The estimate of solution operator for a kind of higher-order Schrödinger equations and its applications

Source abstract

We investigate the Lp−Lq estimates of the solution operator for higher-order Schrödinger equations of the form ut(t,x)=iP(D)u(t,x), where P(D) is a real degenerate elliptic partial differential operator. We use the global point-wise time-space estimates of the fundamental solutions for the above equations to obtain improved Lp−Lq estimates of the solution operator. As an application, we obtain local Strichartz estimates for the above equations and the unique-existence of solutions for higher-order Schrödinger equations with a time-dependent potential.