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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Vydané v:	Mathematics (Basel) Ročník 12; číslo 13; s. 2050
Hlavní autori:	Barbu, Tudor, Miranville, Alain, Moroşanu, Costică
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Basel MDPI AG 01.07.2024
Predmet:	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
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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.
AbstractList	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.
Author	Moroşanu, Costică Barbu, Tudor Miranville, Alain
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Cites_doi	10.1007/s10441-022-09436-4 10.1016/S0167-2789(97)00177-2 10.1109/83.902291 10.3934/dcdss.2022142 10.1109/CVPR.2015.7298965 10.3934/math.2019.3.648 10.1090/surv/165 10.3390/math8122111 10.1109/83.661193 10.1016/j.apnum.2004.05.001 10.1016/j.acra.2009.12.017 10.1016/j.media.2008.11.002 10.2298/CSIS120219060W 10.1016/j.jfa.2007.07.013 10.1007/978-3-642-33905-9 10.1016/S1532-0464(02)00502-6 10.1016/j.inffus.2013.12.002 10.1093/imamat/48.3.249 10.1016/S0167-8655(02)00218-0 10.1007/978-3-0348-0813-2 10.1109/TIP.2003.819229 10.1090/S0273-0979-06-01104-9 10.1109/TIP.2012.2183143 10.4208/eajam.090217.300617a 10.1007/s00245-019-09643-5 10.1016/j.jde.2006.12.002 10.3934/dcdss.2022208 10.1155/2019/3980181 10.1016/j.na.2011.12.019 10.1016/j.physleta.2009.04.076 10.3390/math9010091 10.4208/cicp.221109.290710a 10.1137/060669358 10.1016/j.mri.2013.05.002 10.1016/j.na.2008.02.076
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SubjectTerms	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
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Title	On a Local and Nonlocal Second-Order Boundary Value Problem with In-Homogeneous Cauchy–Neumann Boundary Conditions—Applications in Engineering and Industry
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