A NOTE ON THE INHOMOGENEOUS HARTREE EQUATIONS WITH POTENTIAL

Salah Mahmoud Boulaaras () and Tarek Saanouni
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Salah Mahmoud Boulaaras: Department of Mathematics, College of Science, Qassim University, Buraydah, Kingdom of Saudi Arabia
Tarek Saanouni: Department of Mathematics, College of Science, Qassim University, Buraydah, Kingdom of Saudi Arabia

FRACTALS (fractals), 2025, vol. 33, issue 08, 1-16

Abstract: We consider the focusing non-homogeneous Hartree equation with potential i∂tv −ℋVu = −|v|p−2 1 |x|η 𠒥α ∗|v|p 1 |⋅|η u,ℋV = −Δ + V.Here, the time-space variable is (t,x) ∈ ℠× ℠3, and the potential V satisfies certain conditions that ensure the linear Schrödinger operator ℋV has dispersive and Strichartz estimates. The exponent η > 0 introduces a decaying singular term. The source term is assumed to be inter-critical, i.e. 2−2η+α 3 < p − 1 < 2−2η+α 3. The Riesz potential is defined on ℠3 by 𠒥α(x) := Cα|x|α−3 for some 0 < α < 3.The objectives of this work are two-fold: first, to develop a local well-posedness theory in the energy space HV1 := {f ∈ L2(℠3) |ℋ Vf ∈ L2(℠3)}; and second, to present a dichotomy of global existence and scattering versus blow-up of solutions. Scattering is established using the new approach by Dodson–Murphy. The novelty of this work lies in the inclusion of the potential V.The challenges addressed here include dealing with the issues: a convolution nonlinear term, the singular term 1 |⋅|η that breaks the space translation invariance of the equation, and the presence of the potential V, which leads to a lack of scale invariance in the equation. This study naturally extends previous work by the authors on the same problem for V ∈{0,|x|−2}.

Keywords: Non-homogeneous Generalized Hartree Problem; Nonlinear Equations; Potential; Energy Scattering (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1142/S0218348X25401371

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