Special functions arising in the study of semi-linear equations in circular domains
Rayleigh functions are defined by the formula \(\sigma_{l}(\nu)=\sum_{n=1}^{\infty}\frac{1}{\lambda_{\nu,n}^{2l}}\), where \(l=1,2,\ldots\) and \(\lambda_{\nu,n}\neq 0\) are zeros of the Bessel function \(J_{\nu}(x)\) and \(n=1,2,\ldots\)is the number of the zero. These functions appear in the classical problems of vibrating circular membranes, heat conduction in cylinders and diffraction through circular apertures. In this paper the author studies a new family of special functions, namely, convolutions of Rayleigh functions with respect to the Bessel index, \[ R_{l}(m)=\sum_{p,k=-\infty;\;p+k=m}^{\infty}\,\sum_{q,s=1}^{\infty}\frac{1}{\lambda_{p,q}^{2l}}\,\frac{1}{\lambda_{k,s}^{2l}} \] for \(l=1,2,\ldots ;m=0,\pm1,\pm2,\ldots\) appearing in constructing solutions of semi-linear evolution equations in circular domains. The author considers as an example a forced Cahn-Hilliard equation in a unit disc with homogeneous boundary and initial conditions. Construction of its global in time solutions involves the use of \(R_{1}(m)\) and \(R_{2}(m)\). A general representation of \(R_{l}(m)\) is deduced and on the basis of that a particular result of \(R_{2}(m)\) is obtained convenient for computing its asymptotics as \(| m| \rightarrow \infty\). The latter issue is important for establishing a function space to which the solution of the corresponding problem belongs.
scientific article; zbMATH DE number 5142283
scientific article