Generalized heat kernel related to the operator Lkm and spectrum

Abstract

In this paper, we study the equation with the initial condition u(x, 0) = f(x) for x ∈ R{double-struck}n, where the operator Lkm is defined by, p + q = n is the dimension of the space R{double-struck}n, u(x, t) is an unknown function for (x, t) = (x1, x2,...,xn, t) ∈ R{double-struck}n × (0,∞), f(x) is a given generalized function, k and m are 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 the kernel has interesting properties and also related to the kernel of an extension of the heat equation.