IJPAM: Volume 54, No. 1 (2009)

THE OPERATOR $\otimes$ AND ITS SPECTRUM
RELATED TO HEAT EQUATION

Wanchak Satsanit$^1$, Amnuay Kananthai$^2$
$^{1,2}$Department of Mathematics
Chiangmai University
Chiangmai, 50200, THAILAND
$^2$e-mail: malamnka@science.cmu.ac.th


Abstract.In this paper, we study the equation

\begin{displaymath}\frac{\partial}{\partial t}\,u(x,t)+c^2\otimes u(x,t)=0 \end{displaymath}

with the initial condition

\begin{displaymath}u(x,0)=f(x)\end{displaymath}

for $x\in\mathbb{R}^n$ -- the $n$-dimensional Euclidean space. The operator is
\begin{multline*} \otimes = \left(\sum^{p}_{i=1}\frac{\partial^2}{\partial x^... ...^2\Bigg] =\frac{3}{4}\diamondsuit\triangle+\frac{1}{4}\Box^3\,, \end{multline*}
where

\begin{eqnarray*} \triangle &=& \frac{\partial^2}{\partial x^2_1}+\frac{\part... ... x_{p+2}^2}-\cdots-\frac{\partial^2}{\partial x_{p+q}^2}\,,\\ \end{eqnarray*}

\begin{eqnarray*} \diamondsuit &=& \left(\frac{\partial^2}{\partial x_1^2}+ \... ...+2}^2}+\cdots+\frac{\partial^2}{\partial x_{p+q}^2}\right)^2\,, \end{eqnarray*}

$p+q=n$ is the dimension of the Euclidean space $\mathbb{R}^n$, $u(x,t)$ is an unknown function for $(x,t)=(x_1,x_2,\ldots,x_n,t)\in \mathbb{R}^n\times (0,\infty)$, $f(x)$ 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

\begin{displaymath}\frac{\partial}{\partial t}\, u(x,t)+c^2\triangle^3 u(x,t)=0.\end{displaymath}



Received: June 18, 2009

AMS Subject Classification: 42-XX

Key Words and Phrases: Fourier transform, tempered distribution, diamond operator

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 54
Issue: 1