On the existence of global solutions of the Hartree equation for initial data in the modulation space \(M^{p , q}(\mathbb{R})\)
The author studies the Cauchy problem for Hartree type equations, defined as Schrödinger equations with nonlinear potential involving cubic powers and convolutions with a singular kernel, cf. \textit{P. Cao} and \textit{R. Carles} [Rev. Math. Phys. 23, No. 9, 933--967 (2011; Zbl 1239.35145)]. A global existence and uniqueness result is obtained in the frame of the modulation spaces of \textit{H. G. Feichtinger} and \textit{G. Narimani} [Appl. Comput. Harmon. Anal. 21, No. 3, 349--359 (2006; Zbl 1106.42005)], for large initial data. The proof is based on a splitting method, by decomposing the initial datum as a sum of a square integrable function and a function in modulation spaces. The paper gives evidence of the effectiveness of the use of modulation spaces in the study of nonlinear problems.
scientific article; zbMATH DE number 7483918
scientific article