# zbMATH — the first resource for mathematics

Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions. (English) Zbl 0597.35100
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
Full Text:
##### References:
  Uniform decay estimates and Lorentz invariance of the classical wave equation, preprint.  Von Wahl, Math. Z. 120 pp 93– (1971)  John, Comm. Pure Appl. Math. 37 pp 443– (1984)  Klainerman, Comm. Pure Appl. Math. 33 pp 43– (1980)  John, Comm. Pure Appl. Math. 34 pp 29– (1981)  Shatah, Comm. Pure Appl. Math. 38 pp 685– (1985)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.