Opuscula Math. 26, no. 1 (2006), 13-29

Opuscula Mathematica

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

Wojciech Czernous

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.

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• Wojciech Czernous
• University of Gdańsk, Institute of Mathematics, Wit Stwosz Street 57, 80-952 Gdańsk, Poland
• Received: 2005-10-31.

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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