Special functions arising in the study of semi-linear equations in circular domains

Abstract

Rayleigh functions are defined by the formula σ l ( ν ) = ∑ n = 1 ∞ 1 λ ν , n 2 l , where l = 1 , 2 , 3 , … ; λ ν , n ≠ 0 are zeros of the Bessel function J ν ( x ) and n = 1 , 2 , 3 , … , 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 the present paper it is shown that a new family of special functions, convolutions of Rayleigh functions with respect to the Bessel index, (1) R l ( m ) = ∑ p , k = - ∞ ; p + k = m ∞ ∑ q , s = 1 ∞ 1 λ p , q 2 l 1 λ k , s 2 l for l = 1 , 2 , … ; m = 0 , ± 1 , ± 2 , … , arises in constructing solutions of semi-linear evolution equations in circular domains (see also [V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413–424]). As an example of its application a forced Cahn-Hilliard equation is considered 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 for R 2 ( m ) is obtained convenient for computing its asymptotics as | m | → ∞ . The latter issue is important for establishing a function space to which a solution of the corresponding problem belongs.