Differential Equations and Computational Simulations III J. Graef, R. Shivaji, B. Soni, & J. Zhu (Editors) Electronic Journal of Differential Equations, Conference 01, 1997, pp. 1–9. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp 147.26.103.110 or 129.120.3.113 (login: ftp)

HÖLDER SOLUTIONS FOR THE AMORPHOUS

SILICON SYSTEM AND RELATED PROBLEMS

Walter Allegretto, Yanping Lin, & Aihui Zhou

Abstract. We present existence of solutions and other results for the partial differential equation system with memory which models amorphous silicon devices and related problems in ℝ^{3}. Our approach employs only classical estimates and Degree Theory; it shows the existence of C^{α,α/2} solutions for some α > 0. In view of the mixed boundary conditions, this is the maximum regularity that can be expected.

1. Introduction

In the past few years micro-electronic devices employing amorphous silicon as the semiconductor material have shown promise in a variety of applications such as liquid crystal displays, image sensors and solar cells. The mathematical model usually employed to simulate such devices involves drift-diffusion equations as well as equations describing the density of trapped charges, [3, 8]. The latter may be explicitly integrated in time, giving a drift-diffusion system with integral (i.e. “memory”) terms. Specifically, if we assume only one trapped charge state and set all mathematically irrelevant coefficients to unity, we obtain:

−Δφ = (p − n + C_{1}(x) + ∫_{0}^{t}(p − n)e^{− ∫_{ξ}^{t}(n+p+2)dη}dξ)  (1’)
\frac{∂n}{∂t} − ∇[D_{n}(x, t, n, p, |∇φ|)∇n − nμ_{n}(x, n, p, |∇φ|)∇φ]  (2’)

= 1 − ∫_{0}^{t}(p − n)e^{− ∫_{ξ}^{t}(n+p+2)dη}dξ − n[1 + ∫_{0}^{t}(p − n)e^{− ∫_{ξ}^{t}(n+p+2)dη}dξ]

\frac{∂p}{∂t} − ∇[D_{p}(x, n, p, |∇φ|)∇p + pμ_{p}(x, n, p, |∇φ|)∇φ]  (3’)

= 1 + ∫_{0}^{t}(p − n)e^{− ∫_{ξ}^{t}(n+p+2)dη}dξ − p[1 − ∫_{0}^{t}(p − n)e^{− ∫_{ξ}^{t}(n+p+2)dη}dξ]

to be satisfied in a smooth domain Ω ⊂ R^{3}. We observe that the factor “2” is present in the various integrals in equations (1’)-(3’) to ensure charge conservation, [8]. With equations (1’)–(3’) we associate initial/mixed boundary conditions as follows: Let ∂Ω = Γ_{D} ∪ Γ_{N} with Γ_{D} a smooth nonempty closed sub-manifold in which Dirichlet conditions are to hold:

1991 Mathematics Subject Classifications: 35J60.

Key words and phrases: Reaction, diffusion, semiconductor, Holder continuoussolutions.

\begin{matrix}c \\ ◯ \end{matrix}1998 Southwest Texas State University and University of North Texas.

Published November 12, 1998.

φ(x, t) = \overline{φ}(x), n(x, t) = \overline{n}(x), p(x, t) = \overline{p}(x)

for all t, while Neumann conditions are to hold on Γ_{N} = ∂Ω − Γ_{D} :

\frac{∂φ}{∂ν⃗} = \frac{∂n}{∂ν⃗} = \frac{∂p}{∂ν⃗} = 0 .

With n, p we also associate initial conditions. Both to avoid technical difficulties and in keeping with the situation in the physical problems we also ask that n(x, 0) = \overline{n}(x), p(x, 0) = \overline{p}(x), with \overline{n}(x), \overline{p}(x) ∈ C^{1}(\overline{Ω}) and \overline{n}, \overline{p} ≥ 0. These may be weakened in what follows without essential proof changes, but they do simplify the presentation.

