Electron. J. Diff. Equ., Vol. 2011(2011), No. 04, pp. 1-09.

Solution to the triharmonic heat equation

Wanchak Satsanit

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 7, 2011.
Math Subject Classifications: 46F10, 46F12.
Key Words: Fourier transform; tempered distribution; diamond operator.

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Wanchak Satsanit
Department of Mathematics, Faculty of Science
Maejo University, Chiangmai, 50290, Thailand
email: aunphue@live.com

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