Energy scattering for a Klein–Gordon equation with a cubic convolution

Abstract

In this paper, we study the global well-posedness and scattering problem in the energy space for both focusing and defocusing the Klein-Gordon-Hartree equation in the spatial dimension d 3. The main difficulties are the absence of an interaction Morawetz-type estimate and of a Lorentz invariance which enable one to control the momentum. To compensate, we utilize the strategy derived from concentration compactness ideas, which was first introduced by Kenig and Merle [15] to the scattering problem. Furthermore, employing technique from [37], we consider a virial-type identity in the direction orthogonal to the momentum vector so as to control the momentum in the defocusing case. While in the focusing case, we show that the scattering holds when the initial data (u 0 , u 1 ) is radial, and the energy E(u 0 , u 1 ) < E(W, 0) and ∇u 0 2 2 + u 0 2 2 < ∇W 2 2 + W 2 2 , where W is the ground state.

Global well-posedness and scattering of a generalized nonlinear fourth-order wave equation

“…Their method also holds for the defocusing case. For other study on Klein-Gordon equation, the reader can refer to [7][8][9][10][11][12].…”

Initial boundary value problem for a class of wave equations of Hartree type

“…For the Cauchy problem of Equation (), there is a large volume of literature on the global well‐posedness, the scattering theory, the blowup, and the asymptotical behavior of the solutions (see, e.g., Refs. 3–19 and the references therein). Furthermore, Tatar 20 studied the interaction between a dissipative term and a source term of cubic convolution type for the wave equation in $$R^n$$ $$\begin{eqnarray} &&u_{tt}+mu+\mu u_t(|x|^{-\gamma }*u^2_t)=\Delta u+\lambda h(t)u(|x|^{-\gamma }*u^2), \nobreakspace in\nobreakspace (0, T)\times R^n.$$…”

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

“…is also known as Hardy potential, and this type of potential is important in analyzing many aspects of physical phenomena with singular poles (at origin); see, for example, previous works [15][16][17][18]. There are a lot of studies of evolution equations with this type of potentials, for example, wave equations [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34], Schrodinger equations [35][36][37][38][39][40][41], and parabolic equations [42][43][44][45][46][47]. For wave equations with Hardy potential (or Hartree type), Perla Menzala and Strauss [19] first studied global existence, scattering (existence of asymptotically free solutions) and global nonexistence for the Cauchy problem of equation…”