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

Rayleigh functions σl(ν) are defined as series in inverse powers of the Bessel function zeros λν,n = 0, σl(ν) = ∞ n=1 1 λ2l ν,n , where l = 1, 2, . . . ; ν is the index of the Bessel function Jν(x) and n = 1, 2, . . . is the number of the zeros. Convolutions of Rayleigh functions with respect to the Bessel index, Rl(m) = ∞ k=−∞ σl |m− k| σl |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 R1(m) were investigated. In the present work a general representation of Rl(m) in terms of σl(ν) is deduced. On the basis of this a representation for the function R2(m) is obtained in terms of the ψ-function. An asymptotic expansion is computed for R2(m) as |m|→∞. Such asymptotics are needed for establishing function spaces for solutions of semi-linear equations in boundeddomains with periodicity conditions in one coordinate. As an example of application of Rl(m) a forced Boussinesq equation utt −2bΔut =−αΔ2u +Δu+βΔ u2 + 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 R1(m) and R2(m) which are responsible for the nonlinear smoothing effect. © 2006 Elsevier Inc. All rights reserved.