Inhomogeneous generalized Hartree equation with inverse square potential

• Tarek Saanouni ^[2] ; Mohamed Amine Ben Boubaker ^[1]
  1. [1] University of Carthage

     University of Carthage

     Túnez

     
  2. [2] Department of Mathematics, College of Science and Arts in Uglat Asugour, Qassim University
• Localización: SeMA Journal: Boletín de la Sociedad Española de Matemática Aplicada, ISSN-e 2254-3902, ISSN 2254-3902, Vol. 81, Nº. 4, 2024, págs. 679-706
• Idioma: inglés
• DOI: 10.1007/s40324-023-00342-4
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• Resumen

  • This work studies the inhomogeneous generalized Hartree equation with inverse square potential iu˙ − Kλu = ±|x|−τ |u|p−2(Jα ∗|·|−τ |u|p)u, u : u(t, x) : R × RN → C.

    Here N ≥ 3 and λ > −(N−2)2/4 guarantee via Hardy estimate that Kλ := − + λ|x|2 is a positive operator and satisfies the norm equivalence ||Kλ · ||L2(RN ) ∇ · L2(RN ). The Riesz potential is Jα(x) = Cα,N |x|−(N−α), for certain 0 <α< N. In order to avoidan eventual singularity |u|p−2, one assumes that p ≥ 2. Moreover, taking account of the scaling invariance κ2−2τ+α/2(p−1) u(κ2t, κ·)|| H˙N/2 − 2−2τ+α/2(p−1) = ||u(κ2t)H˙N/2 − 2−2τ+α/2(p−1), one considers the energy sub-critical regime N/2 − 2−2τ+α/2(p−1) < 1, which reads p < 1 + 2−2τ+α/N−2 . The purposeis to extend the previous paper by Alharbi and Saanouni (J Math Phys 60:081514, 2019). Indeed, the novelty is to consider the effect of an inverse square potential, namely λ = 0. In the stationary regime, one obtains a Gagliardo–Nirenberg estimate adapted to the above problem and the existence of ground states. In the evolution regime, one develops a local theory in the energy space by use of Strichartz estimates and the norm equivalence ||∇ · ||Lr(RN ) ||√Kλ · ||Lr(RN ). This gives some complications which yield some restrictions on the parameters. Moreover, one investigates the global theory with small datum. Furthermore, the dichotomy of global/non-global existence of solutions under the ground state threshold is expressed in terms of non-conserved quantities in the spirit of Dinh (Discrete Contin Dyn Syst 40(11):6441–6471, 2020). As a consequence this dichotomy is expressed in terms of the conserved laws in the spirit of Holmer and Roudenko (Commun Math Phys 282:435–467, 2008). The energy scattering of energy solutions is investigated in a paper in progress.