Volume 51, Issue 4
Attractors for a Caginalp Phase-field Model with Singular Potential

Alain Miranville & Charbel Wehbe

J. Math. Study, 51 (2018), pp. 337-376.

Published online: 2018-12

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  • Abstract

We consider a phase field model based on a generalization of the Maxwell Cattaneo heat conduction law, with a logarithmic nonlinearity, associated with Neumann boundary conditions. The originality here, compared with previous works, is that we obtain global in time and dissipative estimates, so that, in particular, we prove, in one and two space dimensions, the existence of a unique solution which is strictly separated from the singularities of the nonlinear term, as well as the existence of the finite-dimensional global attractor and of exponential attractors. In three space dimensions, we prove the existence of a solution.

  • Keywords

Caginalp phase-field system Maxwell-Cattaneo law logarithmic potential Neumann boundary conditions well-posedness global attractor exponential attractor.

  • AMS Subject Headings

35B40 35B41 35K51 80A22 80A20 35Q53 45K05 35K55 35G30 92D50

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

alain.miranville@math.univ-poitiers.fr (Alain Miranville)

charbel_wehbe83@hotmail.com (Charbel Wehbe)

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@Article{JMS-51-337, author = {Miranville , Alain and Wehbe , Charbel }, title = {Attractors for a Caginalp Phase-field Model with Singular Potential}, journal = {Journal of Mathematical Study}, year = {2018}, volume = {51}, number = {4}, pages = {337--376}, abstract = {

We consider a phase field model based on a generalization of the Maxwell Cattaneo heat conduction law, with a logarithmic nonlinearity, associated with Neumann boundary conditions. The originality here, compared with previous works, is that we obtain global in time and dissipative estimates, so that, in particular, we prove, in one and two space dimensions, the existence of a unique solution which is strictly separated from the singularities of the nonlinear term, as well as the existence of the finite-dimensional global attractor and of exponential attractors. In three space dimensions, we prove the existence of a solution.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v51n4.18.01}, url = {http://global-sci.org/intro/article_detail/jms/12915.html} }
TY - JOUR T1 - Attractors for a Caginalp Phase-field Model with Singular Potential AU - Miranville , Alain AU - Wehbe , Charbel JO - Journal of Mathematical Study VL - 4 SP - 337 EP - 376 PY - 2018 DA - 2018/12 SN - 51 DO - http://dor.org/10.4208/jms.v51n4.18.01 UR - https://global-sci.org/intro/jms/12915.html KW - Caginalp phase-field system KW - Maxwell-Cattaneo law KW - logarithmic potential KW - Neumann boundary conditions KW - well-posedness KW - global attractor KW - exponential attractor. AB -

We consider a phase field model based on a generalization of the Maxwell Cattaneo heat conduction law, with a logarithmic nonlinearity, associated with Neumann boundary conditions. The originality here, compared with previous works, is that we obtain global in time and dissipative estimates, so that, in particular, we prove, in one and two space dimensions, the existence of a unique solution which is strictly separated from the singularities of the nonlinear term, as well as the existence of the finite-dimensional global attractor and of exponential attractors. In three space dimensions, we prove the existence of a solution.

Alain Miranville & Charbel Wehbe. (2019). Attractors for a Caginalp Phase-field Model with Singular Potential. Journal of Mathematical Study. 51 (4). 337-376. doi:10.4208/jms.v51n4.18.01
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