Global Existence Result for Pair Diffusion Models

In this paper we prove a global existence result for pair diffusion models in two dimensions. Such models describe the transport of charged particles in semiconductor heterostructures. The underlying model equations are continuity equations for mobile and immobile species coupled with a nonlinear Poisson equation. The continuity equations for the mobile species are nonlinear parabolic PDEs involving drift, diffusion and reaction terms, the corresponding equations for the immobile species are ODEs containing reaction terms only. Forced by applications to semiconductor technology these equations have to be considered with non-smooth data and kinetic coefficients additionally depending on the state variables. Our proof is based on regularizations, on a priori estimates which are obtained by energy estimates and Moser iteration as well as on existence results for the regularized problems. These are obtained by applying the Banach Fixed Point Theorem for the equations of the immobile species, and the Schauder Fixed Point Theorem for the equations of the mobile species. 1. The model. Pair diffusion models describe the transport of charged particles (dopant atoms, point defects, dopant-defect pairs) in semiconductors [4, 7]. In [12] we specified a typical mathematical model of this kind which we shall study in this paper, too. We consider m species Xi. The first l ≤ m species are mobile, the other ones are immobile. We denote by ui, p0i, bi = ui/p0i the density, some reference density, the chemical activity of the i-th species, and by ψ some additional potential. The initial boundary value problem which we are interested in reads as follows: ∂ui ∂t + ∇ · ji + ∑ (α,β)∈RΩ (αi − βi)R αβ = 0 on (0,∞) × Ω , ν · ji − ∑ (α,β)∈RΓ (αi − βi)R αβ = 0 on (0,∞) × Γ , i = 1, . . . , l ; ∂ui ∂t + ∑ (α,β)∈RΩ (αi − βi)R αβ = 0 on (0,∞) × Ω , i = l + 1, . . . ,m ; −∇ · (ε∇ψ) + e(·, ψ) − m ∑ i=1 Qi(ψ)ui = f on (0,∞) × Ω , ν · (ε∇ψ) = 0 on (0,∞) × Γ ; ui(0) = Ui on Ω , i = 1, . . . ,m . Here Γ denotes the boundary of the domain Ω ⊂ R, and ν is the outer unit normal. The transport of the mobile species is governed by the drift-diffusion flux densities ji = −Di(·, b, ψ) p0i (∇bi +Qi(ψ) bi∇ψ) , i = 1, . . . , l , where Qi denotes the charge number of the i-th species which depends on ψ, and Di is the diffusivity which depends on the state variables b = (b1, . . . , bm) and ψ. All 2 A. GLITZKY AND R. HÜNLICH m continuity equations contain volume source terms generated by mass action type reactions of the form α1X1 + . . . + αmXm β1X1 + . . . + βmXm , (α, β) ∈ R , where α, β ∈ Z+ are the vectors of stoichiometric coefficients, and RΩ describes the set of all reactions under consideration. The corresponding reaction rates R αβ are given by R αβ = k Ω αβ(x, b1, . . . , bm, ψ) [ m ∏