Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 76, pp. 1-9.
Title: Solutions of a partial differential equation related to the
oplus operator
Author: Wanchak Satsanit (Maejo Univ., Chiang Mai, Thailand)
Abstract:
In this article, we consider the equation
$$
\oplus^ku(x)=\sum^{m}_{r=0}c_{r}\oplus^{r}\delta
$$
where $\oplus^k$ is the operator iterated k times and
defined by
$$
\oplus^k=\Big(\Big(\sum^p_{i=1}\frac{\partial^2}{\partial
x^2_i}\Big)^{4}-\Big(\sum^{p+q}_{j=p+1}\frac{\partial^2}{\partial
x^2_j}\Big)^{4}\Big)^k,
$$
where $p+q=n$, $x=(x_1,x_2,\dots,x_n)$ is in
the n-dimensional Euclidian space $\mathbb{R}^n$,
$c_{r}$ is a constant, $\delta$ is the
Dirac-delta distribution, $\oplus^{0}\delta=\delta$, and
$k=0,1,2,3,\dots$. It is shown that, depending on the
relationship between k and m, the solution to this equation
can be ordinary functions, tempered distributions,
or singular distributions.
Submitted April 8, 2010. Published June 08, 2010.
Math Subject Classifications: 46F10, 46F12.
Key Words: Ultra-hyperbolic kernel; diamond operator;
tempered distribution.