### On the Operator $\circledast^{k}$ Related to Heat Equation

Wanchak Satsanit, Amnuay Kananthai

#### Abstract

In this paper, we study the equation
$$\frac{\partial}{\partial t}\,u(x,t)+c^2(-\circledast)^{k} u(x,t)=0$$ with the initial condition
$$u(x,0)=f(x)$$
for $x\in\mathbb{R}^n$-the $n$-dimensional Euclidean space. The
operator $(\circledast)^{k}$ is operator iterated $k$ times ,
defined by
\begin{eqnarray*}
\circledast^{k}&=&\left(\left(\sum^{p}_{i=1}\frac{\partial^2}{\partial
x^2_i}\right)^{3}
+\left(\sum^{p+q}_{j=p+1}\frac{\partial^2}{\partial
x^2_j}\right)^{3}\right)^{k}\\
\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 , $k$ is a positive integer
and $c$ is a positive constant. Moreover, if we put $q=0$ and
$k=1$we obtain the solution of equation.
$$\frac{\partial}{\partial t}\,u(x,t)-c^2\triangle^3u(x,t)=0$$ Which is related to the
triharmonic heat equation.\\

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DOI: https://doi.org/10.5539/jmr.v2n2p20