On a nonclassical problem for some quasilinear hyperbolic equation (Q1849043)

The authors study a nonclassical problem for a first-order quasilinear equation \[ u_t+ \left({\lambda(t)\over p+1}|u|^p u\right)_x=Lu \] in the rectangle \(0<x<l\), \(0<t<T\), with the nonlocal boundary conditions \[ u(x,0)=\beta u(x,T), \;u(0,t)=bu(l,t). \] The authors find simple necessary and sufficient conditions for the existence of generalized solutions and show that any generalized solution is necessarily classical, unique (up to a constant multiplier), and can be found explicitly. The investigation is based on techniques of trajectories of solution expansion. Some geometric properties of such trajectories are also described.