Stable equilibria of a singularly perturbed reaction–diffusion equation when the roots of the degenerate equation contact or intersect along a non-smooth hypersurface

Stable transition layer for the Allen–Cahn equation when the spatial inhomogeneity vanishes on a nonsmooth hypersurface in ℝn

“…We still need to prove that u (1) 0 and u (2) 0 (see Theorem 1) are isolated L 1 -local minimizers of E 0 , which corresponds to item (iii) of Theorem 2. This proof can be performed in a similar way to that in [16,Theorem 3.9]. Although the problems are different, the key similarity is that, in both cases, the weight function in the functional Γ-limit (function  in our case) vanishes on a hypersurface 𝛾 ⊂ Ω.…”

“…We still need to prove that $$$ {u}_0^{(1)} $$$ and $$$ {u}_0^{(2)} $$$ (see Theorem 1) are isolated $$$ {L}^1 $$$‐local minimizers of $$$ {E}_0 $$$, which corresponds to item $$$ (iii) $$$ of Theorem 2. This proof can be performed in a similar way to that in [16, Theorem 3.9]. Although the problems are different, the key similarity is that, in both cases, the weight function in the functional $$$ \Gamma $$$‐limit (function $$$ \mathcal{P} $$$ in our case) vanishes on a hypersurface $$$ \gamma \subset \Omega $$$.…”

“…By making certain assumptions, this approach guarantees that if the $$$ \Gamma $$$‐limit of the family of energy functionals associated with our problem has an isolated local minimum $$$ {u}_0 $$$ in the $$$ {L}^1 $$$‐topology, then the family itself has a sequence of minima that converges to $$$ {u}_0 $$$. To achieve this, we draw on some of the results presented in [8, 16, 17] and adapt them to the current problem. However, the presence of spatial inhomogeneity $$$ h $$$—which vanishes on $$$ \gamma $$$—brings about several modifications in the $$$ \Gamma $$$‐convergence computation, making this proof of independent interest.…”

Stable transition layers in an unbalanced bistable equation

“…When functions a, b, c intersect each other. In [9] the problem (1) was studied in an n-dimensional domain Ω with k 2 ≡ 1. It was assumed that the roots of the g (a balanced bistable function, i.e.…”