A Fisher-type evolution equation with applications in image processing

The aim of this paper is to present some applications of the Neumann Laplacian in image processing, along with the necessary mathematical background. We prove weak and strong versions of the maximum principle for weak solutions of elliptic and parabolic problems and apply them to a Fisher K.P.P.-type equation. The original contribution lies in the application of this equation in image processing, where various diffusion-like effects can be achieved. Additionally, a review of the basics of linear and nonlinear PDEs with Neumann boundary conditions is provided, along with updated bibliography and recent qualitative results. There are also some new theoretical results developed in this work.