@Article{Czernous2006,
author = "Czernous, Wojciech",
title = "Classical solutions of initial problems for quasilinear partial functional differential equations of the first order",
journal = "Opuscula Math.",
year = "2006",
volume = "26",
number = "1",
pages = "13--29",
doi = "http://dx.doi.org/10.7494/OpMath.2006.26.1.13",
url = "http://dx.doi.org/10.7494/OpMath.2006.26.1.13",
abstract = "We consider the initial problem for a quasilinear partial functional differential equation of the first order \[ \partial_t z(t,x)+\sum_{i=1}^nf_i(t,x,z_{(t,x)})\partial_{x_i} z(t,x)=G(t,x,z_{(t,x)}),\\ z(t,x)=\varphi(t,x)\;\;((t,x)\in[-h_0,0]\times R^n)\] where \(z_{(t,x)}\colon\,[-h_0,0]\times[-h,h]\to R\) is a function defined by \(z_{(t,x)}(\tau,\xi)=z(t+\tau,x+\xi)\) for \((\tau,\xi)\in[-h_0,0]\times[-h,h]\). 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.",
keywords = "partial functional differential equations, classical solutions, local existence, bicharacteristics",
issn = "1232-9274"
}