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

Rayleigh functions σl(ν)σl(ν) are defined as series in inverse powers of the Bessel function zeros λν,n≠0λν,n≠0,σl(ν)=∑n=1∞1λν,n2l, where l=1,2,…; ν   is the index of the Bessel function Jν(x)Jν(x) and n=1,2,…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)R1(m) were investigated. In the present work a general representation of Rl(m)Rl(m) in terms of σl(ν)σl(ν) is deduced. On the basis of this a representation for the function R2(m)R2(m) is obtained in terms of the ψ  -function. An asymptotic expansion is computed for R2(m)R2(m) as |m|→∞|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 Rl(m)Rl(m) a forced Boussinesq equationutt−2bΔut=−αΔ2u+Δu+βΔ(u2)+futt−2bΔut=−αΔ2u+Δu+βΔ(u2)+f with α,b=const>0α,b=const>0 and β=const∈Rβ=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)R1(m) and R2(m)R2(m) which are responsible for the nonlinear smoothing effect.