Stationary solutions of the non-linear Boltzmann equation in a bounded spatial domain

A contraction mapping (or, alternatively, an implicit function theory) argument is applied in combination with the Fredholm alternative to prove the existence of a unique stationary solution of the non-linear Boltzmann equation on a bounded spatial domain under a rather general reflection law at the piecewise C¹ boundary. The boundary data are to be small in a weighted L_∞-norm.     

Energy-critical Hartree equation with harmonic potential for radial data

In this paper, we consider the defocusing, energy-critical Hartree equation with harmonic potential for the radial data in all dimensions (n≥5) and show the global well-posedness and scattering theory in the space Σ=H¹∩FH¹. We take advantage of some symmetry of the Hartree nonlinearity to exploit the derivative-like properties of the Galilean operators and obtain the energy control as well. Based on Bourgain and Tao’s approach, we use a localized Morawetz identity to show the global well-posedness. A key decay estimate comes from the linear part of the energy rather than the nonlinear part, which finally helps us to complete the scattering theory.     

Stationary solutions for a superlinear Dirac equation

K. Guerlebeck In this paper, we consider the following nonlinear Dirac equation By applying the variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou, we prove the existence of nontrivial and ground state solutions for the aforementioned system under conditions weaker than those in Zhang et al. (Journal of Mathematical Physics, 2013). John Wiley & Sons, Ltd.     

Stationary solutions for the Biswas-Milovic equation

This paper studies the Biswas-Milovic equation by the aid of Lie symmetry analysis. Four types of nonlinearity are being studied for this equation. They are Kerr law, power law, parabolic law and the dual-power law. A closed form stationary solution is obtained for each case.     

Quasiclassical soliton solutions of the Hartree equation

One considers the three-dimensional Hartree equations , ΔU=¦ψ¦². One constructs the asymptotic (h→0) solution ψ of soliton type, localized mod 0 (h^∞) in a compact domain. One finds the corresponding asymptotics of the self-consistent potential U. One obtains quantification conditions on the energy of the soliton.     

Stationary solutions of the vlasov-fokker-planck equation

We study stationary solutions of the Vlasov-Fokker-Planck Equation, a modification of the Vlasov-Poisson Equation, obtained by adding a diffusion term with respect to velocity. From a physical point of view it describes a plasma in thermal equilibrium. We prove existence of stationary solutions; for the mollified equation even an existence- and uniqueness theorem holds for sufficiently high temperature.     

Scattering theory for the Dirac equation of Hartree type and the semirelativistic Hartree equation

Consider a scattering problem for the Dirac equation with a nonlocal term including the Hartree type. We improve the condition of the potential term to show the existence of scattering operators for small initial data in the subcritical Sobolev spaces. Our proofs can be applied to the case of the semirelativistic Hartree equation, and lead to improvement of the condition of the potential.     

Infinitely many solutions for Kirchhoff equations with sign-changing potential and Hartree nonlinearity

This paper is concerned with the following Kirchhoff-type equations:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\big (a+b\int _{\mathbb {R}^{3}}|\nabla u|^{2}\mathrm {d}x\big )\Delta u+ V(x)u+\mu \phi |u|^{p-2}u=f(x, u)+g(x,u), &{} \text{ in } \mathbb {R}^{3},\\ (-\Delta )^{\frac{\alpha }{2}} \phi = \mu |u|^{p}, &{} \text{ in } \mathbb {R}^{3},\\ \end{array} \right. \end{aligned}$$

where \(a>0,~b,~\mu \ge 0\) are constants, \(\alpha \in (0,3)\), \(p\in [2,3+2\alpha )\), the potential V(x) may be unbounded from below and \(\phi |u|^{p-2}u\) is a Hartree-type nonlinearity. Under some mild conditions on V(x), f(x, u) and g(x, u), we prove that the above system has infinitely many nontrivial solutions. Specially, our results cover the general Schrödinger equations, the Kirchhoff equations and the Schrödinger–Poisson system.

     

Asymptotic behavior of solutions for the damped wave equation with slowly decaying data

We consider the Cauchy problem for the damped wave equation

     

Stationary solutions of the linearized Boltzmann equation in a half-space

Stationary half-space solutions of the linearized Boltzmann equation are studied by energy estimates methods. We extend the results of Bardos, Caflisch and Nicolaenko for a gas of hard spheres to a general potential. Asymptotic behaviour is obtained for hard as well as soft potentials and compared to the case of hard spheres.