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

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 co...

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Bibliographic Details
Published in:Mathematics (Basel) Vol. 12; no. 13; p. 2050
Main Authors: Barbu, Tudor, Miranville, Alain, Moroşanu, Costică
Format: Journal Article
Language:English
Published: Basel MDPI AG 01.07.2024
Subjects:
Boundary conditions
Boundary value problems
diffusion processes
fixed points
Function space
Image processing
Image reconstruction
Image segmentation
Leray–Schauder degree theory
nonlinear PDE of parabolic type
Qualitative analysis
qualitative properties of solutions
reaction–diffusion equations
Regularity
Tomography
ISSN:2227-7390, 2227-7390
Online Access:Get full text
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Summary: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.
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ISSN:2227-7390
2227-7390
DOI:10.3390/math12132050