On the operator ⊗<inf>B</inf>^kRelated to the Bessel heat equation

Abstract

In this article, we study the equation ∂/∂t u(x, t) = c 2⊗Bk u(x, t) with the initial condition u(x, 0) = f(x) for x ∈ ℝn+, where the operator ⊗Bk is deflned by ⊗Bk = [(Bx1 +⋯+ Bxp)3 - (BXp+1 +⋯+ BXp+q)3]k, p + q = n is the dimension of the space ℝn+, Bxi = ∂2/∂xi2 + 2vi/x i ∂/ ∂xi, 2vi = 2αi + 1, αi > -1/2, xi > 0, i = 1, 2,... ,n, u(x,t) is an unknown function for (x,t) = (x1, x2,..., xn,t) ∈ ℝn+ × (0, ∞), f(x) is a given 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 kernel. Moreover, such Bessel heat kernel has interesting properties and also related to the kernel of an extension of the heat equation. © 2009 Academic Publications.