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).

The existence, the estimate, and the uniqueness of a solution to a general phase-field system, motivated by Caginalp's model describing the phase changes, are established. The paper extends the results for the already studied types of nonlinearities related to the model. ᮊ

No abstract

No abstract

Two main topics are addressed in the present paper, first, a rigorous qualitative study of a second-order reaction–diffusion problem with non-linear diffusion and cubic-type reactions, as well as inhomogeneous dynamic boundary conditions. Under certain assumptions about the input data: gd(t,x), gfr(t,x), U0(x) and ζ0(x), we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a solution in the space Wp1,2(Q)×Wp1,2(Σ). Here, we extend previous results, enabling new mathematical models to be more suitable to describe the complexity of a wide class of different physical phenomena of life sciences, including moving interface problems, material sciences, digital image processing, automatic vehicle detection and tracking, the spread of an epidemic infection, semantic image segmentation including U-Net neural networks, etc. The second goal is to develop an iterative splitting scheme, corresponding to the non-linear second-order reaction–diffusion problem. Results relating to the convergence of the approximation scheme and error estimation are also established. On the basis of the proposed numerical scheme, we formulate the algorithm alg-frac_sec-ord_dbc, which represents a delicate challenge for our future works. The benefit of such a method could simplify the process of numerical computation.

No 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 v0(x), we prove the well-posedness (the existence, a priori estimates, regularity, uniqueness) of a solution in the Sobolev space Wp1,2(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.

a b s t r a c tThis paper is devoted to the study of a Caginalp phase-field system endowed with non-homogeneous Cauchy-Neumann and nonlinear dynamic boundary conditions. We first prove the existence, uniqueness and regularity of solutions to an Allen-Cahn equation. Our approach allows to consider in the dynamic boundary conditions a nonlinearity of higher order than in the known results. The existence, uniqueness and regularity of solutions to the Caginalp system in this new formulation is also proven.