Existence and stability of transition layers

Abstract

For a second order nonautonomous singularly perturbed ordinary differential equation with Neumann boundary conditions, the existence of single transition layer solutions is proved by using the method of Liapunov-Schmidt. The method also gives the stability of these solutions as an equilibrium point of a parabolic equation.

Transition Layers for a Bistable Reaction-Diffusion Equation with Variable Diffusion

“…See also Angenent, Mallet-Paret and Peletier [2], Hale and Sakamoto [5] and Nakashima [9]. They have discussed the existence and stability of solutions with single-layers for similar problems to (1.2).…”

Radially symmetric internal layers in a semilinear elliptic system

“…We look for a formal approximation to w which, using an idea of Hale and Sakamoto [14], we take of the form (3.50) zeit) = zoit) + ezxit).…”

A Remark on Singular Perturbation Methods via the Lyapunov-Schmidt Reduction

“…We denote a complete orthonormal system in L 2 (0, 1) of eigenfunctions and eigenvalues by (0 n (e, cr), A w (e, cr)} , where /^(e, cr) > A 2 (e, a) > /1 3 (£, a) > •••. The principal eigenvalue A^e, cr) approaches zero as e ->0, and there exists a constant /c/j > 0 such that all other ones are less than -fjt 1 . Let E denote the orthogonal projection onto span {^(e, cr)}, that is, Eu = (tt^jCe, cr)) L 2 (0 >1) 0 1 (e, a).…”

“…Hale and Sakamoto [1] applied Lyapunov-Schmidt reduction to construct singularly perturbed equilibrium solutions to Equation (1) below. This method also gave the stability condition for the solutions simultaneously.…”

“…In the following, we briefly sketch the method of [1], with special attention to the part where our present contribution appears. We consider the following equation To construct unknown equilibrium solution u of (1), we begin with a smooth approximate equilibrium solution U(x, e) of (1) that exhibits a transition layer at x = 0.…”