Consider the Klein-Gordon-Zakharov equations in $\mathbb{R}^{1+2}$, and we are interested in establishing the small global solution to the equations and in investigating the pointwise asymptotic behavior of the solution. The Klein-Gordon-Zakharov equations can be regarded as a coupled semilinear wave and Klein-Gordon system with quadratic nonlinearities which do not satisfy the null conditions, and the fact that wave components and Klein-Gordon components do not decay sufficiently fast makes it harder to conduct the analysis. In order to conquer the difficulties, we will rely on the hyperboloidal foliation method and a minor variance of the ghost weight method. As a side result of the analysis, we are also able to show the small data global existence result for a class of quasilinear wave and Klein-Gordon system violating the null conditions.
We are interested in the Klein-Gordon-Zakharov equations in $\mathbb{R}^{1+3}$, and we aim to show that the energy for the global solution to the equations is uniformly bounded, and we do not require the compactness assumption on the initial data. To achieve these goals, the key is to apply Alinhac's ghost weight energy estimates adapted to the Klein-Gordon equations.
We study the Cauchy problem for the Klein-Gordon-Zakharov system in 3D with low regularity data. We lower down the regularity to the critical value with respect to scaling up to the endpoint. The decisive bilinear estimates are proved by means of methods developed by Bejenaru-Herr for the Zakharov system and already applied by Kinoshita to the Klein-Gordon-Zakharov system in 2D.
We prove small energy scattering for the 3D Klein-Gordon-Zakharov system with radial symmetry. The idea of proof is the same as the Zakharov system studied in \cite{GN}, namely to combine the normal form reduction and the radial-improved Strichartz estimates
We consider the global dynamics below the ground state energy for the Klein-Gordon-Zakharov system in the 3D radial case; and obtain the dichotomy between scattering and finite time blow up.
This paper is concerned with the Cauchy problem of $2$D Klein-Gordon-Zakharov system with very low regularity initial data. We prove the bilinear estimates which are crucial to get the local in time well-posedness. The estimates are established by the Fourier restriction norm method. We utilize the bilinear Strichartz estimates and the nonlinear version of the classical Loomis-Whitney inequality which was applied to Zakharov system.
It is shown that the spatial Sobolev norms of regular global solutions of the (2+1),(3+1) and (4+1)-dimensional Klein-Gordon-Schroedinger system and the (2+1) and (3+1)-dimensional Zakharov system grow at most polynomially with the bound depending on the regularity class of the data. The proof uses the Fourier restriction norm method.
We are interested in the long time asymptotic behavoir of solutions to the scalar Zakharov system \[ i u_{t} + Δu = nu,\] \[n_{tt} - Δn= Δ|u|^2\] and the Klein-Gordon Zakharov system \[ u_{tt} - Δu + u = - nu,\] \[ n_{tt} - Δn= Δ|u|^2\] in one dimension of space. For these two systems, we give two results proving decay of solutions for initial data in the energy space. The first result deals with decay over compact intervals asuming smallness and parity conditions ($u$ odd). The second result proves decay in far field regions along curves for solutions whose growth can be dominated by an incre…
We prove low-regularity global well-posedness for the 1d Zakharov system and 3d Klein-Gordon-Schrödinger system, which are systems in two variables $u:\mathbb{R}_x^d\times \mathbb{R}_t \to \mathbb{C}$ and $n:\mathbb{R}^d_x\times \mathbb{R}_t\to \mathbb{R}$. The Zakharov system is known to be locally well-posed in $(u,n)\in L^2\times H^{-1/2}$ and the Klein-Gordon-Schrödinger system is known to be locally well-posed in $(u,n)\in L^2\times L^2$. Here, we show that the Zakharov and Klein-Gordon-Schrödinger systems are globally well-posed in these spaces, respectively, by using an available conser…
The orbital instability of standing waves for the Klein-Gordon-Zakharov system has been established in two and three space dimensions under radially symmetric condition, see Ohta-Todorova (SIAM J. Math. Anal. 2007). In the one space dimensional case, for the non-degenerate situation, we first check that the Klein-Gordon-Zakharov system satisfies Grillakis-Shatah-Strauss' assumptions on the stability and instability theorems for abstract Hamiltonian systems, see Grillakis-Shatah-Strauss (J. Funct. Anal. 1987). As to the degenerate case that the frequency $|ω|=1/\sqrt{2}$, we follow Wu (ArXiv: 1…
We study the Cauchy problem of the Klein-Gordon-Zakharov system in spatial dimension $d \ge 5$ with initial datum $(u, \partial_t u, n, \partial_t n)|_{t=0} \in H^{s+1}(\mathbb{R}^d) \times H^s(\mathbb{R}^d) \times \dot{H}^s(\mathbb{R}^d) \times \dot{H}^{s-1}(\mathbb{R}^d)$. The critical value of $s$ is $s_c=d/2-2$. By $U^2, V^2$ type spaces, we prove that the small data global well-posedness and scattering hold at $s=s_c$ in $d \ge 5$.
