Abstract
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).