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

Czernous W.

Opuscula Mathematica

2014

Vol. 34, no. 2

291--310

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 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)×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.

Classical solutions of initial problems for quasilinear partial functional differential equations of the first order

Czernous W.

Opuscula Mathematica

2006

Vol. 26, no. 1

13-29

We consider the initial problem for a quasilinear partial functional differential equation of the first order [formula], z(t, x) = varphi(t, x) ((t, x) ∈ [-h0, 0] x Rn) where z(t, x) : [-h0, 0] x [-h, h] → R is a function defined by z(t, x) (τ, ξ) = z(t + τ, + ξ) for (τ, ξ) ∈ [-h0, 0] x [-h, h]. Using the method of bicharacteristics and the fixed-point theorem we prove, under suitable assumptions, a theorem on the local existence and uniqueness of classical solutions of the problem and its continuous dependence on the initial condition.

On the nonlocal boundary value problem for first order quasilinear partial functional differential equations

Człapiński T.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

1999

[Z] 39

57-73

Quasilinear partial functional differential equations of the first order with nonlocal boundary condi-tions are considered. A theorem on the local existence and uniqueness of classical solutions is given. The method of bicharacteristics is used to transform the nonlocal boundary value problem into a functional integral equation and the Banach fixed-point principle to find its solution.