A two-point problem for systems of hyperbolic equations
The work deals with a two-dimensional nonlinear hyperbolic system of the form \[ (1)\quad \partial^ 2u/\partial t\partial x=P(t,x)u(t,x)+f(t,x,u,u_ t), \] where \(x\in [-a,a]\), \(t\in [0,T]\). The initial and boundary conditions for (1) are given by \[ (2)\quad u(0,x)=u_ 0(0)+v(x),\quad Au(0,x)+Cu(T,x)=w(x),\quad u(t,0)=u_ 0(t)+v(0). \] Here \(u,v,u_ 0,w,f\) are vector-valued functions, while P,A,C are matrices. Assuming f is Lipschitz continuous and the matrices P,A,C satisfy suitable assumptions, the authors prove the existence of a solution and construct an approximation sequence of functions tending to the solution. An estimate of the error of the approximation sequence is also established.
scientific article; zbMATH DE number 4210655