On the wave equation with a large rough potential

Abstract

International audienc

Decay estimates for the wave and Dirac equations with a magnetic potential

“…Several results are available for the equations i∂ t u − ∆u + V (x)u = 0, u + V (x)u = 0. We cite among the others [8], [15], [16], [19], [32] and the recent survey [33] for Schrö-dinger; and [5], [6], [10], [12], [13] for the wave equation. We must also mention the wave operator approach of Yajima (see [2], [39], [40], [41]) which permits to deal with the above equations in a unified way, although under nonoptimal assumptions on the potential in dimensions 1 and 3.…”

On some Schrödinger and wave equations with time dependent potentials

“…This is all technically justified by a work of Jensen and Nakamura, [8], (see also a more recent result Theorem 5.2, [1])…”

“…This is usually restricted to either time independent or very small time dependent potentials and in some cases are dimensionally restrictive. This was done (mostly in the physically important 3 D case) in the context of the Schrödinger equation in a series of papers by Rodnianski-Schlag [13], Goldberg-Schlag [6], and in the wave equation context by Georgiev-Visciglia [4], D'Ancona -Pierfelice [1], and others.…”

“…More precisely, for q, r ≥ 2, (q, r, n/2) = (2, ∞, 1), 2/q + n/r = n/2, there exists a constant C = C(q, r, n) 1 , so that…”

“…To fix the ideas, we will consider a linear Schrödinger equation of the form 1) and the wave equation…”