For the same reason, we assume throughout the paper that all equation coefficients are smooth in their variables. Note that since n, p are densities we shall only seek solutions with n, p ≥ 0. The behavior of the equation coefficients in (n, p) for n, p < 0 can thus be chosen for convenience. Some regularity is also needed for ∂(Γ_{N}) ∩ Γ_{D}. Intuitively if x ∈ ∂(Γ_{N}) ∩ Γ_{D} and N is a small neighborhood of x, we require that regularity considerations for N ∩ ∂Ω be reduced via bi-Lipschitz (resp. smooth) coordinate maps to similar problems on quarter-spheres (resp. hemispheres). The reader interested in the explicit formulations of such conditions may find them for example in [7, 11, 13, 16].

In the next sections we introduce and analyze a system of equations which contains as special cases not only (1’)–(3’) but also the standard drift-diffusion equations. While we are not aware of an earlier study of such a system with “memory terms”, we point out there have been numerous results on the non-augmented system in recent years. It is not possible to present a detailed analysis of previous results here, but we refer the interested reader in particular to the papers [2, 4, 5], to the books [9, 10, 12] and the references therein. It is the paper by Fang and Ito [2], and da Veiga, [15] which furnished in part the motivation for this work, and which are closest to the assumptions made. Indeed the regularity of n, p shown here is conjectured in [15]. In general terms, the approaches usually employed in the past are based on time discretization, on semigroup analysis, on fixed point theorems and weak solutions are found in suitable spaces. Often techniques involving maximum principles, the Einstein relations and the introduction of quasi Fermi variables, and coefficient truncation were employed. In this paper we use none of these tools, and it is not clear how useful many of these would be in our situation, given the memory term present in (1′). Instead we employ Degree Theory and work directly with C^{α,α/2} spaces. Not only does this simplify considerably the presentation but the solutions we find are of the regularity one would expect from the physical point of view. More global regularity cannot be realized in general due to the mixed boundary conditions for n, p, φ.

Our procedures are based on simple arguments involving classical results ([6, 17]) which are well-known although in themselves far from simple. As presented, the results are given for Ω ⊂ R^{3} – the physically interesting case. We conjecture that similar results hold for Ω ⊂ R^{N}, N ≠ 3.

As a final observation, note that as mentioned above, we do not make use of the Einstein Relations connecting D_{n}, D_{p} and μ_{n}, μ_{p} as we shall have no need of them. It follows that the results also hold if the system does not admit “ quasi-Fermi ” variables.

II. Analysis

Based on the model system considered in the Introduction, we introduce the following equations:

−Δφ = f(x, t, p − n, h(p, n))  (1)

\frac{∂n}{∂t} − ∇[D_{n}(x, t, n, p, |∇φ|)∇n − nμ_{n}(x, t, n, p, |∇φ|)∇φ]

= 1 − h(p, n) − n[1 + h(p, n)] + R_{n}(x, t, n, p)  (2)

\frac{∂p}{∂t} − ∇[D_{p}(x, t, n, p, |∇φ|)∇p + pμ_{p}(x, t, n, p, |∇φ|)∇φ]

= 1 + h(p, n) − p[1 − h(p, n)] + R_{p}(x, t, n, p)  (3)

with

h(p, n) = ∫_{0}^{t}(p − n)e^{− ∫_{ξ}^{t}(n+p+2)dη}dξ .

We keep the initial/boundary conditions given on (n, p, φ) in the introduction as well as the requirement that D_{n}, D_{p}, μ_{n}, μ_{p}, R_{n}, R_{p} be smooth functions of their respective arguments (at least for n, p ≥ 0). We now introduce the following growth conditions on Ω × (0, T), which may depend on T.

(A) There exist positive constants α, β such that

α ≤ D_{n}(x, t, n, p, |∇φ|), D_{p}(x, t, n, p, |∇φ|) ≤ β

(B) R_{n}(x, t, n, p) = R_{n,1}(x, t, n, p) − nR_{n,2}(x, t, n, p) with R_{n,1}, R_{n,2} nonnegative, bounded, smooth if n, p ≥ 0. We assume that R_{p} admits a similar decomposition into R_{p,1} − pR_{p,2} with and R_{p,1}, R_{p,2} nonnegative, bounded, smooth.

