中文摘要
這篇論文引入V.Gurau, H.Liu, T.Zhou等人的物理模型，簡化至兩個一維模型。我們的探討侷限在質子交換膜燃料電池的觸煤層內解的存在問題，其數學模型為兩個常微分方程系統的邊界值問題。
關於第一個模型我們透過Green’s functions、積分方程及Leray-Schauder固定點的理論證明了古典解的存在。對於第二個模型我們利用偏微分方程的線性理論和先驗估計建構一個疊代然後利用Schauder 固定點定理得到弱解的存在。
另外我們利用Giaquinta 和 Modica的方法得到第二個模型的拋物型方程模擬的一個局部解。

Abstract

In this thesis we discuss the existence of a solution in the cathode catalyst layer of some 1D models of PEM fuel cells. The first model is governed by the equations

T′′-f(T)Φ′ = 0
(f(T)Φ′)′+g(T)Y = 0 , x∈(a,b)
Y′′-h(T)Y = 0,

with the boundary condition

μ₁T(a)-μ₂T′(a) = 1,T′(b)=0
α₁Y(a)-α₂Y′(a) = 1,Y′(b)=0
Φ(b)+βf(T(b))⋅Φ′(b) = 0,Φ′(a)=0,

where μ₁, μ₂, α₁, α₂ are positive constants and β>0, and x ∈(a,b). Assume f,g,h∈C_{b}¹(R), k∈C_{b}¹(R),and f≥δ₁>0, h≥δ₂>0.
The second model is governed by the equations

T′′-k(T)+λf(T)(Φ′)² = 0
(f(T)Φ′)+g(T)Y = 0
Y′′-h(T)Y = 0,

with the boundary condition

μ₁T(a)-μ₂T′(a) = 1, T′(b)=0
Φ(b)+βf(T(b))⋅Φ′(b) = 0, Φ′(a)=0
α₁Y(a)-α₂Y′(a) = 1, Y′(b)=0,

where μ₁, μ₂, α₁, α₂ are positive constants and β>0, and for x ∈ (a,b). Assume f,g,h∈C_{b}¹(R), k∈C_{b}¹(R),and f≥δ₁>0, h≥δ₂>0.
For the first model, we use Green's functions to rewrite the problem into an integral equation, then we apply the Leray-Schauder fixed point theorem to show the existence of a classical solution for the first model. For the second model, we use the linear theory to construct an iteration process, and apply the Schauder fixed point theorem to show the existence of a weak solution for the second model.
We also prove a local existence result for a parabolic analogy of the second model by applying Banch contraction principle (Following Giaquinta and Modica).

References
[1] Adams, R. A. Sobolev Spaces, Academic Press, 1975.
[2] Chen, Y. Z. and Wu, L. C., Second Order Elliptic Equations and Elliptic Systems, American
Mathematical Society Providence, Rhode Island, 1998.
[3] Chen, Y. Z., Second Order Differential Equations of Parabolic Type, Peking University Press,
2002.
[4] Evans, L. C., Partial Differential Equations, American Mathematical Society, Providence,
Rhode Island,. 1999.
[5] Fuller, T. F., and Newman, J., Water and Thermal Management in Solid-Polymer-Electrolyte
Fuel-Cells, J. Electrochem. Soc., 140(5), (1993), pp1218-1225.
[6] Giaquinta, M., Multiple Integrals In The Calculus Of Variations And Nonlinear Elliptic Systems,
Princeton University Press, Princeton, New Jersey 1983.
[7] Giaquinta, M. and Modica, G., Local Existence for Quasilinear Parabolic Systems under
Nonlinear Boundary Conditions, Annali di Matematica Pura ed Applicata, 1987.
[8] Gilbarg, D. and Trudinger, N. S. Elliptic Partial Differential Equations of Second Order, 2nd
ed., Springer-Verlag, 1983.
[9] V. Gurau, F. Barbir, H. Liu, An analytic solution of a half-cell model for PEM fuel cells,
Journal of The Electrochemical Society, 147 (2000) 2468-2477.
[10] V. Gurau, H. Liu, S. Kakac, Two-dimensional model for proton exchange membrane fuel
cells, AIChE Journal 44 (1998) 2410-2422.
[11] Ladyzenskaya, O. A., Solonnikov, V. A. and Uralceva, N. N., Linear and Quasi-linear Equations
of Parabolic type, American Mathematical Society, Providence, Rhode Island,. 1968.
[12] Ladyzenskaya, O. A. and Uralceva, N. N., Linear and Quasilinear Elliptic Equations, Academic
Press New York and London 1968.
72