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

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.