(C) μ_{n} = μ_{n,1} + μ_{n,2}, with μ_{n,1} a positive constant and μ_{n,2} = μ_{n,2}(x, t, n, p, |∇φ|) such that |μ_{n,2}∇φ| ≤ a_{n} for some positive constants a_{n} if (x, t) are bounded. Similarly, μ_{p} = μ_{p,1} + μ_{p,2} with 0 < μ_{p,1} constant and |μ_{p,2}∇φ| ≤ a_{p}.

(D) There exist positive smooth functions M_{1}, M_{2} of (x, t) such that

|f(x, t, ξ_{1}, ξ_{2}) − M_{1}|ξ_{1}|^{α_{2}} sign ξ_{1}| ≤ M_{2}

for some α_{2} ≥ 1 and all (x, t) ∈ Ω × [0, T], 0 ≤ ξ_{2} ≤ 1. Observe that system (1)–(3) with conditions (A – D) includes both the standard

Drift-Diffusion model and the amorphous silicon model.

We choose and fix a parameter τ with 3 < τ < 4, set Q_{T} = Ω × (0, T) and recall Ω ⊂ R^{3}. We observe the following results

Lemma 0.

(a) Let −Δu(x) = f_{1}(x) in Ω, with f_{1} ∈ L^{τ} (Ω). If u = \overline{u}(x) ∈ C^{1} in Γ_{D}, \frac{∂u}{∂n} = 0 on Γ_{N} then u ∈ H^{1,τ} (Ω) and

∥∇u∥_{L^{τ} (Ω)} ≤ C[∥f_{1}∥_{L^{τ} (Ω)} + ∥\overline{u}∥_{C^{1}(Ω)}]

(b) Let v be a generalized solution of

v_{t} − ∇[w∇v + δ⃗v] + mv = f_{2}  (4)

with 0 < α < w(x, t) < β (α, β constants) and |δ⃗|^{2}, m, f_{2} in L^{q,r}(Q_{T}) for some q ∈ (\frac{n}{2}, ∞], r ∈ (1, ∞], \frac{1}{r} + \frac{n}{2q} < 1. Suppose v satisfies the initial/boundary conditions: v = \overline{v}(x) ∈ C^{1} on \{Γ_{D} × (0, T)\} ∪ \{Ω × \{0\}\}, \frac{∂v}{∂ν⃗} = 0 on Γ_{N} × (0, T) and v is bounded in L^{2}(Q_{T}). Then there exists an α_{0} > 0 (independent of v) such that v ∈ C^{α_{0},α_{0}/2}(\overline{Q}_{T}).

(c) If v solves (4) with the given initial/boundary conditions and ∥v∥_{L^{2}(Ω)}(t) is bounded, then v is globally bounded in L^{∞}.

Proof. Part (a) is immediate from the results of Shamir, [13].

Part (b) follows from e.g. [6, Theorem 10.1, p. 205] (see also [1]) and a reflection process to establish the needed regularity on \overline{Γ}_{N} ∩Γ_{D}, [7], [16], and Part (c) follows from [6, p. 192]. More explicitly, let v satisfy (4) and suppose first that Ω_{0} ⊂⊂ Ω. Then for Ω_{0} Parts (b), (c) are found explicitly in [6]. Next, if P ∈ Γ_{N} then we map a neighborhood N of P by a bi-Lipschitz map L to a sphere S with L(Γ_{N} ∩ N) ⊂ {x | x_{3} = 0}, L(P) = 0 and L(Ω ∩ N) ⊂ {x | x_{3} > 0}. We extend v as an even function to the whole of S and the coefficients as in [14], [16] so that the extended function v̂ satisfies (the extended) (4) in S. We can now use the interior/initial results to conclude first that v̂ is bounded in L^{∞} and then that v̂ ∈ C^{α_{0},α_{0}/2} in a neighborhood of 0 × [0, T]. Applying L^{−1} then shows the result for N. If P ∈ \overline{Γ}_{N} ∩ Γ_{D} then the process is the same except now L(Ω ∩ N) ⊂ \{x | x_{2} > 0, x_{3} > 0}, L(Γ_{N} ∩ N } ⊂ {x | x_{2} = 0}, L(Γ_{D} ∩ N) ⊂ {x | x_{3} = 0}. We first extend v as an even function to the upper hemisphere and then apply the Dirichlet problem results. The results for P on the Dirichlet Boundary are in [6].

