Search

This work studies an inhomogeneous generalized Hartree equation with inverse square potential. The purpose is to prove the existence and strong instability of inter-critical standing waves. This means that there are infinitely many data near to the ground state, such that the associated solution blows-up in finite time. The proof combines a variational analysis with the standard variance identity. The challenge is to deal with three difficulties: the singular potential | x | − 2 , an inhomogeneous term | x | − λ , and a non-local source term. [ABSTRACT FROM AUTHOR]

This work studies the focusing inhomogeneous nonlinear equation of Hartree type і∂tu - Кλu +|х|-τ-|u|p-2)(Jα*|.|-τ|u|P)u = 0, u(t.x): R x R³ → C. Here, the linear Schrödinger operator reads Kλ:= - Δ+ λ/|x|² for some λ > - 1/4. The Riesz potential is Jα (x) = Cα|x|-(3-α) for certain 0 <α< 3. The singular decaying term |x|-τ for some τ > 0 gives an inhomogeneous non-linearity. One considers the inter-critical regime, namely, 1 +2 (1-τ)+α /3 0 because there is no dispersive estimate L¹→ L∞ for λ<0. Contrarily to the homogeneous case τ = 0, for λ> 0, there is a ground state which minimizes the associated Gagliardo-Nirenberg-type estimate. The purpose is to investigate the energy scattering of global solutions under the ground state threshold. One uses the method of Dodson-Murphy based on Tao's scattering criteria and Morawetz estimates. The decay of the inhomogeneous term |x|-τ avoids any spherically symmetric assumption. [ABSTRACT FROM AUTHOR]

This paper studies the asymptotic behavior of energy solutions to a non-linear generalized Hartree equation. Indeed, in the inter-critical regime, one revisits the scattering versus finite time blow-up of energy solutions with non-necessarily spherically symmetric datum. Here, one uses the new approach due to Dodson and Murphy. The novelty in this work is to express the scattering threshold in terms of some non-conserved quantities. The main result of this note seems to be stronger than the classical scattering versus finite time blow-up dichotomy given in terms of the conserved mass and energy by Holmer and Roudenko. Indeed, as an application, one investigates the scattering in three different regimes: under, at and beyond the ground state threshold. The main result given here, which can be seen as a criteria of scattering versus finite time blow-up of energy solutions, enables us to give a unified approach to deal with the above generalised Hartree equation in different regimes. This paper follows some new ideas presented recently in the classical Schrödinger equation with a local source term, by V.D. Dinh. [ABSTRACT FROM AUTHOR]

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 . [ABSTRACT FROM AUTHOR]

This note studies the fourth-order generalized Hartree equation$ i\dot u+\Delta^2 u\pm|u|^{p-2}(J_\gamma*|u|^p)u = 0,\quad p\geq2. $Indeed, for both attractive and repulsive sign, the scattering is obtained in the inter-critical regime, which is given by $ 0

This note studies the linearly damped generalized Hartree equation i u ˙ − (− Δ) s u + i a u = ± | u | p − 2 (J γ ∗ | u | p) u , 0 < s < 1 , a > 0 , p ≥ 2. Indeed, one proves an exponential scattering of the energy global solutions, with spherically symmetric datum. This means that, for large time, the solution goes exponentially to the solution of the associated free problem i u ˙ − (− Δ) s u + i a u = 0 , in H s norm. The radial assumption avoids a loss of regularity in Strichartz estimates. The exponential scattering, which means that v : = e a t u scatters in H s , is proved in the energy sub-critical defocusing regime and in the mass-sub-critical focusing regime. This result is presented because of the gap due to the lack of scattering in the mass sub-critical regime, which seems not to be well understood. In this manuscript, one needs to overcome three technical difficulties which are mixed together: the first one is a fractional Laplace operator, the second one is a Choquard (non-local) source term, including the Hartree-type term when p = 2 and the last one is a damping term i a u . In a work in progress, the authors investigate the exponential scattering of global solutions to the above Schrödinger problem, with different kind of damping terms. [ABSTRACT FROM AUTHOR]

It is the purpose of this work to study the Choquard equation i u ˙ - (- Δ) s u = ± | x | γ (I α ∗ | · | γ | u | p ) | u | p - 2 u in the space H ˙ s ∩ H ˙ s c , where 0 < s c < s corresponds to the scale invariant homogeneous Sobolev norm. Here, one considers to two separate cases. The first one is the classical case s = 1 and the second one is the fractional regime 0 < s < 1 with radial data. One tries to develop a local theory using a new adapted sharp Gagliardo–Nirenberg estimate. Moreover, one investigates the concentration of non-global solutions in L T ∗ ∞ (H ˙ s c) . One needs to deal with the lack of a mass conservation, since the data are not supposed to be in L 2 . This note gives a complementary to the previous works about the same problem in the energy space H 1 . [ABSTRACT FROM AUTHOR]

This document is a correction notice for an article titled "On the inter-critical inhomogeneous generalized Hartree equation" published in the Arabian Journal of Mathematics. The correction addresses incorrect affiliation details for the authors. The correct affiliations are provided, including the Department of Mathematics at Qassim University in Saudi Arabia and the Faculty of Science of Tunis at the University of Tunis El Manar in Tunisia. The original article has been corrected. The publisher, Springer Nature, remains neutral regarding jurisdictional claims and institutional affiliations. The authors of the article are Tarek Saanouni and Talal Alharbi. [Extracted from the article]

This note studies the non-linear fractional Schrödinger equation i u ̇ − (− Δ) s u + ( I α * | u | p ) | u | p − 2 u = 0. In the mass super-critical and energy sub-critical regimes, the local solutions exist globally and scatter in the energy space or blow-up in finite time if the data belong to some stable sets. [ABSTRACT FROM AUTHOR]

It is the purpose of this work to obtain a sharp threshold of global existence vs blow-up dichotomy for mass-super-critical and energy subcritical solutions to an inhomogeneous Choquard equation. [ABSTRACT FROM AUTHOR]

In this article, we study the initial value problem for a semi-linear damped Schrödinger equation with exponential growth nonlinearity in two space dimensions. We show global well-posedness and exponential decay. [ABSTRACT FROM AUTHOR]

We investigate the initial value problem for a semilinear damped Schrödinger equation with exponential growth nonlinearity in two space dimensions. We obtain global well-posedness in the energy space. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]