Opuscula Mathematica

Opuscula Math.
 26
, no. 1
 (), 13-29
Opuscula Mathematica

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.
Keywords: partial functional differential equations, classical solutions, local existence, bicharacteristics.
Mathematics Subject Classification: 35R10, 35L45.
Cite this article as:
Wojciech Czernous, Classical solutions of initial problems for quasilinear partial functional differential equations of the first order, Opuscula Math. 26, no. 1 (2006), 13-29
 
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ISSN 1232−9274, e-ISSN 2300−6919, DOI https://doi.org/10.7494/OpMath
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