Limiting classification on linearized eigenvalue problems for 1-dimensional Allen-Cahn equation. II: Asymptotic profiles of eigenfunctions. (Q324564)

The present paper is a continuation of the authors' paper [ibid. 258, No. 11, 3960--4006 (2015; Zbl 1319.34152)].NEWLINENEWLINEThe authors study the small diffusion (\(\varepsilon \to 0\)) limit of the eigenvalue problem NEWLINE\[NEWLINE\begin{aligned}\varepsilon^2 \varphi_{xx}(x)+f_u(u_n(x)) \varphi(x)+\lambda_j^n\varphi(x)=0,\\ \varphi_x(0)=\varphi_x(1)=0,\end{aligned}NEWLINE\]NEWLINE where \(u_n\) is an \(n\)-nodal solution of the nonlinear boundary value problem NEWLINE\[NEWLINE\begin{cases}\varepsilon^2 u_{xx}(x)+f(u(x)) =0,\\ u_x(0)=u_x(1)=0. \end{cases}NEWLINE\]NEWLINE They set \(f(u)=u-u^3\) and call the latter equation the Allen-Cahn equation.NEWLINENEWLINEWhile in the previous paper the author computed the dominant term in the asymptotic expansion of the eigenvalues \(\lambda_j^n\), in the present paper they study the asymptotics of the eigenfunctions. They show that there are three different asymptotic regimes, \(0\leq j < n\), \(n\leq j < 2n\) and \(j \geq 2n\), and for each regime they compute the limiting form of the eigenfunctions.