Convolutions of Rayleigh functions and their application to semi-linear equations in circular domains
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.
scientific article; zbMATH DE number 5083363
scientific article