Well-Posedness of a Class of Radial Inhomogeneous Hartree Equations

Saleh Almuthaybiri, Radhia Ghanmi and Tarek Saanouni ()
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Saleh Almuthaybiri: Department of Mathematics, College of Science and Arts in Uglat Asugour, Qassim University, Buraydah 51452, Saudi Arabia
Radhia Ghanmi: LR03ES04 Partial Differential Equations and Applications, Faculty of Sciences of Tunis, University of Tunis El Manar, 2092 Tunis, Tunisia
Tarek Saanouni: Department of Mathematics, College of Science and Arts in Uglat Asugour, Qassim University, Buraydah 51452, Saudi Arabia

Mathematics, 2023, vol. 11, issue 23, 1-25

Abstract: The present paper investigates the following inhomogeneous generalized Hartree equation i u ˙ + Δ u = ± | u | p − 2 | x | b ( I α ∗ | u | p | · | b ) u , where the wave function is u : = u ( t , x ) : R × R N → C , with N ≥ 2 . In addition, the exponent b > 0 gives an unbounded inhomogeneous term | x | b and I α ≈ | · | − ( N − α ) denotes the Riesz-potential for certain 0 < α < N . In this work, our aim is to establish the local existence of solutions in some radial Sobolev spaces, as well as the global existence for small data and the decay of energy sub-critical defocusing global solutions. Our results complement the recent work (Sharp threshold of global well-posedness versus finite time blow-up for a class of inhomogeneous Choquard equations, J. Math. Phys. 60 (2019), 081514). The main challenge in this work is to overcome the singularity of the unbounded inhomogeneous term | x | b for certain b > 0 .

Keywords: inhomogeneous Hartree problem; nonlinear equations; global/non-global existence; decay (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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