A Qualitative Analysis of a Nonlinear Second-Order Anisotropic Diffusion Problem with Non-homogeneous Cauchy–Stefan–Boltzmann Boundary Conditions

Abstract

The paper is concerned with a qualitative analysis for a nonlinear second-order parabolic problem, subject to non-homogeneous Cauchy-Stefan-Boltzmann boundary conditions, extending the types already studied. Under certain assumptions, we prove the existence, a priori estimates, regularity and uniqueness of a solution in the class W 1,2 p (Q). Here we extend the results already proven by the authors for a nonlinearity of cubic type, making the present mathematical model to be more capable for describing the complexity of certain wide classes of real physical phenomena (phase separation and transition, for instance).

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

“…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.…”

“…is a positive and bounded nonlinear real function of class C 1 (Q) with bounded derivatives (see [1]), having the role of controlling the speed of the diffusion process and enhances the edges (e.g., in the evolving image); • Ψ(v x (t, x)) is the mobility; •q(t, x) is a positive and bounded real function; • f (t, x) ∈ L p (Q) is the distributed control (a given function), where…”

“…Concerning Equation 51 , we recall that it is of quasi-linear type with principal part in divergence form (see [1]), with a i , i = 1, 2, given by (4) and…”

“…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…”

“…The Lions and Peetre embedding theorem (see [1] and references therein) to ensure the existence of a continuous embedding…”