Abstract
Our aim in this article is to study generalizations of the nonconserved Caginalp phase-field system based on the Maxwell-Cattaneo law with two temperatures for heat conduction and with logarithmic nonlinear terms. We obtain well-posedness results and study the asymptotic behavior of the system. In particular, we prove the existence of the global attractor. Furthermore, we give some numerical simulations, obtained with the $\mathtt{FreeFem++}$ software [ 24 ], comparing the nonconserved Caginalp phase-field model with regular and logarithmic nonlinear terms.
Highlights
The nonconserved Caginalp phase field system ∂u− ∆u + f (u) = T, (1) ∂t ∂T − ∆T = −, (2)2010 Mathematics Subject Classification. 35B45, 35K55, 35L15
In order to compare the logarithmic potentials with the cubic ones in the numerical simulations that we will perform, we will choose a cubic polynomial which has the same extrema as the logarithmic potential
ΦM (= ΦM1,M2 ) = {(φ, θ, ξ) ∈ H2(Ω)3; φ L∞(Ω) < 1, | θ + ξ | M1, | ξ | M2}. This family of operators forms a semigroup which is continuous for the L2(Ω) × L2(Ω) × L2(Ω)-topology
Summary
UMR CNRS 7348, SP2MI Boulevard Marie et Pierre Curie - Teleport 2 F-86962 Chasseneuil Futuroscope Cedex, France. Laboratoire de Mathematiques et Applications, UMR CNRS 7348 Boulevard Marie et Pierre Curie - Teleport 2
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