We are interested in the Klein-Gordon-Zakharov system in $\mathbb{R}^{1+2}$, which is an important model in plasma physics with extensive mathematical studies. The system can be regarded as semilinear coupled wave and Klein-Gordon equations with nonlinearities violating the null conditions. Without the compactness assumptions on the initial data, we aim to establish the existence of small global solutions, and in addition, we want to illustrate the optimal pointwise decay of the solutions. Furthermore, we show that the Klein-Gordon part of the system enjoys linear scattering while the wave par…
This paper studies the regularity of solutions to the Zakharov and Klein-Gordon-Schrödinger systems at low regularity levels. The main result is that the nonlinear part of the solution flow falls in a smoother space than the initial data. This relies on a new bilinear $X^{s,b}$ estimate, which is proved using delicate dyadic and angular decompositions of the frequency domain. Such smoothing estimates have a number of implications for the long-term dynamics of the system. In this work, we give a simplified proof of the existence of global attractors for the Klein-Gordon-Schrödinger flow in the …
The scattering problem for the Klein-Gordon equation with cubic convolution nonlinearity is considered. Based on the Strichartz estimates for the inhomogeneous Klein-Gordon equation we prove the existence of the scattering operator.
Roughly speaking a solitary wave is a solution of a field equation whose energy travels as a localized packet and which preserves this localization in time. A solitary wave which has a non-vanishing angular momentum is called vortex. We know (at least) three mechanisms which might produce solitary waves and vortices: 1) Complete integrability, (e.g. Kortewg-de Vries equation) 2) Topological constraints, (e.g. Sine-Gordon equation); 3) Ratio energy/charge: (e.g. the nonlinear Klein-Gordon equation). The third type of solitary waves or solitons will be called hylomorphic. This class includes the…
We show asymptotic completeness for the charged Klein-Gordon equation in the exterior De Sitter-Reissner-Nordström spacetime when the product of the charge of the black hole with the charge of Klein-Gordon field is small enough. We then interpret scattering as asymptotic transports along principal null geodesics in a Kaluza-Klein extension of the original spacetime.
We extend our previous result on the focusing cubic Klein-Gordon equation in three dimensions to the non-radial case, giving a complete classification of global dynamics of all solutions with energy at most slightly above that of the ground state.
We study the one-dimensional nonlinear Klein-Gordon (NLKG) equation with a convolution potential, and we prove that solutions with small analytic norm remain small for exponentially long times. The result is uniform with respect to $c \geq 1$, which however has to belong to a set of large measure.
We study the one-dimensional nonlinear Klein-Gordon (NLKG) equation with a convolution potential, and we prove that solutions with small $H^s$ norm remain small for long times. The result is uniform with respect to $c \geq 1$, which however has to belong to a set of large measure.
We consider waves, which obey the semilinear Klein-Gordon equation, propagating in the Friedmann-Lemaitre-Robertson-Walker spacetimes. The equations in the de Sitter and Einstein-de Sitter spacetimes are the important particular cases. We show the global in time existence in the energy class of solutions of the Cauchy problem.
In this article we construct the fundamental solutions for the Klein-Gordon equation in de Sitter spacetime. We use these fundamental solutions to represent solutions of the Cauchy problem and to prove $L^p-L^q$ estimates for the solutions of the equation with and without a source term.
This paper deals with the Klein-Gordon-Maxwell system in a bounded spatial domain. We study the existence of solutions having a specific form, namely standing waves in equilibrium with a purely electrostatic field. We prescribe Dirichlet boundary conditions on the matter field, and either Dirichlet or Neumann boundary conditions on the electric potential.
We study the Cauchy problem for the Klein-Gordon-Zakharov system in spatial dimension $d \ge 4$ with radial or non-radial initial datum $(u, \partial_t u, n, \partial_t n)|_{t=0}\in H^{s+1}(\mathbb{R}^d) \times H^s(\mathbb{R}^d) \times \dot{H}^s(\mathbb{R}^d) \times \dot{H}^{s-1}(\mathbb{R}^d)$. The critical value of $s$ is $s=s_c=d/2-2$. If the initial datum is radial, then we prove the small data global well-posedness and scattering at the critical space in $d \ge 4$ by applying the radial Strichartz estimates and $U^2, V^2$ type spaces. On the other hand, if the initial datum is non-radial,…
In this article we will develop some techniques aimed at the strong couplings in two-dimensional wave-Klein-Gordon system. We distinguish the roles of different type of decay factors and develop a method which permits us to "exchange" one type of decay into the other. Then a global existence result of a model problem is established. We also give a sketch of the Klein-Gordon-Zakharov system and establish the associate global existence result.
