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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

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