Convolutions of Rayleigh functions and their application to semi-linear equations in circular domains
type: Item
instance of: scholarly article
title: Convolutions of Rayleigh functions and their application to semi-linear equations in circular domains
zbMATH Open document ID: 1107.33013
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-01-09T00:00:00Z
review text: Convolutions of Reyleigh functions \(\sigma_l(\nu)\) with respect to the Bessel index \[ R_l(m)=\sum_{k=-\infty}^\infty\sigma_l(| m-k| )\sigma_l(| k| ), \quad l\in\mathbb N,\;m\in\mathbb Z, \] where \(\sigma_l(\nu)= \sum_{n=1}^\infty1/\lambda_{\nu,n}^{2l}\) and \(\lambda_{\nu,n}\not=0\) are the Bessel functions zeros are considered. The properties of \(R_1(m)\) were investigated earlier by the author in [\textit{V.~Varlamov}, J. Math. Anal. Appl. 306, No. 2, 413--424 (2005; Zbl 1068.35115)]. General representation of special function \(R_l(m)\) in terms of the Reyleigh sums \(\sigma_i(k)\) is given in Theorem 2. On the basis of this result in Theorem 3 the function \(R_2(m)\) is computed and its asymptotics for \(| m| \to\infty\) is obtained. As an application of the functions \(R_1(m)\) and \(R_2(m)\) the initial-boundary value problem in \(\mathbb R^+\times\{(x,y)| x^2+ y^2<1\}\) for forced Boussinesq equation \(u_{tt}-2b\Delta u_t=-\alpha \Delta^2 u +\Delta u+\beta \Delta(u^2)+f\), where \(\alpha,b=\text{const}>0\) and \(\beta\in\mathbb R\) is considered. In Theorem 1 the existence and uniqueness conditions for a mild solution of the above problem are given.
reviewed by: Valery V. Karachik
Mathematics Subject Classification ID: 33C67
Mathematics Subject Classification ID: 33C10
Mathematics Subject Classification ID: 35L75
Mathematics Subject Classification ID: 35Q72
zbMATH DE Number: 5083363
zbMATH Keywords: convolution
zbMATH Keywords: Reyleigh function
zbMATH Keywords: Boussinesq equation
zbMATH Keywords: Bessel function
zbMATH Keywords: asymptotic
zbMATH Keywords: eigenfunction
author: V. V. Varlamov
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.05.016
OpenAlex ID: W2001060905
cites work: The zeta function constructed from the zeros of the Bessel function
cites work: Q4658073
cites work: Riccati equations and convolution formulae for functions of Rayleigh type
cites work: Sum rules for zeros of Bessel functions and an application to spherical Aharonov-Bohm quantum bags
cites work: The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types
cites work: The Rayleigh Function
cites work: Instability and Nonexistence of Global Solutions to Nonlinear Wave Equations of the Form Pu tt = -Au + ℱ(u)
cites work: Q5628582
cites work: Q3231359
cites work: Convolution formulae for functions of Rayleigh type
cites work: Q4065978
cites work: Q3028909
cites work: Dynamics of evolutionary equations
cites work: Series associated with the zeta and related functions
cites work: On the damped Boussinesq equation in a circle
cites work: On the spatially two-dimensional Boussinesq equation in a circular domain
cites work: Convolution of Rayleigh functions with respect to the Bessel index
cites work: Q5562752
DOI: 10.1016/J.JMAA.2006.05.016
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description: scientific article; zbMATH DE number 5083363