Lp-Lq Estimates for higher order perturbed hyperbolic equation

In this paper L ^p -L ^q estimates for the solution u(κ,t) to the following perturbed higher order hyperbolic equation are considered, We assume that the potential V(x) and the initial data f (x) are compactly supported, and V(x) is sufficiently small, then the solution u(x,t) of the above problem satisfies the same L ^p -L ^q estimates as that of the unperturbed problem.