Multi-layered stationary solutions for a spatially inhomogeneous Allen--Cahn equation.
Consider the Neumann problem \[ \varepsilon^2u''+h^2(x)f(u)=0, \quad u'(0)=0,\,\,u'(1)=0, \tag{1} \] where \(h\in\mathcal{C}([0,1])\) is positive and \(f\in\mathcal{C}^1(\mathbb{R})\) is derived from a double well potential with equal minimums at points \(\alpha^-<0\) and \(\alpha^+>0\). The author studies \(n\)-mode solutions to (1), i.e., solutions which have precisely \(n\) zeros in the interval \(]0,1[\). As \(\varepsilon\) goes to zero, these solutions oscillate from one of the constants \(\alpha^-\) or \(\alpha^+\) to the other with precisely \(n\) transitions. The existence of such solutions is considered. A first problem concerns the localization of the transition layers. It is proved that they only take place near extremum points of \(h(x)\). Single layers appear near local minimum points, while multiple layers can take place near local maximum points. Internal and boundary layers are considered. The author proves also the existence of solutions with clustering layers, i.e., multiple layers that appear within a very small distance from one another. The existence of solutions with an arbitrary number of clustering layers is considered. Solutions to (1) are stationary solutions to the Allen-Cahn equation \[ \varepsilon^2(u_t-u_{xx})=h^2(x)f(u), \quad u_x(0,t)=u_x(1,t)=0, \quad u(x,0)=u_0(x). \] The stability of the \(n\)-modes solutions is considered. This follows as these solutions are local minimizers of the action corresponding to (1). It is also established that the Morse index of the \(n\)-modes solutions is the total number of layers that appear near the local maximum points of \(h(x)\).
scientific article; zbMATH DE number 1915673