Electronic Journal of Differential Equations,
Vol. 2011(2011), No. 04, pp. 1-09.

Title: Solution to the triharmonic heat equation

Author: Wanchak Satsanit (Maejo  Univ., Chiangmai, Thailand)

Abstract:
 In this article, we study the equation
 $$
 \frac{\partial}{\partial t}\,u(x,t)-c^2\circledast u(x,t)=0
 $$
 with initial condition $u(x,0)=f(x)$. Where
 x is in the Euclidean space $\mathbb{R}^n$,
 $$
 \circledast=\Big(\sum^p_{i=1}\frac{\partial^2}{\partial x^2_i}\Big)^3
 +\Big(\sum^{p+q}_{j=p+1}\frac{\partial^2}{\partial x^2_j}\Big)^3
 $$
 with $p+q=n$, $u(x,t)$ is an unknown function,
 $(x,t)=(x_1,x_2,\dots,x_n,t)\in \mathbb{R}^n\times (0,\infty)$,
 $f(x)$ is a generalized function, and $c$ is a positive
 constant.
 Under suitable conditions on f and u, we obtain a
 unique solution. Note that for $q=0$, we
 have the triharmonic  heat equation
 $$
 \frac{\partial}{\partial t} u(x,t)-c^2\Delta^3 u(x,t)=0\,.
 $$

Submitted June 14, 2010. Published January 07, 2011.
Math Subject Classifications: 46F10, 46F12.
Key Words: Fourier transform; tempered distribution; diamond operator.