Electro‐reaction‐diffusion systems with nonlocal constraints

Abstract

The paper deals with equations modelling the redistribution of charged particles by reactions, drift and diffusion processes. The corresponding model equations contain parabolic PDEs for the densities of mobile species, ODEs for the densities of immobile species, a possibly nonlinear, nonlocal Poisson equation and some nonlocal constraints. Based on applications to semiconductor technology these equations have to be investigated for non-smooth data and kinetic coefficients which depend on the state variables.In two space dimensions we discuss the steady states of the system, we prove energy estimates, global a priori estimates and give a global existence result. The modelWe consider a widely general electro-reaction-diffusion system for m species X i . Let z 0 denote the electrostatic potential, and additionally let the time functions z 1 , . . . , z k be internal parameters (e.g. electrochemical potentials of species, which have been eliminated by foregoing considerations). For examples of such model equations see §8. We write z = (z 0 , z 1 , . . . , z k ). Moreover, let p i (x, z) be suitable chosen reference densities such thatwhere the functions P i depend on z only. We call the quantities We assume that the species X 1 , . . . , X l are mobile which means that they underlie drift-diffusion processes. The second group of species X l+1 , . . . , X m is assumed to be immobile, no drift-diffusion processes take place. The spatial and temporal variation of their densities is realised by reactions with mobile species and their transport processes. If the mobile and immobile species are charged they contribute to the charge density and influence the electrostatic potential z 0 . The transport of the mobile species is based on the drift-diffusion flux densitieswith diffusion coefficients D i depending on the spatial variable and on the state (b, z) and with charge numbers(1.1)The continuity equations for the mobile and immobile species contain source terms resulting from reversible mass action type reactions of the form

On Uniform Decay of the Entropy for Reaction–Diffusion Systems

“…It was observed in [10,13,19] that under certain structural assumption (see [19, Assumption (I,iv)] for a general sufficient condition) the exponential decay to equilibrium persists even in the case when some diffusion constants δ i vanish. Our method can also be used in such cases.…”

“…This is the much simpler, finite-dimensional analog of the results in [19,22]. We need the additional assumption that there are no steady states ofȧ = −R(a) on the boundary of C q .…”

“…Moreover, Theorem 4.5 covers the case δ 2 = 0 and provides the q-dependent lower bound λ(q) ≥ 1 100 min{r 1 , 2k} min{10q, 7}. Such systems with a degenerate diffusion tensor D were also considered in [10,13,19].…”

“…Because of the coupling to the electrostatic interaction, their analysis is restricted to two-dimensional domains Ω, but it is clear that without this coupling the result of [22,Thm. 5.2] or [19,Thm. 4.16] transfers to our general setting.…”

“…The methods developed originally in [24][25][26] and further refined in [19,22] show that under quite general conditions there exists a λ(q) > 0, depending only on the conserved quantities q, such that…”