Classical solutions of mixed problems for quasilinear first order PFDEs on a cylindrical domain

Abstract

We abandon the setting of the domain as a Cartesian product of real intervals, customary for first order PFDEs (partial functional differential equations) with initial boundary conditions. We give a new set of conditions on the possibly unbounded domain \(\Omega\) with Lipschitz differentiable boundary. Well-posedness is then reliant on a variant of the normal vector condition. There is a neighbourhood of \(\partial\Omega\) with the property that if a characteristic trajectory has a point therein, then its every earlier point lies there as well. With local assumptions on coefficients and on the free term, we prove existence and Lipschitz dependence on data of classical solutions on \((0,c)\times\Omega\) to the initial boundary value problem, for small \(c\). Regularity of solutions matches this domain, and the proof uses the Banach fixed-point theorem. Our general model of functional dependence covers problems with deviating arguments and integro-differential equations.