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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: $$(\square+1)u=F(u,u_t,u_x,u_{tx},u_{xx})\quad \text{for}\quad (t,x)\in(0,\infty)\times{\Bbb R},\tag 1$$ $$u(0,x)=\varepsilon f(x),\quad u_t(0,x)=\varepsilon g(x)\quad \text{for}\quad x\in{\Bbb R}, \tag 2$$ where $\square=(\partial^2/\partial t^2)-(\partial^2/\partial x^2)$, $u_t=(\partial/\partial t)u$, $u_x=(\partial/\partial x)u$, etc., $\varepsilon$ is a small positive parameter, $f$, $g\in C_0^\infty({\Bbb R})$ and the nonlinear function $F(\dots)=F(\lambda)$ with $\lambda=(u,u_t,u_x,u_{tx},u_{xx})$ can be written in the form $F(\lambda)=\sum_{i=1}^{10}c_iG_i(\lambda) +N(\lambda)+H(\lambda)$ in a neighborhood of $\lambda=0$; here $\{c_i\}_{i=1}^{10}\subset{\Bbb R}$, $\{G_i\}_{i=1}^{10}$ are linearly independent polynomials of $\lambda$ defined by $$\align G_1(\lambda)&=u(-u^2+3u_t^2-3u_x^2),\\ G_2(\lambda)&=u_t(-3u^2+u_t^2-u_x^2)+2u(u_tu_{xx}-u_xu_{tx}),\\ G_3(\lambda)&=u_x(-u^2+u_t^2-u_x^2)+2u(u_tu_{tx}-u_xu_{tx}),\\ G_4(\lambda)&=u^3-2u^2u_{xx}-3uu_t^2+2u_t^2u_{xx}-2u_tu_xu_{tx} -u(u_{tx}^2-u_{xx}^2),\\ G_5(\lambda)&=(-u^2+u_t^2-u_x^2)u_{tx}-2uu_tu_x,\\ G_6(\lambda)&=-uu_x^2+2u_x(u_tu_{tx}-u_xu_{xx})+u(u_{tx}^2-u_{xx}^2),\\ G_7(\lambda) &=3u^2u_t-6uu_tu_{xx}-u_t^3-3u_t(u_{tx}^2-u_{xx}^2),\\ G_8(\lambda)&=u^2u_x-2uu_tu_{tx}-2uu_xu_{xx}-u_t^2u_x-u_x(u_{tx}^2-u_{xx}^2),\\ G_9(\lambda)&=-2uu_xu_{tx}-u_tu_x^2+u_t(u_{tx}^2-u_{xx}^2),\\ G_{10}(\lambda)&=-u_x^3+3u_x(u_{tx}^2-u_{xx}^2),\endalign$$ $N$ is of the form $N(\lambda)=P_1(\lambda)(u_tu_{tx}-u_xu_{xx}+uu_x)+ P_2(\lambda)(u_tu_{xx}-u_xu_{tx})+P_3(\lambda)(u_{tx}^2-u_{xx}^2+uu_{xx})$, $\{P_i(\lambda)\}_{i=1}^3$ are homogeneous polynomials of degree $1$, and $H(\lambda)$ is a smooth function of $\lambda$ of degree $4$, i.e. $H(\lambda)=O(|\lambda|^4)$ near $\lambda=0$. The main result of the paper is the following: Theorem 1.1. For any integer $k\ge 15$ there exists a positive constant $\varepsilon_0$ such that for any $\varepsilon\in(0,\varepsilon_0]$ the Cauchy problem \text{(1)}, \text{(2)} has a unique classical solution $u=u(t,x)\in C^\infty([0,\infty)\times{\Bbb R})$. Moreover, the solution $u$ has a free profile in the sense that there exist $u_0\in H^{k+1}({\Bbb R})$ and $u_1\in H^k({\Bbb R})$ ($H^k({\Bbb R})$ is the usual Sobolev space) such that $$\|(u-U)(t,\cdot)\|_{H^{k+1}({\Bbb R})}+\|\partial_t(u-U)(t,\cdot)\|_{H^k({\Bbb R})}\to 0\quad\text{as}\quad t\to+\infty,$$ where $U$ is the solution to the Cauchy problem for the linear Klein-Gordon equation $(\square+1)U=0$ in $(0,\infty)\times{\Bbb R}$ with initial data $U(0,x)=u_0(x)$ and $U_t(0,x)=u_1(x)$, $x\in{\Bbb R}$.

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