Qualitative properties of solutions to mixed-diffusion bistable equations

TL;DR

This paper investigates a fourth-order extension of the Allen-Cahn model with mixed diffusion, establishing key properties of solutions such as existence, uniqueness, stability, and symmetry, and constructs a specific saddle solution in the plane.

Contribution

It introduces a novel fourth-order mixed-diffusion Allen-Cahn model and provides comprehensive analysis of solution properties using variational and bifurcation methods.

Findings

Proved existence and uniqueness of solutions.

Established stability and symmetry properties.

Constructed a nontrivial saddle solution in the plane.

Abstract

We consider a fourth-order extension of the Allen-Cahn model with mixed-diffusion and Navier boundary conditions. Using variational and bifurcation methods, we prove results on existence, uniqueness, positivity, stability, a priori estimates, and symmetry of solutions. As an application, we construct a nontrivial bounded saddle solution in the plane.