The Generalized $otimes^{k}$ operator related to Triharmonic Wave Equation

In this paper, we study the generalized wave equation of the form $frac{partial^2}{partial t^2}u(x,t)+c^2(otimes)^ku(x,t)=0$ with the initial conditions $u(x,0)=f(x),~~ frac{partial}{partial t}u(x,0)=g(x)$ where $u(x,t)in mathbb{R}^n imes [0,infty)$, $mathbb{R}^n$ is the $n$-dimensional Euclidean space, $otimes^k$ is the operator iterated $k$-times defined by $otimes^k= displaystyle left[left(sum^{p}_{i=1}frac{partial^2}{partial x^2_i} ight)^3 -left(sum^{p+q}_{j=p+1}frac{partial^2}{partial x^2_j} ight)^3 ight]^k $ $c$ is a positive constant, $k$ is a nonnegative integer, $f$ and $g$ are continuous and absolutely integrable functions. We obtain $u(x,t)$ as a solution for such equation. Moreover, by $epsilon$-approximation we also obtain the asymptotic solution $u(x,t)=O(epsilon^{-n/3k})$. In particularly, if we put $k=1$ and $q=0$, the $u(x,t)$ reduces to the solution of the wave equation $frac{partial^2}{partial t^2}u(x,t)+c^2( riangle)^3u(x,t)=0$ which is related to the triharmonic wave equation.