In summary, for each P ∈ \overline{Ω}, there exists a neighborhood M such that u ∈ C^{α_{0},α_{0}/2}(M×[0, T]) and thus u ∈ C^{α_{0},α_{0}/2}(\overline{Q}_{T}) by boundary regularity. The same arguments also show Part (c).

Theorem 1. There exist α_{1} > 0 and K > 0 such that all solutions of (1 – 3) in C^{α,α/2}(\overline{Q}_{T}) with 0 < α < α_{1} and n, p ≥ 0 actually satisfy

∥n∥_{C^{α_{1},α_{1}/2}} + ∥p∥_{C^{α_{1},α_{1}/2}} + ∥φ∥_{C^{α_{1},α_{1}/2}} ≤ K.

Proof. Let (n, p, φ) represent a solution in C^{α,α/2} for some α > 0. First note that

\frac{∂h}{∂t} + (n + p + 2)h = p − n.

Since p, n are assumed nonnegative and h(x, 0) = 0, we immediately conclude that |h| ≤ 1. We next show that p, n are bounded in L^{ξ}(L^{ξ}) for some large ξ. Assume without loss of generality that μ_{n,1} = μ_{p,1} = μ_{1}. Otherwise we multiply the “n equation” in procedures that follow by \frac{μ_{p,1}}{μ_{n,1}} and repeat.

Put E = max [∥\overline{n} + \overline{p}∥_{L^{∞}}, 1] and let n = Ew, p = Ez in equations (1 – 3). We then have 0 < w, z < 1 on Γ_{D} and equations (1 - 3) yield

−Δφ = f(x, t, E(z − w), h(p, n))  (5)
\frac{∂w}{∂t} − ∇[D_{n}∇w − wμ_{n}∇φ] ≤ \frac{R_{n,1} + 2}{E}  (6)
\frac{∂z}{∂t} − ∇[D_{p}∇z + zμ_{p}∇φ] ≤ \frac{R_{p,1} + 2}{E}.  (7)

Let g^{(1)} = (g − 1)^{+} for any function g, C denote an arbitrary constant and use the Steklov average of [(w^{(1)})]^{θ}, [(z^{(1)}]^{θ} as test functions in equations (6), (7) respectively for some θ > 1. Since n, p are assumed of class C^{α,α/2}, these are suitable test functions. We find from assumptions (A), (B), (C) that:

\frac{1}{(θ + 1)} ∫_{Ω}[w^{(1)}]^{θ+1}|_{t_{1}}^{t_{2}} + \frac{4θ}{(θ + 1)^{2}} ∫_{t_{1}}^{t_{2}} ∫_{Ω} α|∇ ([w^{(1)}]^{\frac{θ+1}{2}} ) |^{2}

− ∫_{t_{1}}^{t_{2}} ∫_{Ω} a_{n}θ{[w^{(1)}]^{θ} + [w^{(1)}]^{θ−1}}|∇w^{(1)}|

− ∫_{t_{1}}^{t_{2}} ∫_{Ω} μ_{1}θ[(w^{(1)})^{θ} + (w^{(1)})^{θ−1}]∇φ∇w^{(1)}

≤ ∫_{t_{1}}^{t_{2}} ∫_{Ω} \frac{C}{E} (w^{(1)})^{θ}.

