The operator ⊗ and its spectrum related to heat equation

Abstract

In this paper, we study the equation - ∂/∂t u(x, t) + c 2 ⊗ u(x, t) = 0 with the initial condition u(x,0)=f(x) for x ε ℝn -the n-dimensional Euclidean space. The operator is Equation presented where Equation presented p+q = n is the dimension of the Euclidean space ℝn, u(x, t) is an unknown function for (x,t) = (x1, x2,..., xn, t) ε ℝn × (0,∞), f(x) is the given generalized function and c is a positive constant. On the suitable conditions for f and u, we obtain the uniqueness solution of such equation. Moreover, if we put q = 0 we obtain the solution of heat equation - ∂/∂t u(x,t) + c2Δ3u(x,t) = 0. © 2009 Academic Publications.