Global smooth solvability of a nonlinear boundary value problem for a quasilinear hyperbolic system (Q1063180)

The paper deals with the system (1) \(u_ t+uu_ x=0\), \(v_ t-vv_ x=0\) for \(0\leq x\leq 1\), \(t\geq 0\), with boundary conditions \((2)\quad u(0,t)=v(0,t),\quad v(1,t)=\phi (u(1,t)),\) \(t\geq 0\), and usual initial conditions. Using the method of characteristics the problem is transformed to the functional equation \(z(t+z(t))=\psi (z(t)),\quad \psi (z)=2/\phi (2/z),\) which is used for the proof of global existence and uniqueness of the continuously differentiable solution of the problem for a certain class of functions \(\phi\) and initial conditions. The asymptotic behaviour of the solution is studied. Analogous results are proved for the system (1) with perturbed boundary condition \((2')\quad u(0,t)=v(0,t),\quad \epsilon v_ t(1,t)+v(1,t)=\phi (u(1,t)),\) \(t\geq 0\), \(\epsilon >0\).