We repeat with equation (7) and add to obtain

\frac{1}{(θ + 1)} ∫_{Ω} \{(w^{(1)})^{θ+1} + (z^{(1)})^{θ+1}\}|_{t_{1}}^{t_{2}}

+ \frac{4θ}{(θ + 1)^{2}} ∫_{t_{1}}^{t_{2}} ∫_{Ω} α \{|∇([w^{(1)}]^{\frac{θ+1}{2}} )|^{2} + |∇([z^{(1)}]^{\frac{θ+1}{2}} )|^{2}\}

− ∫_{t_{1}}^{t_{2}} ∫_{Ω} θ(a_{n} + b_{n})[{[w^{(1)}]^{θ} + [w^{(1)}]^{θ−1}}|∇w^{(1)}| + {[z^{(1)}]^{θ} + [z^{(1)}]^{θ−1}}|∇z^{(1)}|]

− ∫_{t_{1}}^{t_{2}} ∫_{Ω} μ_{1}θ∇φ∇[\frac{(w^{(1)})^{θ+1}}{θ + 1} + \frac{(w^{(1)})^{θ}}{θ} − \frac{(z^{(1)})^{θ+1}}{θ + 1} − \frac{(z^{(1)})^{θ}}{θ} ]

≤ ∫_{t_{1}}^{t_{2}} ∫_{Ω} \frac{C}{E} [(w^{(1)})^{θ} + (z^{(1)})^{θ}].  (8)

Let I_{1}, I_{2}, I_{3}, I_{4} denote the four integrals on the left hand side of (8). I_{3}, I_{4} can be estimated by elementary means as follows. The first part of I_{3} can be estimated by

θa_{n} ∫_{t_{1}}^{t_{2}} ∫_{Ω}{[w^{(1)}]^{θ} + [w^{(1)}]^{θ−1}}|∇w^{(1)}|

≤ \frac{2θa_{n}}{θ + 1} ∫_{t_{1}}^{t_{2}} ∫_{Ω}\{[w^{(1)}]^{\frac{θ+1}{2}} + [w^{(1)}]^{\frac{θ−1}{2}} \}|∇[(w^{(1)})^{\frac{θ+1}{2}} ]|

≤ \frac{2θa_{n}}{θ + 1}[\frac{1}{2ε} ∫_{t_{1}}^{t_{2}} ∫_{Ω}\{[w^{(1)}])^{θ+1} + [w^{(1)}]^{θ−1}\} + \frac{ε}{2} ∫_{t_{1}}^{t_{2}} ∫_{Ω} |∇[(w^{(1)})^{\frac{θ+1}{2}} ]|^{2}].

If we choose ε small enough (depending on a_{n}, α, θ) then the second integral on the right hand side has coefficient smaller than the corresponding term in I_{2}. Observe that if \frac{α}{a_{n}(θ+1)} is big enough, then we can also employ to advantage the estimate

∫_{Ω}(w^{(1)})^{θ+1} ≤ \frac{1}{ρ_{1}} ∫_{Ω} |∇[(w^{(1)})^{\frac{θ+1}{2}} ]|^{2}

where ρ_{1} denotes the least eigenvalue of −Δ with mixed boundary conditions. While this comment is irrelevant here, it is useful both for the existence of steady state solutions and of an absorbing set. The second part of I_{3} is treated identically, with z replacing w. Next:

−I_{4} = ∫_{t_{1}}^{t_{2}} ∫_{Ω} μ_{1}θ\{− \frac{(w^{(1)})^{θ+1}}{θ + 1} − \frac{(w^{(1)})^{θ}}{θ} + \frac{(z^{(1)})^{θ+1}}{θ + 1} + \frac{(z^{(1)})^{θ}}{θ} \}f(x, t, E(z −w), h)

Without loss of generality, at any given point (x, t) we may first assume z^{(1)} > w^{(1)} with z^{(1)} > 0 and also note that (z − w)^{α_{2}} ≥ (z^{(1)} − w^{(1)})^{α_{2}} and recall α_{2} ≥ 1. We then have from (5)

\frac{1}{θ + 1}[(z^{(1)})^{θ+1} − (w^{(1)})^{θ+1}][M_{1}E^{α_{2}}(z^{(1)} − w^{(1)})^{α_{2}} − M_{2}]

≥ \frac{1}{θ + 1}[(z^{(1)})^{θ+1} − (w^{(1)})^{θ+1}][M_{1}E^{α_{2}}(z^{(1)} − w^{(1)}) − (M_{2} + M_{1}E^{α_{2}})]

