Rigorous Mathematical Investigation of a Nonlocal and Nonlinear Second-Order Anisotropic Reaction-Diffusion Model: Applications on Image Segmentation

Costică Moroşanu and Silviu Pavăl
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Costică Moroşanu: Department of Mathematics, “Alexandru Ioan Cuza” University, Bd. Carol I, 11, 700506 Iaşi, Romania
Silviu Pavăl: Faculty of Automatic Control and Computer Engineering, Technical University “Gheorghe Asachi” of Iaşi, Dimitrie Mangeron, nr. 27, 700050 Iaşi, Romania

Mathematics, 2021, vol. 9, issue 1, 1-23

Abstract: In this paper we are addressing two main topics, as follows. First, a rigorous qualitative study is elaborated for a second-order parabolic problem, equipped with nonlinear anisotropic diffusion and cubic nonlinear reaction, as well as non-homogeneous Cauchy-Neumann boundary conditions. Under certain assumptions on the input data: f ( t , x ) , w ( t , x ) and v 0 ( x ) , we prove the well-posedness (the existence, a priori estimates, regularity, uniqueness) of a solution in the Sobolev space W p 1 , 2 ( Q ) , facilitating for the present model to be a more complete description of certain classes of physical phenomena. The second topic refers to the construction of two numerical schemes in order to approximate the solution of a particular mathematical model (local and nonlocal case). To illustrate the effectiveness of the new mathematical model, we present some numerical experiments by applying the model to image segmentation tasks.

Keywords: nonlinear anisotropic reaction-diffusion; well-posedness of solutions; Leray-Schauder degree theory; finite difference method; explicit numerical approximation scheme; image segmentation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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