Global solution to the wave and Klein-Gordon system under null condition in dimension two

Abstract

We are interested in studying the coupled wave and Klein-Gordon equations with null quadratic nonlinearities in R 2+1 . We want to establish the small data global existence result, and in addition, we also demonstrate the pointwise asymptotic behaviour of the solution to the coupled system. The initial data are not required to have compact support, and this is achieved by applying the Alinhac's ghost weight method to both the wave and the Klein-Gordon equations.

Global solution to the Klein-Gordon-Zakharov equations with uniform energy bounds

“…Originally, the ghost weight method was applied to wave equations with null nonlinearities [1], and allows one to benefit from the t − r decay when some good derivatives (x a /r)∂ t + ∂ a acting on the wave components. But we find in [6] that the ghost weight energy estimates on Klein-Gordon equations provide us with some new and strong result, i.e. we can benefit from the t − r decay for the Klein-Gordon components without any derivatives (or with good derivatives (x a /r)∂ t + ∂ a ).…”

“…In order to prove the energy for E, n is uniformly bounded, the key is to apply Alinhac's ghost weight method adapted to Klein-Gordon equations, which was used in [6] by the author when dealing with a coupled wave and Klein-Gordon system in R 1+2 . Originally, the ghost weight method was applied to wave equations with null nonlinearities [1], and allows one to benefit from the t − r decay when some good derivatives (x a /r)∂ t + ∂ a acting on the wave components.…”

Hidden structure and sharp asymptotics for the Dirac--Klein-Gordon system in two space dimensions

“…This was initiated by Ma for quasilinear wave-Klein-Gordon systems [28,29] and has been succeeded by Ma [30,31,32] and the present authors [12]. There has also been work by Stingo [37] and the first author [10] which does not require compactly supported data. Other work has also looked at the Klein-Gordon-Zakharov model in 1 + 2 dimensions [11,16,32] and the wave map model derived in [1] has been studied in the critical case of 1 + 2 dimensions in the recent works [13,16].…”

Global existence for systems of nonlinear wave and Klein-Gordon equations in two space dimensions under a kind of the weak null condition

“…To illustrate it, let us consider the system (2.4) with a = 0 and c = 1 in Example 2.2. For single wave equation w = −(∂ t w) 3 , it is known that the energy decay occurs (see [16,17,18]). We show that the same is true for the wave component w in (2.4).…”

“…Finally, for systems of nonlinear wave and Klein-Gordon equations (namely the case where 1 ≤ N 0 < N ), we also need some restriction on the nonlinearity to obtain the small data global existence as these systems contain nonlinear wave equations. Motivated by the previous works [8,14,15,24] for systems with quadratic nonlinearity in three space dimensions, Aiguchi [1] investigated the Cauchy problem for (1.2)-(1.3), and proved the small data global existence, assuming that the interaction between wave components in the wave equations satisfies the null condition (recently systems with quadratic nonlinearities in two space dimensions are also widely studied; see for example, Dong [3,4], Duan-Ma [5], Dong-Wyatt [6] and Ma [26,27,28,29,30]).…”