On the operator bk related to the bessel heat equation

Abstract

In this paper, we study the equation ∂/∂t u(x, t) = c2Bk u(x, t) with the initial condition u(x, 0) = f(x) for x ∈ ℝn+, where the operator Bk is defined by Bk = [(Bx1 + ⋯ B xp)3 + (Bxp+1 + ⋯ + B xp-q)3]k, p+q = n is the dimension of the space ℝ2+ = {x = (x1, x2,. . . xn) : x1 > 0, x2 > 0, . . . ,xn > 0}, Bx1 = ∂2/∂xi2 + 2v i/xi ∂/∂xi, 2vi = 2αi + 1, αi > -1/2, i = 1, 2,..., n, u(x, t) is an unknown function for (x, t) = [x1, x2,...,x n, t) ∈ ℝn+×(0, ∞), f(x) is a generalized function, k is a positive integer and c is a positive constant. We obtain the solution of such equation which is related to the spectrum and the heat kernel. Moreover, such Bessel heat kernel has interesting properties and also related to the kernel of an extension of the heat equation. © 2010 Academic Publications.