Mixed states for an Allen‐Cahn type equation

Boundary-clustered interfaces for the Allen–Cahn equation

“…(For the Allen-Cahn equation with inhomogeneity, u + a(x)(u − u 3 ) = 0 in ‫ޒ‬ 2 , see [Rabinowitz and Stredulinsky 2003;.) Here, we continue our study, initiated in [Malchiodi et al 2005], of clustered layered solutions for semilinear elliptic equations, and show that the homogeneous Allen-Cahn equation itself can generate multiple clustered interfaces near the boundary.…”

Multibump solutions and asymptotic expansions for mesoscopic Allen-Cahn type equations

“…Analogous layered and multibump solutions have been studied in [1,23,24] and multiplicity results are also in [7]: differently from those results, the multibumps are here obtained not by perturbing the potential F (t) into Q(x)F (t), but by using the mesoscopic term H(x). A more formal description of Theorem 1.2 will be given in the subsequent Section 2.…”

“…IV), it does not lie in the essential spectrum of −Δ + F (u ± ), hence it belongs to the discrete spectrum. Let now w ± be the eigenvector corresponding to λ ± such that 23) i.e. there holds…”

Spatially heteroclinic solutions for a semilinear elliptic P.D.E.

“…So as not to delay the exposition, the details of some of the steps will be postponed until Section 3. Although the setting is different, the structure of the argument is very close to that of [7] which we will strongly follow.…”

“…Thus another approach is needed here. For such an approach, we were motivated by some recent work [7] on an Allen-Cahn model equation which in turn has antecedents in work of Moser [5] and Bangert [1] on minimal laminations of a torus and of Bosetto and Serra [2] on an ODE problem related to [1]. Γ will be replaced by a class of functions asymptotic from v to w and having an additional monotonicity property and J by a related function J * .…”