Small data scattering for semi-relativistic equations with Hartree type nonlinearity
We prove that the initial value problem for the equation i partial derivative tu + root m2-Delta u = (e-mu broken vertical bar x broken vertical bar/broken vertical bar x broken vertical bar* broken vertical bar u broken vertical bar) u in R1+3, m >= 0, mu 0 > 0 is globally well-posed and the solution scatters to free waves asymptotically as t oo if we start with initial data which is small in FP (R3) for s> and if m > 0. Moreover, if the initial data is radially symmetric we can improve the above result to m > 0 and s > 0, which is almost optimal, in the sense that L-2(R-3) is the critical space for the equation. The main ingredients in the proof are certain endpoint Strichartz estimates, L-2(R1+3) bilinear estimates for free waves and an application of the UP and VP function spaces. (C) 2015 Elsevier Inc. All rights reserved.
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5510-5532
5510-5532
Elsevier