ON THE GENERALIZED BESSEL HEAT EQUATION RELATED TO THE GENERALIZED BESSEL DIAMOND OPERATOR

Abstract

In this article, we study the equation∂∂tu(x, t) = c2 ⊗m,kB u(x, t)with the initial condition u(x, 0) = f(x) for x ∈ R+n . Here the operator⊗m,kB is called the Generalized Bessel Diamond Operator, iterated ktimes, and is defined by⊗m,kB = Bx1 + Bx2 + · · · + Bxpm−Bxp+1 + · · · + Bxp+qmk,where k and m are positive integers, p + q = n, Bxi =∂2∂x2i+2vixi∂∂xi,2vi = 2αi + 1, αi > −12, xi > 0, i = 1, 2, . . . , n, n being the dimensionof the space R+n , u(x, t) is an unknown function of the form (x, t) =(x1, . . . , xn, t) ∈ R+n ×(0, ∞), f(x) is a given generalized function and ca constant. We obtain the solution of this equation, which is related tothe spectrum and the kernel, the so called generalized Bessel diamondheat kernel. Moreover, the generalized Bessel diamond heat kernel isshown to have interesting properties and to be related to the kernel ofan extension of the heat equatio