Copyright © 2010 Wanchak Satsanit. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the equation (∂/∂t)u(x,t)=c2⊕Bku(x,t) with the initial condition u(x,0)=f(x) for x∈Rn+. The operator ⊕Bk is the operator iterated k-times and is defined by ⊕Bk=((∑i=1pBxi)4-(∑j=p+1p+qBxi)4)k, where p+q=n is the dimension of the Rn+, Bxi=∂2/∂xi2+(2vi/xi)(∂/∂xi), 2vi=2αi+1, αi>-1/2, i=1,2,3,…,n, and k is a nonnegative integer, u(x,t) is an unknown function for (x,t)=(x1,x2,…,xn,t)∈Rn+×(0,∞), f(x) is a given generalized function, 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 Bessel heat kernel. Moreover, such Bessel heat kernel has interesting properties and also related to the kernel of an extension of the heat equation.