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A higher order nonlinear hyperbolic system of generalized sine-Gordon type. (Chinese) Zbl 0538.35055
Consider high order hyperbolic partial differential equations $$u_{tt}+(-1)^ MAu_{x^{2M}}=f(x,t,u,u_ x,...,u_{xM},u_ t)$$ where u is a J-vector, M a $$J\times J$$ matrix and the J-vector $$f=(f_ j)$$ is of the form: $f_ j=\sum^{[M/2]}_{m=0}(-1)^{1+m}D^ m_ x(\partial F/\partial p_{mj})+g_ j$ for some function $$F=F(u,u_ x,...,u_ x^{[M/2]})$$, $$p_{mj}\equiv D^ m_ xu_ j$$. The above system generalizes the following sine-Gordon equation and also nonlinear forced string equations $$u_{tt}-u_{xx}=\sin u,\quad u_{tt}- u_{xx}+u^ 3=0.$$ Using Galerkin approximation and energy estimates, the existence and regularity of periodic solutions for initial value problems $$u(x+2D,t)=u(x,t),$$ $$t\geq 0$$; $$u(x,0)=\phi(x),\quad u_ t(x,0)=\psi(x), \phi$$ and $$\psi$$ given periodic data, are obtained. More general initial value problem is also solved.
Reviewer: T.-P.Liu

##### MSC:
 35L75 Higher-order nonlinear hyperbolic equations 35A15 Variational methods applied to PDEs 35B10 Periodic solutions to PDEs 35L30 Initial value problems for higher-order hyperbolic equations 35B45 A priori estimates in context of PDEs 35A35 Theoretical approximation in context of PDEs 35B65 Smoothness and regularity of solutions to PDEs