We consider the Klein--Gordon equation associated with the Laplace--Beltrami operator $Δ$ on real hyperbolic spaces of dimension $n\!\ge\!2$; as $Δ$ has a spectral gap, the wave equation is a particular case of our study. After a careful kernel analysis, we obtain dispersive and Strichartz estimates for a large family of admissible couples. As an application, we prove global well--posedness results for the corresponding semilinear equation with low regularity data.
In this paper we prove standard and reversed Strichartz estimates for the Klein--Gordon equation in $\mathbb{R}^{3+1}$. Instead of the Fourier theory, our analysis is based on fundamental solutions of the free equations and fractional integrations. In the final part of this paper, we apply Strichartz estimates in the study of a semilinear Klein--Gordon equation.
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 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 study linear Klein-Gordon equations with moving potentials motivated by the stability analysis of traveling waves and multi-solitons. In this paper, Strichartz estimates, local energy decay and the scattering theory for these models are established. The results and estimates obtained in this paper will be used to study the interaction of solitons and the stability of multisolitons of nonlinear Klein-Gordon equations.
We present a set of time quasi-periodic solutions to a nonlinear Klein-Gordon equation with a decaying nonlinear term on the torus in arbitrary dimensions. This paper generalizes the bifurcation method developed in [W2].
In this paper, we consider a semiclassical version of the fractional Klein-Gordon equation on the lattice, $h{\mathbb{Z}}^n.$ Contrary to the Euclidean case that was considered in [2], the discrete fractional Klein-Gordon equation is well-posed in $\ell^2(h{\mathbb{Z}}^n).$ However, we also recover the well-posedness results in the certain Sobolev spaces in the limit of the semiclassical parameter $h\rightarrow 0.$
We obtain explicit characterization of spectral and orbital stability of solitary wave solutions to the $\mathbf{U}(1)$-invariant Klein--Gordon equation in one spatial dimension coupled to an anharmonic oscillator. We also give the complete analysis of the spectrum of the linearization at a solitary wave.
In this paper, developing a new approach based on Fourier analysis methods for dispersive PDEs, we establish a low regularity NLS approximation for the one-dimensional cubic Klein-Gordon equation. Our main result includes energy class solutions which are formally asymptotically in $L^2(\mathbb{R})$. A precise rate of convergence is also obtained assuming more regularity.
The Cauchy problem for the Klein-Gordon equation under the quartic potential is considered in the de Sitter spacetime. The existence of the global solution is shown based on the mechanism of the spontaneous symmetry breaking for the small positive Hubble constant. The effects of the spatial expansion and contraction on the problem are considered.
