# On a partial differential equation related to the diamond Bessel Klein-Gordon operator

Sudprathai Bupasiri and Krailikhit Latpala – GJM, Volume 8, Issue 1 (2023), 58-68.

In this paper, we consider the equation $\diamondsuit_{B,m} ^{k}u(x)=\sum_{r=0}^{t}c_{r}\diamondsuit_{B,m}^r \delta$ where $\diamondsuit_{B,m}^{k}$ is the operator iterated $k$-times and is defined by $\diamondsuit_{B,m}^{k}=\left( \left(\left(\sum_{i=1}^{p} B_{x_i}\right)^2+\frac{m^2}{2}\right)^2 - \left(\left( \sum_{j=p+1}^{p+q} B_{x_j}\right)^2 -\frac{m^2}{2} \right)^2\right )^k,$ where $p+q=n, x=(x_{1},\ldots , x_{n})\in \mathbb{R}^{+}_n, B_{x_{i}}=\frac{\partial ^{2}}{\partial x_{i}^{2}}+ \frac{2v_{i}}{x_{i}}\frac{\partial }{\partial x_{i}}, v_{i}=2\alpha _{i}+1, \alpha _{i}>-\frac{1}{2}\;\;, x_{i}>0, i=1,2,\ldots,n, c_{r}$ is a constant, $k$ is a nonnegative integer, $\delta$ is the Dirac-delta distribution, $\diamondsuit_{B,m} ^{0}\delta =\delta$ and $n$ is the dimension of $\mathbb{R}^{+}_n$. It is shown that, depending on the relationship between $k$ and $t$, the solution to this equation can be ordinary functions, tempered distributions, or singular distributions.

Categories: 2023, Issue1