On some Schrödinger and wave equations with time dependent potentials

Abstract

Abstract. The existence and uniqueness of solutions in the initial value problem for Schröding-er and wave equations in the presence of a (large) time dependent potential is studied. The usual Strichartz estimates for such linear evolutions are shown to hold true with optimal assumptions on the potentials. As a byproduct, one obtains a counterexample to the two dimensional double endpoint inhomogeneous Strichartz estimate.

On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications

“…We obtain the following well-posedness for (1.14) withḢ γ initial data f below L 2 and potentials V ∈ L r t L w x under the scaling-critical range α/r + n/w = α. The Cauchy problem (1.14) was studied in [7,19] particularly when α = 2 and γ = 0.…”

On the bilinear control of the Gross-Pitaevskii equation

“…Concerning the study of the well-posedness of Schrödinger equations with potentials, we refer to [11,17,8].…”

Space–time estimates for the wave equations with singular potentials

“…Existence of the solution was first established in [24] and recently considered in [14,17]. The main differences are the choice of the space and the regularity of the potentials as well as the initial data.…”