Existence for weakly coercive nonlinear diffusion equations via a variational principle

TL;DR

This paper establishes the existence of solutions for weakly coercive nonlinear diffusion equations using a variational principle, broadening the conditions under which solutions can be guaranteed.

Contribution

It introduces a novel variational approach inspired by Brezis-Ekeland to prove existence of solutions under minimal growth and coercivity assumptions.

Findings

Solutions can be obtained as null minimizers of a convex functional.

The method applies to equations with superlinear growth nonlinearities.

New existence results are established under weaker conditions.

Abstract

We are concerned with the study of the well-posedness of a nonlinear diffusion equation with a monotonically increasing multivalued time-dependent nonlinearity derived from a convex continuous potential having a superlinear growth to infinity. The results in this paper state that the solution of the nonlinear equation can be retrieved as the null minimizer of an appropriate minimization problem for a convex functional involving the potential and its conjugate. This approach, inspired by the Brezis-Ekeland variational principle, provides new existence results under minimal growth and coercivity conditions.