Let Lk= （-△）k ＋ Vk be a SchrSdinger type operator, where k ≥1 is a positive integer and V is a nonnegative polynomial. We obtain the Lp estimates for the operators △2kLk-1 and △kLk-1/2
In this paper Lp Lq estimates for the solution u(x,t) to the following perturbed higher order hyperbolic equation are considered,(οtt-aΔ)(οtt-bΔ)u+V(x)u=0,\ x∈Rn,n≥6, οjtu(x,0)=0,\ ο3tu(x,0)=f(x),\ (j=0,1,2).W...In this paper Lp Lq estimates for the solution u(x,t) to the following perturbed higher order hyperbolic equation are considered,(οtt-aΔ)(οtt-bΔ)u+V(x)u=0,\ x∈Rn,n≥6, οjtu(x,0)=0,\ ο3tu(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 Lp Lq estimates as that of the unperturbed problem. Received November 25,1996. Revised April 14,1997.1991 MR Subject Classification:35L05,35B20,35B45.展开更多
In this paper similarly to the second-order case, we give an elementary and straightforward proof of global Lp estimates for the initial-value parabolic problem of the bi-harmonic type. Moreover, we obtain the existen...In this paper similarly to the second-order case, we give an elementary and straightforward proof of global Lp estimates for the initial-value parabolic problem of the bi-harmonic type. Moreover, we obtain the existence and uniqueness of the solutions in the suitable space using the potential theory, Marcinkiewicz interpolation theorem and approximation argument. Meanwhile, by the same approach we can deal with the general polyharmonic cases.展开更多
基金Supported by the National Natural Science Foundation of China(10901018,11001002)the Beijing Foundation Program(201010009009,2010D005002000002)the Fundamental Research Funds for the Central Universities
文摘Let Lk= （-△）k ＋ Vk be a SchrSdinger type operator, where k ≥1 is a positive integer and V is a nonnegative polynomial. We obtain the Lp estimates for the operators △2kLk-1 and △kLk-1/2
文摘In this paper Lp Lq estimates for the solution u(x,t) to the following perturbed higher order hyperbolic equation are considered,(οtt-aΔ)(οtt-bΔ)u+V(x)u=0,\ x∈Rn,n≥6, οjtu(x,0)=0,\ ο3tu(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 Lp Lq estimates as that of the unperturbed problem. Received November 25,1996. Revised April 14,1997.1991 MR Subject Classification:35L05,35B20,35B45.
文摘In this paper similarly to the second-order case, we give an elementary and straightforward proof of global Lp estimates for the initial-value parabolic problem of the bi-harmonic type. Moreover, we obtain the existence and uniqueness of the solutions in the suitable space using the potential theory, Marcinkiewicz interpolation theorem and approximation argument. Meanwhile, by the same approach we can deal with the general polyharmonic cases.
