Asymptotic behaviour of nonlinear dispersive models with variable coefficients
type: Item
instance of: scholarly article
title: Asymptotic behaviour of nonlinear dispersive models with variable coefficients
zbMATH Open document ID: 0863.35028
DOI: 10.1007/BF01759262
author: Vanilde Bisognin
author: G. Perla Menzala
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1997-06-03T00:00:00Z
review text: The paper is concerned with the asymptotic behavior of solutions of the nonlinear dispersive equation \[ Mu_t+a(t)u_t+b(t)u^pu_x=0\quad\text{for}\quad (x,t)\in\mathbb{R}\times\mathbb{R}^+\quad\text{and}\quad u(x,0)=\varphi(x),\tag{\(*\)} \] where \(M\) is a Fourier multiplier operator defined as \(\widehat{Mf}(y)=m(y)\widehat f(y)\), and \(p\) is a positive integer. The authors were motivated by the decay estimate of solutions of the generalized Benjamin-Bona-Mahony equation: \(u_t+u_x+u^pu_x-u_{xxt}=0\) in [\textit{J. Albert}, J. Math. Anal. Appl. 141, No. 2, 527-537 (1989; Zbl 0697.35116)], and derived an analogous one of solutions \(u(x,t)\) of \((*)\) with sufficiently small initial data \(\varphi(x)\) under suitable assumptions on \(m(y)\), \(a(t)\), \(b(t)\) and \(p\). Roughly, the obtained estimate is of the type \[ |u(\cdot,t)|_{L^\infty}=O\Biggl(\Biggl(1+\int^t_0 a(s)ds\Biggr)^{-\alpha}\Biggr)\quad\text{as}\quad t\to+\infty, \] where \(\int^\infty_0f(t)dt\) is assumed to be positively infinite and \(\alpha\) is a positive constant related to constants in the assumptions.
Mathematics Subject Classification ID: 35F25
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 878450
zbMATH Keywords: weakly nonlinear dispersive long wave
zbMATH Keywords: Fourier multiplier operator
zbMATH Keywords: decay estimate
MaRDI profile type: Publication
cites work: Dispersion of low-energy waves for the generalized Benjamin-Bona-Mahony equation
cites work: On the decay of solutions of the generalized Benjamin-Bona-Mahony equation
cites work: Model equations for long waves in nonlinear dispersive systems
cites work: Decay rates of the solutions of nonlinear dispersive equations
cites work: Q5670314
cites work: Instability of a class of dispersive solitary waves
cites work: Dispersion of low-energy waves for two conservative equations
cites work: Decay and asymptotics for \(\square u = F(u)\)
full work available at URL: https://doi.org/10.1007/bf01759262
OpenAlex ID: W2088003944
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description: scientific article; zbMATH DE number 878450