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

Wojciech Czernous
2014 Opuscula Mathematica  
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 Ω with Lipschitz differentiable boundary. Well-posedness is then reliant on a variant of the normal vector condition. There is a neighbourhood of ∂Ω with the property that if a characteristic trajectory has a point therein, then its every
more » ... 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) × Ω 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.
doi:10.7494/opmath.2014.34.2.291 fatcat:nptx32vekngyblvznaicllu3xq