Hartree Type Nonlinearity Research Articles

Global existence and blowup of solutions to a class of wave equations with Hartree type nonlinearity

Abstract In this paper, we consider a class of wave-Hartree equations on a bounded smooth convex domain with Dirichlet boundary condition. We prove the local existence of solutions in the natural energy space by using the standard Galërkin method. The results on global existence and nonexistence of solutions are obtained mainly by means of the potential well theory and concavity method.

Mixed local and nonlocal elliptic equation with singular and critical Choquard nonlinearity

In this article, we study a class of elliptic problems involving both local and nonlocal operators with different orders and a singular nonlinearity in combination with critical Hartree type nonlinearity (see problem ( P λ ) below). Using variational methods together with the critical point theory of nonsmooth analysis, we show the existence, the regularity and the multiplicity of weak solutions with respect to the parameter λ.

Strichartz estimates for the 2D and 3D massless Dirac-Coulomb equations and applications

In this paper we continue the analysis of the dispersive properties of the 2D and 3D massless Dirac-Coulomb equations that has been started in [7] and [8]. We prove a priori estimates of the solution of the mentioned systems, in particular Strichartz estimates with an additional angular regularity, exploiting the tools developed in the previous works. As an application, we show local well-posedness results for a Dirac-Coulomb equation perturbed with Hartree-type nonlinearities.

Sign-changing solutions to critical Schrödinger equation with Hartree-type nonlinearity

Sign-changing solutions to critical Schrödinger equation with Hartree-type nonlinearity

Normalized Ground States for the Schrödinger Equation with Hartree Type and Square-Root Nonlinearities

Normalized Ground States for the Schrödinger Equation with Hartree Type and Square-Root Nonlinearities

Strong ill-posedness for fractional Hartree and cubic NLS equations

We consider fractional Hartree and cubic nonlinear Schrödinger equations on Euclidean space Rd and on torus Td. We establish norm inflation (a stronger phenomena than standard ill-posedness) at every initial data in Fourier amalgam spaces with negative regularity. In particular, these spaces include Fourier-Lebesgue, modulation and Sobolev spaces. We further show that this can be even worse by exhibiting norm inflation with an infinite loss of regularity. To establish these phenomena, we employ a Fourier analytic approach and introduce new resonant sets corresponding to the fractional dispersion (−Δ)α/2. In particular, when dispersion index α is large enough, we obtain norm inflation above scaling critical regularity in some of these spaces. It turns out that our approach could treat both equations (Hartree and power-type NLS) in a unified manner. The method should also work for a broader range of nonlinear equations with Hartree-type nonlinearity.

A remark on Chern-Simons-Schrödinger equations with Hartree type nonlinearity

A remark on Chern-Simons-Schrödinger equations with Hartree type nonlinearity

Classification of solutions for mixed order conformally system with Hartree-type nonlinearity in ℝn

In this paper, we consider the following mixed order conformally invariant system with Hartree-type nonlinearity : [Formula: see text] where [Formula: see text], [Formula: see text] is a integer, [Formula: see text], [Formula: see text], [Formula: see text]. We first prove the equivalence of the PDEs system and the IEs system. Then we give the classification of the nonnegative solutions to the system ( 0.1 ) by using the method of moving spheres. Finally, we prove Liouville-type theorems results for system ( 0.1 ) in the critical and supercritical-order cases (i.e. [Formula: see text]), respectively.

Modified scattering for the higher-order nonlinear Schrödinger equation with the Hartree-type nonlinearity

Modified scattering for the higher-order nonlinear Schrödinger equation with the Hartree-type nonlinearity

Ground State Solution of Kirchhoff Problems with Hartree Type Nonlinearity

Ground State Solution of Kirchhoff Problems with Hartree Type Nonlinearity

Existence and concentration of ground state solutions for critical Kirchhoff-type equation involving Hartree-type nonlinearities

Existence and concentration of ground state solutions for critical Kirchhoff-type equation involving Hartree-type nonlinearities

Three-dimensional stochastic cubic nonlinear wave equation with almost space-time white noise

We study the stochastic cubic nonlinear wave equation (SNLW) with an additive noise on the three-dimensional torus {mathbb {T}}^3. In particular, we prove local well-posedness of the (renormalized) SNLW when the noise is almost a space-time white noise. In recent years, the paracontrolled calculus has played a crucial role in the well-posedness study of singular SNLW on {mathbb {T}}^3 by Gubinelli et al. (Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity, 2018, arXiv:1811.07808 [math.AP]), Oh et al. (Focusing Phi ^4_3-model with a Hartree-type nonlinearity, 2020. arXiv:2009.03251 [math.PR]), and Bringmann (Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: Dynamics, 2020, arXiv:2009.04616 [math.AP]). Our approach, however, does not rely on the paracontrolled calculus. We instead proceed with the second order expansion and study the resulting equation for the residual term, using multilinear dispersive smoothing.

