Qualitative properties of solutions to elliptic mixed-diffusion bistable equations

Nonlinear fourth-order PDEs usually have a richer and more
complex set of solutions when compared to its second-order
counterpart. In this sense, many models exhibit behaviors that could
be better described with fourth-order equations, like ocean and
atmosphere dynamics, bridges, and pattern formation , just to mention
some of them. Higher-order models, however, are far less developed and
understood than their second-order analogue and many basic questions
remain open.  Lack of maximum principles and oscillatory behavior of
solutions are two of the main difficulties in the study of such
problems.

I this talk I consider a fourth-order extension of the Allen-Cahn
model with mixed-diffusion and Navier boundary conditions. I present
results on existence, uniqueness, positivity, stability, a priori
estimates, and symmetry. As an application, the construction of a
saddle solution and a periodic solution in the whole space is
shown. The proofs rely on variational and bifurcation methods. Some
numerical approximations of solutions will also be discussed.