We consider a collisionless ensemble of classical particles coupled to a Klein-Gordon field. For the resulting nonlinear system of partial differential equations, the relativistic Vlasov-Klein-Gordon system, we prove local-in-time existence of classical solutions and a continuation criterion which says that a solution can blow up only if the particle momenta become large. We also show that classical solutions are global in time in the one-dimensional case.
We obtain sharp decay estimates and asymptotics for small solutions to the one-dimensional Klein-Gordon equation with constant coefficient cubic and spatially localized, variable coefficient cubic nonlinearities. Vector-field techniques to deal with the long-range nature of the cubic nonlinearity become problematic in the presence of variable coefficients. We introduce a novel approach based on pointwise-in-time local decay estimates for the Klein-Gordon propagator to overcome this impasse.
We study small data scattering of solutions to Nonlinear Klein-Gordon equations with suitable pure power nonlinearities, posed on $\mathbb{R}^d\times \mathcal{M}^k$ with $k\leq2$ and $d\geq1$ and $\mathcal{M}^k$ a compact Riemannian manifold. As a special case we cover the $H^1-$critical NLKG on $\mathbb{R} \times M^2$.
We show asymptotic completeness for a class of superradiant Klein-Gordon equations. Our results are applied to the Klein-Gordon equation on the De Sitter Kerr metric with small angular momentum of the black hole. For this equation we obtain asymptotic completeness for fixed angular momentum of the field.
In this paper we present explicit formulas for the fundamental solution to the Klein-Gordon operator on some higher dimensional generalizations of the Möbius strip and the Klein bottle with values in distinct pinor bundles. The fundamental solution is described in terms of generalizations of the Weierstraß $\wp$-function that are adapted to the context of these geometries. The explicit formulas for the kernel then allow us to express all solutions to the homogeneous and inhomogeneous Klein-Gordon problem with given boundary data in the context of these manifolds. In the case of the Klein bottl…
We report on the well-posedness of the Feynman problem for the Klein-Gordon equation on asymptotically Minkowski spacetimes. The main result is the invertibility of the Klein-Gordon operator with Feynman conditions at infinite times. Furthermore, the inverse is shown to coincide with the Duistermaat-Hörmander Feynman parametrix modulo smoothing terms.
We consider the stability of the toroidal AdS-Schwarzshild black holes as solutions of the Einstein--Klein-Gordon system, with Dirichlet or Neumann boundary conditions for the scalar field. Restricting to perturbations that respect the toroidal symmetry we show both orbital and asymptotic stability for the full nonlinear problem, for a range of choices of the Klein-Gordon mass. The solutions we construct with Neumann boundary conditions have a Hawking mass which diverges towards infinity, reflecting the infinite energy of the Klein-Gordon field for perturbations satisfying these boundary condi…
We construct one-parameter families of solutions to the Einstein--Klein--Gordon equations bifurcating off the Kerr solution such that the underlying family of spacetimes are each an asymptotically flat, stationary, axisymmetric, black hole spacetime, and such that the corresponding scalar fields are non-zero and time-periodic. An immediate corollary is that for these Klein--Gordon masses, the Kerr family is not asymptotically stable as a solution to the Einstein--Klein--Gordon equations.
We prove that finite energy solutions to the nonlinear Schroedinger equation and nonlinear Klein--Gordon equation which have the compact time spectrum have to be one-frequency solitary waves. The argument is based on the generalization of the Titchmarsh convolution theorem to partial convolutions.
This note aims to give prominence to some new results on the absence and localization of eigenvalues for the Dirac and Klein-Gordon operators, starting from known resolvent estimates already established in the literature combined with the renowned Birman-Schwinger principle.
In this note, we consider discrete nonlinear Klein-Gordon equations with potential. By the pioneering work of Sigal, it is known that for the "continuous" nonlinear Klein-Gordon equation, no small time periodic solution exists generically. However, for the discrete nonlinear Klein-Gordon equations, we show that there exist small time periodic solutions.
