Convolutions of Rayleigh functions and their application to semi-linear equations in circular domains

Abstract

Rayleigh functions σ l ( ν ) are defined as series in inverse powers of the Bessel function zeros λ ν , n ≠ 0 , σ l ( ν ) = ∑ n = 1 ∞ 1 λ ν , n 2 l , where l = 1 , 2 , … ; ν is the index of the Bessel function J ν ( x ) and n = 1 , 2 , … is the number of the zeros. Convolutions of Rayleigh functions with respect to the Bessel index, R l ( m ) = ∑ k = − ∞ ∞ σ l ( | m − k | ) σ l ( | k | ) for l = 1 , 2 , … ; m = 0 , ± 1 , ± 2 , … , are needed for constructing global-in-time solutions of semi-linear evolution equations in circular domains [V. Varlamov, On the spatially two-dimensional Boussinesq equation in a circular domain, Nonlinear Anal. 46 (2001) 699–725; V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413–424]. The study of this new family of special functions was initiated in [V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413–424], where the properties of R 1 ( m ) were investigated. In the present work a general representation of R l ( m ) in terms of σ l ( ν ) is deduced. On the basis of this a representation for the function R 2 ( m ) is obtained in terms of the ψ-function. An asymptotic expansion is computed for R 2 ( m ) as | m | → ∞ . Such asymptotics are needed for establishing function spaces for solutions of semi-linear equations in bounded domains with periodicity conditions in one coordinate. As an example of application of R l ( m ) a forced Boussinesq equation u t t − 2 b Δ u t = − α Δ 2 u + Δ u + β Δ ( u 2 ) + f with α , b = const > 0 and β = const ∈ R is considered in a unit disc with homogeneous boundary and initial data. Construction of its global-in-time solutions involves the use of the functions R 1 ( m ) and R 2 ( m ) which are responsible for the nonlinear smoothing effect.