5. Contrast Structures of Alternating Type
A. B. Vasilieva · Society for Industrial and Applied Mathematics eBooks · 2005
The paper is devoted to spike-type solutions (contrast structures) of singularly perturbed parabolic equations. 5.1 Equation without Explicit Dependence on ux We consider the singularly perturbed parabolic equation (ε is a small parameter) ε2 ( uxx − ut )=F (u,x,t) , (x,t) ∈ Ω≔ {x ∈ [0,1],t ∈ ℝ} , 5.1 under the boundary conditions u (0,t,ε)=0, u (1,t,ε)=0, 5.2 and the 2π-periodicity condition u (x,t,ε) =u (x,t+2π,ε) . 5.3 Remark. We assume that the boundary conditions (5.2) are homogeneous for simplicity of presentation of our results. However, one can consider − ∞ < a < b < +∞, u ∈ [a, b] and the boundary conditions can have the form u(0, t, ε) = u0, u(1, t, ε) = u1 (see [1]). Let the following assumptions hold: 1) The function F(u, x, t) ∈ C2(G), where G ≔ {ℝ × Ω}.