Asymptotic properties and classification of bistable fronts with Lipschitz level sets
The authors deal with the study of classical bounded solutions of the following elliptic equation \[ \Delta u- c\partial_yu+ f(u)= 0\text{ in }\mathbb R^d= \{z=(x,y)\mid x=(x_1,\dots, x_{d-1})\in \mathbb R^{d-1},\,y\in\mathbb R\},\tag{1} \] where the function \(f\) is of class \(C^2\) and ``bistable'' type. They are interested in solutions \(u\) with cylindrical symmetry, that is, \(u(x,y)= \widetilde u(|x|,y)\), and satisfying the following condition at infinity \[ \limsup_{A\to+\infty,y\geq A+\varphi(|x|)}|u(x,y)- 1|= 0,\quad \limsup_{A\to-\infty, y\leq A+\varphi(|x|)}|u(x,y)|= 0.\tag{2} \] In (2) the notation \(|x|\) stands for the Euclidean norm of \(x\) and \(\varphi: \mathbb R_t\to\mathbb R\) is a globally Lipschitz-continuous function. The authors present uniqueness and classification results in dimension 2 and higher.
scientific article; zbMATH DE number 5125745