L~p-L~q estimates for higher order perturbed hyperbolic equation

摘要

In this paper L~p-L~q estimated for the solution u(x, t)to the following perturbed high- er order hyperbolic equation are considered. (α_tt-aΔ)(α_tt-bΔ)u+V(x)u = 0, x∈R~n, n≥6, α~j_tu(x,0)=0, α~3_tu(x, 0)=f(x), (j = 0,1,2). 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 unpreturbed problem.