Philip Brenner

OnLp−Lp′ estimates for the wave-equation

The purpose of this note is to present a short proof of some of the results announced by Littman [4], and in particular to obtain a fundamental Lp-Lp , estimate for a class of evolution equations with constant coefficients, in particular containing the stimate for the wave-equation due to Strichartz [-7, 8]. The method used here is particularly apt to deal with approximations to the wave-equation, such as those obtained by finite element or finite difference methods. Details of such applications will appear elsewhere.

On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations

Asymptotic properties of solutions of the nonlinear Klein-Gordon equation ∂t2u − Δu + m2u + f(u) = 0 (NLKG) u¦0 = θ, ∂tu¦0 = Ψ, are investigated, which are inherited from the corresponding solutions v of the (linear) Klein-Gordon equation ∂t2v − Δv + m2v = 0v¦0 = θ, ∂tv¦0 = Ψ, (KG) In particular, the finiteness of time-integrals in Lq over R+ of certain Sobolevnorms in space of the solution is proved to be such a hereditary property. Together with a device by W. A. Strauss and a weak decay result for the (KG) due to R. S. Strichartz, this is used to prove that under suitable restrictions on the nonlinearity, the scattering operator for the (NLKG) is defined on all of L21 × L2 for n = 3.