Semiclassical states for Schrödinger-Poisson system with Hartree-type nonlinearity

Semiclassical states for Schrödinger-Poisson system with Hartree-type nonlinearity

INFINITELY MANY SOLUTIONS FOR A QUASILINEAR KIRCHHOFF-TYPE EQUATION WITH HARTREE-TYPE NONLINEARITIES

In this paper, we consider a new kind of Kirchhoff-type equation with Hartree-type nonlinearities which is stated in the introduction. Under certain assumptions on <i>g</i>(<i>u</i>), we prove that the equation has infinitely many solutions by variational methods.

Global well-posedness for Klein-Gordon-Hartree and fractional Hartree equations on modulation spaces

We study the Cauchy problems for the Klein-Gordon (HNLKG), wave (HNLW), and Schrodinger (HNLS) equations with cubic convolution (of Hartree type) nonlinearity. Some global well-posedness and scattering are obtained for the (HNLKG) and (HNLS) with small Cauchy data in some modulation spaces. Global well-posedness for fractional Schrodinger (fNLSH) equation with Hartree type nonlinearity is obtained with Cauchy data in some modulation spaces. Local well-posedness for (HNLW), (fHNLS) and (HNLKG) with rough data in modulation spaces is shown. As a consequence, we get local and global well-posedness and scattering in larger than usual \(L^p\)-Sobolev spaces. For more information see https://ejde.math.txstate.edu/Volumes/2021/101/abstr.html

Small data scattering of 2d Hartree type Dirac equations

In this paper, we study the Cauchy problem of 2d Dirac equation with Hartree type nonlinearity c(|⋅|−γ⁎〈ψ,βψ〉)βψ with c∈R∖{0}, 0<γ<2. Our aim is to show the small data global well-posedness and scattering in Hs for s>γ−1 and 1<γ<2. The difficulty stems from the singularity of the low-frequency part |ξ|−(2−γ)χ{|ξ|≤1} of potential. To overcome it we adapt Up−Vp space argument and bilinear estimates of [27,25] arising from the null structure. We also provide nonexistence result for scattering in the long-range case 0<γ≤1.

Liouville Theorems for a p-Laplace Equation with Hartree Type Nonlinearity

Liouville Theorems for a p-Laplace Equation with Hartree Type Nonlinearity

Global Large Data Solutions for 2D Dirac Equation with Hartree Type Interaction

Abstract Local and global well posedness of the solution to the two space dimensional Dirac equation with Hartree type nonlinearity is established with the initial datum in the Sobolev space of order $s&amp;gt;0.$

Small Data Scattering of Hartree Type Fractional Schrödinger Equations in a Scaling Critical Space

In this paper we study the small-data scattering of Hartree type fractional Schrödinger equations in space dimension 2, 3. It has Lévy index α between 1 and 2, and Hartree type nonlinearity F(u) = μ(|x|-γ * |u|2)u with 2d/(2d-1) < γ < 2, γ ≥ α > 1. This equation is scaling-critical in Ḣsc with sc = (γ-α)/2. We show that the solution scatters in Ḣsc,1, where Ḣsc,1 is also a scaling critical space of Sobolev type taking in angular regularity with norm defined by ||ϕ||Ḣsc,1 = ||ϕ||Ḣsc + ||∇Sϕ||Ḣsc. For this purpose we use the recently developed Strichartz estimate which is Lθ2-averaged on the unit sphere Sd-1.

On fractional Schrödinger equations with Hartree type nonlinearities

&lt;abstract&gt;&lt;p&gt;Goal of this paper is to study the following doubly nonlocal equation&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document} $(- \Delta)^s u + \mu u = (I_\alpha*F(u))F'(u) \quad {\rm{in}}\;{\mathbb{R}^N}\qquad\qquad\qquad\qquad ({\rm{P}}) $ \end{document} &lt;/tex-math&gt; &lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;in the case of general nonlinearities $ F \in C^1(\mathbb{R}) $ of Berestycki-Lions type, when $ N \geq 2 $ and $ \mu &amp;gt; 0 $ is fixed. Here $ (-\Delta)^s $, $ s \in (0, 1) $, denotes the fractional Laplacian, while the Hartree-type term is given by convolution with the Riesz potential $ I_{\alpha} $, $ \alpha \in (0, N) $. We prove existence of ground states of (P). Furthermore we obtain regularity and asymptotic decay of general solutions, extending some results contained in &lt;sup&gt;[&lt;xref ref-type="bibr" rid="b23"&gt;23&lt;/xref&gt;,&lt;xref ref-type="bibr" rid="b61"&gt;61&lt;/xref&gt;]&lt;/sup&gt;.&lt;/p&gt;&lt;/abstract&gt;