Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions. (English) Zbl 0597.35100

Klainerman, Sergiu

Commun. Pure Appl. Math. 38, 631-641 (1985).

It is shown that the initial value problem \[ (\partial^ 2_ t- \partial^ 2_ 1-\partial^ 2_ 2-\partial^ 2_ 3)u(t,x)+u(t,x)=F(u,u',u'');\quad u(x,0)=\epsilon f(x),\quad u_ t=\epsilon g(x) \] with \(x\in R^ 3\) and \(t\geq 0\) possesses a unique solution \(u\in C^{\infty}((0,\infty)\times {\mathbb{R}}^ 3)\) for all \(\epsilon\) with \(0<\epsilon <\epsilon_ 0\), if \(\epsilon_ 0\) is sufficiently small. Here F(u,u’,u”) is a smooth function of the solution u and its first derivatives, and if is assumed that F together with its first derivatives vanishes at \((u,u',u'')=0\). Moreover, it is shown that u(t,x) decays as \(t^{-5/4}\) for large t, uniformly in \(x\in {\mathbb{R}}^ 3\). The number \(\epsilon_ 0\) only depends on a finite number of derivatives of f, g and F.
The proof relies essentially on the following result for solutions of the linear Klein-Gordon equation derived in the paper. The solution u of \[ (\partial^ 2_ t-\partial^ 2_ 1-\partial^ 2_ 2-\partial^ 2_ 3)u+u=g;\quad u(x,0)=u_ 0(x),\quad u_ t(x,0)=u_ 1(x) \] subject to the condition \(u,g=0\) for \(| x| \geq t+1\) satisfies \(| u(t,x)| \leq c(1+t)^{-5/4} K\) where K can be estimated by suitable Sobolev norms of the initial values and of g.

Reviewer: H.D.Alber

MSC:

35Q99	Partial differential equations of mathematical physics and other areas of application
35L05	Wave equation
35B40	Asymptotic behavior of solutions to PDEs
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

nonlinear wave equations; Lorentz invariance properties; nonlinear; Klein-Gordon equation; initial value problem; linear Klein-Gordon equation; Sobolev norms

References:

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