Stability of Solutions to a Caginalp Phase-Field Type Equations

Authors

  • Mohamed Ali Ipopa Laboratory of Mathematics and Applied Physics, Masuku University of Science and Technology, BP 943, Franceville, Gabon
  • Brice Landry Doumbé Bangola Laboratory of Mathematics and Applied Physics, Masuku University of Science and Technology, BP 943, Franceville, Gabon
  • Armel Andami Ovono Normal Superieur School, BP 17 009, Libreville, Gabon https://orcid.org/0000-0002-0533-9877

DOI:

https://doi.org/10.37256/cm.5120242725

Keywords:

Caginalp phase-field system, two temperatures, well-posedness, dissipativity, global attractor, Lojasiewicz-Simon inequality, convergence to steady states

Abstract

This paper is concerned with the study of the asymptotic behavior of a generalization of the Caginalp phase-field model subject to homogeneous Neumann boundary conditions and regular potentials involving two temperatures. This work follows on from a paper in which the well-posedness of the problem, the dissipativity of the system, and the existence of global and exponential attractors were demonstrated. In addition, a study on the semi-infinite cylinder was also carried out. Indeed, if it is true that the existence of a global attractor makes it possible to predict the asymptotic behavior of solutions on a bounded domain, it does not say that these solutions converge. After having shown the existence of the global attractor, it is therefore important to look at the convergence of the solutions over time. There are several methods for determining the asymptotic behavior of the solutions of a differential system. We can mention the one that consists of transforming the given differential equations into integral equations and then applying the classical Picard successive approximation procedure to them. This work is devoted to the study of the convergence of solutions to steady states, adapting a well-known result concerning Lojasiewicz-Simon’s inequality.

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Published

2024-03-13

How to Cite

1.
Ipopa MA, Doumbé Bangola BL, Andami Ovono A. Stability of Solutions to a Caginalp Phase-Field Type Equations. Contemp. Math. [Internet]. 2024 Mar. 13 [cited 2024 May 28];5(1):949-58. Available from: https://ojs.wiserpub.com/index.php/CM/article/view/2725