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

Abstract

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 $u(x,t)\in \mathbb{R}^n\times [0,\infty)$, $\mathbb{R}^n$ 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}\right)^3 -\left(\sum^{p+q}_{j=p+1}\frac{\partial^2}{\partial x^2_j}\right)^3\right]^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

$\frac{\partial^2}{\partial t^2}u(x,t)+c^2(\triangle)^3u(x,t)=0$

which is related to the triharmonic wave equation.