≥ −\frac{1}{θ + 1} \frac{[(z^{(1)})^{θ+1} − (w^{(1)})^{θ+1}]}{(z^{(1)} − w^{(1)})} \frac{(M_{1}E^{α_{2}} + M_{2})^{2}}{4M_{1}E^{α_{2}}}

≥ −\frac{(M_{1}E^{α_{2}} + M_{2})^{2}}{4M_{1}E^{α_{2}}} [(z^{(1)})^{θ} + (w^{(1)})^{θ}].

An identical estimate, with θ replaced by θ − 1, holds for the other two terms in the integrand of I_{4} and for the points where w^{(1)} > z^{(1)}. Thus:

−I_{4} ≥ −C ∫_{t_{1}}^{t_{2}} ∫_{Ω}{(z^{(1)})^{θ} + (w^{(1)})^{θ} + (z^{(1)})^{θ−1} + (w^{(1)})^{θ−1}}

with a calculable constant C. In summary, setting s = (w^{(1)})^{\frac{θ+1}{2}} and r = (z^{(1)})^{\frac{θ+1}{2}} , we obtain from equation (8):

∫_{Ω}(s^{2} + r^{2})|_{t_{1}}^{t_{2}} + c_{0} ∫_{t_{1}}^{t_{2}} ∫_{Ω}[|∇s|^{2} + |∇r|^{2}] ≤ c_{1} ∫_{t_{1}}^{t_{2}} ∫_{Ω}(s^{2} + r^{2}) + c_{2}

with calculable positive constants c_{0}, c_{1}, c_{2}. We thus have that (z^{(1)})^{(θ+1)/2} and (w^{(1)})^{(θ+1)/2} are bounded in C(L^{2}) ∩ L^{2}(H^{1,2}) and thus, see e.g. [6], w^{θ+1}, z^{θ+1} are bounded in L^{10/3}(L^{10/3}), i.e., n, p are bounded in L^{ξ}(L^{ξ}) for any large chosen ξ. In particular, f is bounded in L^{ξ}(L^{ξ}) and thus |∇φ| is bounded in L^{ξ}(L^{τ}) for ξ large, where we recall 3 < τ < 4. We now employ [6] and Lemma 0 to conclude that n, p (and thus φ) are bounded in C^{α_{1},α_{1}/2} with α_{1} and bound independent of n, p.

It is useful to embed (1 – 3) and the associated boundary/initial conditions in the following system:

−Δφ = λf(x, t, p − n, h(p^{+}, n^{+}))  (9)
\frac{∂n}{∂t} − ∇[D_{n}∇n − nμ_{n}∇φ] = λ\{1 − h(p^{+}, n^{+}) − n^{+}[1 + h(p^{+}, n^{+})] + R̃_{n}\} (10)
\frac{∂p}{∂t} − ∇[D_{p}∇p + pμ_{p}∇φ] = λ\{1 + h(p^{+}, n^{+}) − p^{+}[1 − h(p^{+}, n^{+})] + R̃_{p}\} (11)

with boundary/initial Dirichlet conditions

φ = λ \overline{φ}, n = λ \overline{n}, p = λ \overline{p} on Γ_{D}; n = λ\overline{n}, p = λ\overline{p} at t = 0 (12)

and R̂_{n} = R_{n}(x, t, n^{+}, p^{+}), R̂_{p} = R_{p}(x, t, n^{+}, p^{+}). Observe that for λ = 1 and n, p ≥ 0, this reduces to the original problem, and the solutions n, p must be nonnegative by the weak maximum principle and equation (9).

Theorem 2. There exists a C^{α,α/2} solution (n, p, φ) of system (1,3) with the associated boundary/initial conditions for some α > 0 independent of (n, p, φ), with (n, p) nonnegative. If D_{n}, D_{p}, μ_{n}, μ_{p} are only functions of (x, t), the solution is unique.

