 05185 Enno Lenzmann
 WellPosedness for SemiRelativistic Hartree Equations of Critical Type
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May 24, 05

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Abstract. We prove local and global wellposedness for semirelativistic, nonlinear Schr\"odinger equations $i \partial_t u = \sqrt{\Delta + m^2} u + F(u)$ with initial data in $H^s(\mathbb{R}^3)$, $s \geq 1/2$. Here $F(u)$ is a critical Hartree nonlinearity that corresponds to Coulomb or Yukawa type selfinteractions. For focusing $F(u)$, which arise in the quantum theory of boson stars, we derive a sufficient condition for globalintime existence in terms of a solitary wave ground state. Our proof of wellposedness does not rely on Strichartz type estimates, and it enables us to add external potentials of a general class.
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