On the generalized ultra-hyperbolic heat kernel related to the spectrum

Abstract

In this paper, we study the equation ∂/∂t-u(x,t) = c2k u(x,t) with the initial condition u(x, 0) = f(x) for x ∈ ℝn - the n-dimensional Euclidean space. The operator k is named the ultra-hyperbolic operator iterated k -times, defined by k=(∂2/∂x12+ ⋯ + ∂2/∂xp2-∂2/ ∂xp+12-⋯-∂2/∂x p+q2 p + q = n is the dimension of the Euclidean space ℝn, u(x,t) is an unknown function for (x,t) = (x 1,...,xn,t) ∈ ℝn × (0,∞), f(x) 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 which is so called the generalized ultra-hyperbolic heat kernel. Moreover, such the generalized ultra-hyperbolic heat kernel has interesting properties and also related to the the kernel of an extension of the heat equation.