Proof. We transform (9)–(12) into an operator equation in the usual way. Let (λ_{0}, n_{0}, p_{0}) be given with (n_{0}, p_{0}) ∈ C^{α,α/2} with α > 0 chosen, evaluate f at this point and calculate φ_{0} from (9). Evaluate the coefficients D_{n}, D_{p}, μ_{n}, μ_{p}, h, boundary/initial conditions and the right hand sides of (10), (11) at (λ_{0}, n_{0}, p_{0}, φ_{0}) and solve the now linear equations to obtain the new (n, p). We may express this process in the form:

λ = λ_{0}
φ = λT_{0}(n_{0}, p_{0})
(n, p) = T_{1}(n_{0}, p_{0}, T_{0}(n_{0}, p_{0}), λ).

Observe that T_{1} : C^{α,α/2} × [0, 1] → C^{α_{1},α_{1}/2} whence if we choose some α < α_{1} we have compactness, since C^{α_{1},α_{1}/2} ⊂⊂ C^{α,α/2} and the earlier estimates of Theorem 1 still hold (indeed the presence of λ helps as 0 ≤ λ ≤ 1). Finally, that T_{1} is continuous can be seen from lengthy but routine arguments. Note in particular that the compactness of T_{1} implies that continuity need only be shown C^{α,α/2} × [0, 1] → L^{2} and that the coefficients D_{n}, μ_{n}, D_{p}, μ_{p} are assumed bounded and thus the Lebesgue Convergence Theorem can be applied, much as for example, was done in [2]. Once again the framework of C^{α,α/2} spaces makes this process easier. The existence of a solution is then immediate by the Leray-Schauder Degree using λ as a homotopy parameter, [17]. The uniqueness of (n, p, φ) under the extra assumption on D_{n}, D_{p}, μ_{n}, μ_{p}, is immediate from Gronwall’s Lemma (see once again, e.g., [2]), since if we let (n, p, φ) and (n̂, p̂, φ̂) denote two solutions we then observe the estimate:

|h(n, p) − h(n̂, p̂)| = | ∫_{0}^{t}((p − p̂) − (n − n̂))e^{− ∫_{ξ}^{t}(p+n+2)dξ}
+ ∫_{0}^{t}(p̂− n̂)[e^{− ∫_{ξ}^{t}(p+n+2)} − e^{− ∫_{ξ}^{t}(p̂+n̂+2)}]|
≤ [C_{0} + C_{1}(n, p, n̂, p̂)t] ∫_{0}^{t} |p − p̂| + |n − n̂|

for some constant C_{0}. Choose T and let 0 ≤ t ≤ t_{1} < T. In view of the assumed regularity of the coefficient functions and employing the equations solved by φ, φ̂, we have

\frac{1}{2} ∫_{Ω}\{(n − n̂)^{2} + (p − p̂)^{2}\}|_{t_{1}} − C_{2} ∫_{0}^{t_{1}} ∫_{Ω}\{(n − n̂)^{2} + (p − p̂)^{2}\}
≤ ∫_{Ω} C_{3}[ ∫_{0}^{t_{1}}(|n − n̂| + |p − p̂|)( ∫_{0}^{t} |n − n̂| + |p − p̂|)dt]
≤ C_{4} ∫_{0}^{t_{1}} ∫_{Ω}{|n − n̂|^{2} + |p − p̂|^{2}}

with the constants C_{i} depending on n, p, n̂, p̂, T. We then have n ≡ n̂ and p ≡ p̂ for t < T and thus for all t.

III. Global Results

In the earlier section, we cannot exclude the possibility that n, p → ∞ as t → ∞. However, it is easy to give conditions which ensure global boundedness and the existence of steady state solutions. Indeed, we need only show the local boundedness of (n, p) in L^{ξ} (L^{ξ}) for large ξ. After that, classical results (see again [6]) will ensure the conclusion. Observe in this regard that if a_{n} + b_{n} are sufficiently small then from simple modifications of the proof of Theorem 1, as mentioned above, it follows that we can estimate all of the “negative” integrals in (8) in terms of the positive ones and by repeating obtain the estimate

