Journal of Mathematical Physics, Analysis, Geometry 2021, vol. 17, No 3, pp. 295-325   https://doi.org/10.15407/mag17.03.295     ( to contents , go back )

### A Note on a Damped Focusing Inhomogeneous Choquard Equation

Department of Mathematics, College of Science and Arts in Uglat Asugour, Qassim University, Buraydah, Kingdom of Saudia Arabia

Preparatory Institute for Engineering Studies of Elmanar, University Campus, BP 244 CP 2092, Elmanar 2, Tunis, Tunisia
E-mail: L.CHERGUI@qu.edu.sa

Received August 27, 2020, revised November 5, 2020.

Abstract

This paper is devoted to the focusing inhomogeneous Choquard equation with linear damping:
$$i\dot{u}+\triangle u+ia u=-|x|^{-\gamma}(I_{\alpha}\ast|u|^p)|u|^{p-2}u \quad \text{on} \ \mathbb R^N,$$
where $a\geq 0$ and $0<\gamma<\inf(N,2+\alpha)$. Global existence and scattering are proved for sufficiently large damping. For arbitrary damping, global existence of solutions is obtained if the initial data belong to some invariant sets.

Mathematics Subject Classification 2010: 35Q55
Key words: damped Choquard equation, large damping, global existence, scattering, invariant sets