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A note on global existence of solutions to nonlinear Klein-Gordon equations in one space dimension. (English) Zbl 0945.35056

In this paper the following Cauchy problem for the nonlinear Klein-Gordon equation is considered:

(+1)u=F(u,u t ,u x ,u tx ,u xx )for(t,x)(0,)×,(1)
u(0,x)=εf(x),u t (0,x)=εg(x)forx,(2)

where =( 2 /t 2 )-( 2 /x 2 ), u t =(/t)u, u x =(/x)u, etc., ε is a small positive parameter, f, gC 0 () and the nonlinear function F()=F(λ) with λ=(u,u t ,u x ,u tx ,u xx ) can be written in the form F(λ)= i=1 10 c i G i (λ)+N(λ)+H(λ) in a neighborhood of λ=0; here {c i } i=1 10 , {G i } i=1 10 are linearly independent polynomials of λ defined by

G 1 (λ)=u(-u 2 +3u t 2 -3u x 2 ),G 2 (λ)=u t (-3u 2 +u t 2 -u x 2 )+2u(u t u xx -u x u tx ),G 3 (λ)=u x (-u 2 +u t 2 -u x 2 )+2u(u t u tx -u x u tx ),G 4 (λ)=u 3 -2u 2 u xx -3uu t 2 +2u t 2 u xx -2u t u x u tx -u(u tx 2 -u xx 2 ),G 5 (λ)=(-u 2 +u t 2 -u x 2 )u tx -2uu t u x ,G 6 (λ)=-uu x 2 +2u x (u t u tx -u x u xx )+u(u tx 2 -u xx 2 ),G 7 (λ)=3u 2 u t -6uu t u xx -u t 3 -3u t (u tx 2 -u xx 2 ),G 8 (λ)=u 2 u x -2uu t u tx -2uu x u xx -u t 2 u x -u x (u tx 2 -u xx 2 ),G 9 (λ)=-2uu x u tx -u t u x 2 +u t (u tx 2 -u xx 2 ),G 10 (λ)=-u x 3 +3u x (u tx 2 -u xx 2 ),

N is of the form N(λ)=P 1 (λ)(u t u tx -u x u xx +uu x )+P 2 (λ)(u t u xx -u x u tx )+P 3 (λ)(u tx 2 -u xx 2 +uu xx ), {P i (λ)} i=1 3 are homogeneous polynomials of degree 1, and H(λ) is a smooth function of λ of degree 4, i.e. H(λ)=O(|λ| 4 ) near λ=0. The main result of the paper is the following:

Theorem 1.1. For any integer k15 there exists a positive constant ε 0 such that for any ε(0,ε 0 ] the Cauchy problem (1), (2) has a unique classical solution u=u(t,x)C ([0,)×). Moreover, the solution u has a free profile in the sense that there exist u 0 H k+1 () and u 1 H k () (H k () is the usual Sobolev space) such that

(u-U)(t,·) H k+1 () + t (u-U)(t,·) H k () 0ast+,

where U is the solution to the Cauchy problem for the linear Klein-Gordon equation (+1)U=0 in (0,)× with initial data U(0,x)=u 0 (x) and U t (0,x)=u 1 (x), x.

MSC:
35L70Nonlinear second-order hyperbolic equations
35L15Second order hyperbolic equations, initial value problems