On a Local and Nonlocal Second-Order Boundary Value Problem with In-Homogeneous Cauchy–Neumann Boundary Conditions—Applications in Engineering and Industry

Abstract

A qualitative study for a second-order boundary value problem with local or nonlocal diffusion and a cubic nonlinear reaction term, endowed with in-homogeneous Cauchy–Neumann (Robin) boundary conditions, is addressed in the present paper. Provided that the initial data meet appropriate regularity conditions, the existence of solutions to the nonlocal problem is given at the beginning in a function space suitably chosen. Next, under certain assumptions on the known data, we prove the well posedness (the existence, a priori estimates, regularity, uniqueness) of the classical solution to the local problem. At the end, we present a particularization of the local and nonlocal problems, with applications for image processing (reconstruction, segmentation, etc.). Some conclusions are given, as well as new directions to extend the results and methods presented in this paper.

Keywords

• qualitative properties of solutions
• nonlinear PDE of parabolic type
• reaction–diffusion equations
• fixed points
• Leray–Schauder degree theory
• diffusion processes