∫_{Ω}[n^{θ} + p^{θ}]|_{t} ≤ C_{0}

for some C_{0} > 0. Obviously a similar estimate holds for ∥n∥_{L^{∞}} + ∥p∥_{L^{∞}}. Furthermore a similar proof shows that in this case there exists at least one steady state solution n̂, p̂, φ̂ in C^{α_{1},α_{1}/2}(Ω), with h = \frac{p−n}{n+p+2},

Absorbing set considerations can also be based in this case directly on the proof of Theorem 1. Indeed, choose E = sup \{∥\overline{n} + \overline{p}∥_{L^{∞}(∂Ω_{D})}, 1\}. We then repeat and find that there exist a K, t_{0} such that for t ≥ t_{0} we have ∥n + p∥_{L^{∞}} ≤ K, where K depends only on ∥\overline{n} + \overline{p}∥_{L^{∞}(∂Ω_{D})} and t_{0} on ∥\overline{n} + \overline{p}∥_{L^{∞}(Ω)}. Some idea of the precise nature of the bounds K and t_{0} can be obtained by following the various proofs in [6], [13] and Theorem 1. In general, however, precise estimates seem extremely difficult to obtain due to the difficulty in estimating the various constants.

References

[1] E. DiBendetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993.

[2] W. Fang and K. Ito, Global solutions of the time-dependent drift-diffusion semiconductor equations, J. Differential Equations 123 (1995), 523–566.

[3] J. Furlan, Charge carrier dynamic nonequilibrium in amorphous semiconductors, IEEE Transactions on Electron Devices 39(2) (1992), 448–450.

[4] H. Gajewski and K. Gröger, Semiconductor equations for variable mobilities based on Boltzmann Statistics or Fermi-Dirac Statistics, Math. Nachr. 140, 7–36.

[5] J. Jerome, Evolution systems in semiconductor modeling: a cyclic uncoupled line analysis for Gummel map, Math. Methods Appl. Sci. 9 (1987), 455–492.

[6] O. Ladyzenskaja, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, Translation of Mathematical Monographs, Volume 23, American Mathematical Society, Providence, 1968.

HÖLDER SOLUTIONS FOR THE AMORPHOUS SILICON SYSTEM 9

[7] G. Lieberman, Intermediate Schauder theory for second order parabolic equations IV: time irregularity and regularity, Diff. Int. Eq. 5 (1992), 1219–1236.

[8] J. R. F. McMacken and S. G. Chamberlain, “A numerical model for two-dimensional transient simulation of amorphous silicon thin-film transistors’, IEEE Transactions on Computer Aided Design 11(5) (1992), 629–637.

[9] P. Markowich, The Stationary Semiconductor Equations, Springer-Verlag, New York, 1986.

[10] P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, New York, 1986.

[11] M. Murthy and G. Stampacchia, A variational inequality with mixed boundary conditions, Israel J. Math. 13 (1972), 189–224.

[12] S. Selberher, Analysis and Simulation of Semiconductor Devices, Springer-Verlag, New York, 1984.

[13] E. Shamir, Regularization of mixed second order boundary value problems, Israel J. Math. 8 (1968), 151–168.

[14] G. M. Troianiello, Elliptic Differential Equations and Obstacle Problems, Plenum Press, New York, 1987.

[15] H. Beirao da Veiga, Remarks on the flow of holes and electrons in crystalline semiconductors, Navier-Stokes Equations and Related Nonlinear Problems (A. Sequeira, ed.), Plenum, New York, 1995, pp. 291-306.

[16] H. Xie, L^{2,μ}-estimate to the mixed boundary value problem for second order elliptic equations and applications in thermistor problems, Nonl. Anal. TMA 24, 9–28.

[17] E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. I, Springer-Verlag, New York, 1986.

Walter Allegretto Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

   E-mail address: retl@retl.math.ualberta.ca

Yanping Lin Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

   E-mail address: ylin@hilbert.math.ualberta.ca

Aihui Zhou Institute of Systems Science, Academia Sinica, Beijing 100080, China

   E-mail address: azhou@bamboo.iss.ac.cn