Abstract
We consider the Cauchy problem for coupled systems of nonlinear wave and Klein-Gordon equations in three space dimensions. The author previously proved the small data global existence for rapidly decreasing data under a certain condition on nonlinearity. In this paper, we show that we can weaken the condition, provided that the initial data are compactly supported.
Highlights
We consider the Cauchy problem for the following system of nonlinear wave and Klein-Gordon equations:( + m2j )uj = Fj(u, ∂u) in (0, ∞) × R3, (1)uj(0, x) = εfj(x), (∂tuj)(0, x) = εgj(x), x = (x1, x2, x3) ∈ R3 (2)for j = 1, 2, . . . , N, where u =1≤j≤N is an RN -valued unknown function of (t, x) ∈ (0, ∞) × R3, := ∂t2 − ∆, and mj ≥ 0. ∂u stands for the first derivatives of u, that is to say, ∂u := (∂auj)1≤j≤N,0≤a≤3, where ∂0 := ∂t and ∂k := ∂xk for 1 ≤ k ≤ 3
For j = 1, 2, . . . , N, where u =1≤j≤N is an RN -valued unknown function of (t, x) ∈ (0, ∞) × R3, := ∂t2 − ∆, and mj ≥ 0. ∂u stands for the first derivatives of u, that is to say, ∂u := (∂auj)1≤j≤N,0≤a≤3, where ∂0 := ∂t and ∂k := ∂xk for 1 ≤ k ≤ 3
(we understand this relation as mj > 0 for all j if N0 = N, and mj = 0 for all j if N0 = 0)
Summary
We consider the Cauchy problem for the following system of nonlinear wave and Klein-Gordon equations:. N , where u = (uj)1≤j≤N is an RN -valued unknown function of (t, x) ∈ (0, ∞) × R3, := ∂t2 − ∆, and mj ≥ 0. ∂u stands for the first derivatives of u, that is to say, ∂u := (∂auj)1≤j≤N,0≤a≤3, where ∂0 := ∂t and ∂k := ∂xk for 1 ≤ k ≤ 3. The nonlinear term F = (Fj)1≤j≤N is a smooth function of (u, ∂u) with. (1) is called a nonlinear wave equation when mj = 0, and a nonlinear Klein-. We call vj as the Klein-Gordon component, and wj as the wave component. Klein-Gordon equation, global existence, null condition, small initial data.
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