1). Moreover, there exists a constant A such that the solution has the following large time asymptotics

uniformly with respect to the spatial variable x ∈ (0, a), where = e -i /2 cos /2 i +i∞ 0 e -K(z) dz.