On nonlinear systems consisting of different types of differential equations
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ON NONLINEAR SYSTEMS CONSISTING OF DIFFERENT TYPES OF DIFFERENTIAL EQUATIONS ´szlo ´ Simon1 La
Dedicated to the memory of Professor Mikl´ os Farkas
Institute of Mathematics, E¨ otv¨ os Lor´ and University H-1117 Budapest, P´ azm´ any P. s´et´ any 1/c E-mail: [email protected] (Received August 3, 2007; Accepted October 15, 2007)
Abstract
We consider a system consisting of a quasilinear parabolic equation and a first order ordinary differential equation where both equations contain functional dependence on the unknown functions. Then we consider a system which consists of a quasilinear parabolic partial differential equation, a first order ordinary differential equation and an elliptic partial differential equation. These systems were motivated by models describing diffusion and transport in porous media with variable porosity.
Introduction It is well known that in different models of diffusion, population dynamics give rise to nonlinear parabolic differential equations and systems of such equations. Sometimes the diffusion coefficient depends on a nonlocal quantity or on the distribution of other materials, species, respectively. The first case was considered by M. Chipot, L. Molinet, B. Lovat in [5], [6]. M. Farkas considered in [9] crossdiffusion which is the second case. J. I. D´ıaz and G. Hetzer studied a climate model which was a quasilinear functional reaction-diffusion equation containing a delay term [8]. U. Hornung, W. J¨ ager, A. Mikeli˘c in [10], [11] considered a system of Mathematics subject classification numbers: 35Q35, 35R10. Key words and phrases: diffusion and transport in porous media, functional partial differential equations, systems of differential equations. 1 Supported by the Hungarian NFSR under grant OTKA T 049819. 0031-5303/2008/$20.00
c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
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L. SIMON
nonlinear parabolic functional differential equations when modelling diffusion, convection, absorption reaction of chemicals in porous media and reactive transport through an array of cells with semipermeable membranes. The present paper is a summary of works on systems consisting of different types of PDEs, functional PDEs and ODEs, based on the theory of monotone type operators. These works were motivated by models which arise when modelling diffusion and transport in porous media with variable porosity. In [7] S. Cinca investigated the following model: div[K(x, w)(∇p(t, x) + g(t, x))] = 0 in Ω, ∂(n(t, x)u(t, x)) ∂t = div[D(x, w)∇u(t, x) + K(x, w)(∇p(t, x) + g(t, x))u(t, x)] in (0, T ) × Ω,
(0.1) (0.2)
∂w = n(t, x)u(t, x)F (u, w) in (0, T ) × Ω (0.3) ∂t with some initial and boundary conditions where u(t, x) is the concentration of the solute in x ∈ Ω ⊂ RN , in time t, n(t, x) is the porosity, w = 1 − n and p(t, x) is the pressure (these are considered unknown functions). The hydraulic conductivity matrix K, the diffusion matrix D and the functions g, F are considered known. The aim of [7] was to prove the existence of a solution of system (0.1)–(0.3) and to show a numeric
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