The Cauchy problem for the classical Dirac-Klein-Gordon system in two space dimensions is globally well-posed for L^2 Schoedinger data and wave data in H^{1/2} \times H^{-1/2}. In the case of smooth data there exists a global smooth (classical) solution. The proof uses function spaces of Bourgain type based on Besov spaces - previously applied by Colliander, Kenig and Staffilani for generalized Benjamin-Ono equations and also by Bejenaru, Herr, Holmer and Tataru for the 2D Zakharov system - and the null structure of the system detected by d'Ancona, Foschi and Selberg, and a refined bilinear St…
In this article we will prove the global existence of a type of wave-Klein-Gordon system in $2+1$ spacetime dimension. Some technical tools such as conformal energy estimate on hyperboloid, normal form transform on Klein-Gordon equations will be adapted to this $2+1$ dimensional case.
By exploiting estimates on Bloch functions obtained in a previous paper, we prove decay estimates for Klein Gordon equations with a time independent potential periodic in space in 1D and with generic mass
In this paper we consider the Klein-Gordon-Maxwell system in the electrostatic case, assuming the fall-off large-distance requirement on the gauge potential. We are interested in proving the existence of finite energy (and finite charge) standing waves, having the phase corresponding to the mass coefficient in the Klein-Gordon Lagrangian.
We study a Klein-Gordon-Maxwell system, in a bounded spatial domain, under Neumann boundary conditions on the electric potential. We allow a nonconstant coupling coefficient. For sufficiently small data, we find infinitely many standing waves.
We prove the existence of a mountain-pass solution and the a priori bound property for the electrostatic Klein-Gordon-Maxwell equations in high dimension.
We consider the decay rate of solutions to nonlinear Klein-Gordon systems with a critical type nonlinearity. We will specify the optimal decay rate for a specific class of Klein-Gordon systems containing the dissipative nonlinearites. It will turn out that the decay rate which is previously found in some models is optimal.
We consider a system of two coupled non-linear Klein-Gordon equations. We show the existence of standing waves solutions and the existence of a Lyapunov function for the ground state.
Static Klein-Gordon-Maxwell-Proca systems are massive versions of the electrostatic Klein-Gordon-Maxwell Systems. The vector field in these systems inherits a mass and is governed by the Proca action which generalizes that of Maxwell. Following previous work by Druet & al. we prove existence and uniform bounds for such systems in the double critical case of dimension 4.
We prove global in time dispersion for the wave and the Klein-Gordon equation inside the Friedlander domain by taking full advantage of the space-time localization of caustics and a precise estimate of the number of waves that may cross at a given, large time. Moreover, we uncover a significant difference between Klein-Gordon and the wave equation in the low frequency, large time regime, where Klein-Gordon exhibits a worse decay that the wave, unlike in the flat space.
We consider long time evolution of small solutions to general multispeed Klein-Gordon systems in 3+1 dimensions. We prove that such solutions are always global and scatter to a linear flow, thus extending previous partial results. The main new ingredients of our method is an improved linear dispersion estimate exploiting the asymptotic spherical symmetry of Klein-Gordon waves, and a corresponding bilinear oscillatory integral estimate.
We prove that in the nonrelativistic limit, solutions of the Klein-Gordon-Maxwell system in 1+3 dimensions converge in the energy space to solutions of a Schrodinger-Poisson system, under appropriate conditions on the initial data. This requires the splitting of the scalar Klein-Gordon field into a sum of two fields, corresponding, in the physical interpretation, to electrons and positrons.
In this paper, we prove the exponential decay of local energy for the Klein-Gordon equation with localized critical nonlinearity. The proof relies on generalized Strichartz estimates, and semi-group of Lax-Phillips.
We consider the Cauchy problem for the 2D and 3D Klein-Gordon-Schrödinger system. In 2D we show local well-posedness for Schrödinger data in H^s and wave data in H^σ x H^{σ-1} for s=-1/4 + and σ= -1/2, whereas ill-posedness holds for s<- 1/4 or σ<-1/2, and global well-posedness for s\ge 0 and s- 1/2 \le σ< s+ 3/2. In 3D we show global well-posedness for s \ge 0, s - 1/2 < σ\le s+1. Fundamental for our results are the studies by Bejenaru, Herr, Holmer and Tataru, and Bejenaru and Herr for the Zakharov system, and also the global well-posedness results for the Zakharov and Klein-Gordon-Schröding…
We prove small data global existence and scattering for quasilinear systems of Klein-Gordon equations with different speeds, in dimension three. As an application, we obtain a robust global stability result for the Euler-Maxwell equations for electrons.
We prove existence and uniform bounds for electrostatic Klein-Gordon-Maxwell systems in the inhomogeneous context of a compact Riemannian manifold when the mass potential, balanced by the phase, is small in a quantified sense.
We derive a uniform exponential decay of the total energy for the nonlinear Klein-Gordon equation with a damping around spatial infinity in the whole space or in the exterior of a star shaped obstacle.
