For a diagonal first-order system (d_t + Lambda d_x)u = f(x,t,u) on a bounded space interval, the inflowing components at each end may be prescribed by boundary functions that are both nonlinear and nonlocal, depending on the traces of every component at both ends. Given zero-order and first-order compatibility between the initial and boundary data, such a problem has a unique classical C1 solution for all time, obtained by rewriting it as an integral operator along characteristics and contracting on short time slabs that are then iterated to arbitrary horizons. The same argument survives the loss of the Lipschitz hypothesis: gradients growing like the fourth root of a double logarithm of a polynomial still give global solutions, which places the frontier between regular behaviour and finite-time blow-up between norm(u) log log norm(u) and quadratic growth.
A first-order hyperbolic system here is n coupled PDEs (d_t + Lambda(x,t) d_x)u = f(x,t,u) on the strip 0<x<1, t>0, with Lambda diagonal and k of its characteristic speeds negative and the remaining …