Nonlinear nonlocal Whitham equation on a segment
type: Item
instance of: scholarly article
title: Nonlinear nonlocal Whitham equation on a segment
zbMATH Open document ID: 1064.45008
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2004-11-22T00:00:00Z
review text: The author studies global existence and asymptotic behavior of solutions to the initial-boundary value problem for the nonlinear nonlocal Whitham equation on a segment \[ \begin{cases} u_t+uu_x+{K}u=0, &t>0,\;x\in(0,a),\\ u(x,0)=u_0(x), &x\in(0,a),\\ u(a,t)=0, &t>0, \end{cases} \tag{1} \] where the pseudodifferential operator \({K}u\) is defined on a segment \([0,a]\) by \[ {K}u= \frac{1}{2\pi i}\int_{-i\infty}^{i\infty}e^{px}K(p) \times\left(\widehat{u}(p,t)-\frac{u(0,t)-e^{-pa}\,u(a,t)}{p}\right)\,dp, \] where \(K(p)=C_\alpha p^\alpha\), \(\alpha\in(1,3/2)\) and \(C_\alpha\) is chosen such that \(\text{Re}\,K(p)>0\) for \(\text{Re}\,p=0\). She proves that, for initial data \(u_0\in {L}^\infty\) and \(\| u_0\| _{L^\infty}<\varepsilon\), the initial-boundary value problem ({1}) has a unique solution in \({C}([0,\infty);{L}^2(0,a))\). The solution has the following large time asymptotics \[ u(x,t)=At^{-1/\alpha}\Lambda+O(t^{-(1+\delta)/\alpha}), \] for some \(A\) and uniformly with respect to the spatial variable \(x\in(0,a)\), where \[ \Lambda=\frac{e^{-i\pi\alpha/2}\cos\pi\alpha/2}{\pi i} \int_0^{+i\infty} e^{-K(z)}\,dz. \]
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 45M05
zbMATH DE Number: 2117562
zbMATH Keywords: initial-boundary value problem
zbMATH Keywords: nonlinear nonlocal Whitham equation
zbMATH Keywords: pseudodifferential operator
zbMATH Keywords: global existence
zbMATH Keywords: large time asymptotic behavior
zbMATH Keywords: dissipative nonlinear evolution equation
author: Elena I. Kaikina
reviewed by: S. Z. Németh
MaRDI profile type: Publication
full work available at URL: j.na.2004.07.002
OpenAlex ID: W1970158995
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DOI: 10.1016/J.NA.2004.07.002
description: scientific article; zbMATH DE number 2117562