Article Contents

2023, Volume 16, Issue 1: 148-186. Doi: 10.3934/dcdss.2022203

This issue Previous Article On a nonlocal and nonlinear second-order anisotropic reaction-diffusion system with in-homogeneous Cauchy-Neumann boundary conditions. Applications on epidemic infection spread Next Article Erratum to: Remarks on mean curvature flow solitons in warped products

A qualitative analysis of a second-order anisotropic phase-field transition system endowed with a general class of nonlinear dynamic boundary conditions

1.
Department of Mathematics and Computer Science, North University Center at Baia Mare, Technical University of Cluj-Napoca, Victoriei 76, 430122 Baia Mare, Romania
2.
Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348, SP2MI, 86962 Chasseneuil Futuroscope Cedex, France
3.
University "Al. I. Cuza" of Iaşi, 700506 Iaşi, Romania
4.
Academy of Romanian Scientists, Ilfov Str. no. 3, 050045 Bucharest, Romania

Received: April 2022

Revised: September 2022

Early access: December 2022

Published: January 2023

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Abstract

The paper is concerned with the study of a nonlinear second-order anisotropic phase-field transition system of Caginalp type, subject to nonlinear and in-homogeneous dynamic boundary conditions (in both unknown functions). Under certain hypothesis on the input data: $ f_{_1}(t,x) $, $ f_{_2}(t,x) $, $ w_{_1}(t,x) $, $ w_{_2}(t,x) $, $ u_0(x) $, $ \alpha_0(x) $, $ \varphi_0(x) $ and $ \xi_0(x) $, we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a classical solution in the Sobolev space $ W^{1,2}_p(Q)\times W^{1,2}_p(\Sigma) $, $ W^{1,2}_\nu(Q)\times W^{1,2}_p(\Sigma) $. Here we extend the previous results concerned with nonlinearity of cubic type, allowing to the present mathematical model to be more capable for describing the complexity of a wide class of real physical phenomena (moving interface problems, image processing, the phase changes at the boundary $ \partial\Omega $, etc.).

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Mathematics Subject Classification: Primary: 35Bxx; Secondary: 35K55, 35K60, 35Qxx.

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