Stationary layered solutions in ${\mathbb{R}}^2$ for a class of non autonomous Allen-Cahn equations

Abstract

We consider a class of non autonomous Allen-Cahn equations

Mixed states for an Allen‐Cahn type equation

“…The original motivation for this paper was work of Alessio, Jeanjean, and Montecchiari [1]. They studied (PDE) for F(x, y, z) = a(x)G(z) where a is 1-periodic in x and G has a pair of nondegenerate local minima at, e.g., z = 0 and 1.…”

Complex Dynamics in a ODE Model Related to Phase Transition

“…We present below some results which will be then applied to the Poincaré map associated to (1). For the sake of completeness in the exposition, we also introduce the set Σ m := {0, .…”

“…Our interest for equation (1) is motivated by recent works by Byeon and Rabinowitz [6], [7], [8], [9] concerning the equation…”

“…. , x N and such that the support of A| [0,1] N is contained in ]0, 1[ N . It was shown in [6] that there is an infinitude of mixed states that shadow 0 and 1 in any prescribed way on a spatially periodic array of sets (from the Introduction in [8]).…”

“…The presence of nonconstant weight functions accounts for models describing heterogeneous materials. In recent years a great deal of interests has been devoted to the study of multiple solutions satisfying different boundary conditions (see, for instance [1], [2], [3], [5], [6], [9], [17], [18] and the references therein).…”