A qualitative analysis and numerical simulations of a nonlinear second-order anisotropic diffusion problem with non-homogeneous Cauchy–Neumann boundary conditions

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

“…In Theorem 1, we prove the existence, regularity and uniqueness of solution for (1). (see [15] for a numerical study of Equation (1) corresponding to a linear reaction term v(t, x) − v 0 (x), with homogeneous Neumann boundary condition).…”

“…Let us now show that H is continuous and compact. The sketch of the proof is the same as in [1,15]. However, for reader convenience, we present details in the sequel.…”

“…In addition, the nonlinear problems of type (1) 1 , occur in the phase-field transition system (e.g., [6]) where the phase function v(t, x) describes the transition between the solid and liquid phases in the solidification process of a material occupying a region Ω. For more general assumptions and with various types of boundary conditions, Equation (5) has been numerically investigated (e.g., [6][7][8][9][10][11][12][13][14][15][16][17]). The error analysis for the implicit backward Euler approximation is presented in [16], and computations with several different higher-order time-stepping schemes are used in [11].…”

“…Another important novelty in our paper concerns the non-homogeneous Cauchy-Neumann boundary conditions, which can be seen as boundary control in industry. Thus, as applications of problem (1), we indicate the moving interface problems, e.g., phase separation and transition (see [3,8,12,17,18,[22][23][24][25][26][27]), anisotropy effects (see [15,[28][29][30]), image denoising and segmentation (see [15,24,26,[30][31][32][33][34][35][36][37][38][39] and references therein), etc.…”

“…The Leray-Schauder principle (see [1,4,[11][12][13][14][15][19][20][21] and reference therein); • The L p -theory of linear and quasi-linear parabolic equations; • Green's first identity…”