Electron. J. Diff. Equ., Vol. 2010(2010), No. 76, pp. 1-9.

Solutions of a partial differential equation related to the oplus operator

Wanchak Satsanit

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 8, 2010.
Math Subject Classifications: 46F10, 46F12.
Key Words: Ultra-hyperbolic kernel; diamond operator; tempered distribution.

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

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