Quarterly of Applied Mathematics

Author(s): Maurizio Grasselli; Hana Petzeltová; Giulio Schimperna
Journal: Quart. Appl. Math. 65 (2007), 451-469.
MSC (2000): Primary 34D05, 35B40, 35Q99, 80A22
Posted: July 19, 2007
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Abstract: We study a phase-field system where the energy balance equation has the standard (parabolic) form, while the kinetic equation ruling the evolution of the order parameter $\chi$ is a nonlocal and nonlinear second-order ODE. The main features of the latter equation are a space convolution term which models long-range interactions of particles and a singular configuration potential that forces $\chi$ to take values in

. We first prove the global existence and uniqueness of a regular solution to a suitable initial and boundary value problem associated with the system. Then, we investigate its long time behavior from the point of view of $\omega$ -limits. In particular, using a nonsmooth version of the $\L$ ojasiewicz-Simon inequality, we show that the $\omega$ -limit of any trajectory contains one and only one stationary solution, provided that the configuration potential in the kinetic equation is convex and analytic.

1.: H. Amann, ``Linear and Quasilinear Parabolic Problems'', Birkhäuser Verlag, Basel-Boston-Berlin, 1995. MR 1345385 (96g:34088)
2.: V. Barbu, ``Nonlinear Semigroups and Differential Equations in Banach Spaces'', Noordhoff, Leyden, 1976. MR 0390843 (52:11666)
3.: P.W. Bates, F. Chen, Traveling wave solutions for a nonlocal phase-field system, Interfaces Free Bound., 4 (2002), 227-238. MR 1914622 (2003f:35275)
4.: P.W. Bates, F. Chen, Spectral analysis and multidimensional stability of traveling waves for nonlocal Allen-Cahn equation, J. Math. Anal. Appl., 273 (2002), 45-57. MR 1933014 (2003h:35104)
5.: P.W. Bates, J. Han, The Neumann boundary problem for a nonlocal Cahn-Hilliard equation, J. Differential Equations, 212 (2005), 235-277. MR 2129092 (2005m:35141)
6.: M. Brokate, J. Sprekels, ``Hysteresis and Phase Transitions'', Springer, New York, 1996. MR 1411908 (97g:35127)
7.: G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. MR 816623 (87c:80011)
8.: J.C. Cahn, J.E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.
9.: R. Chill, On the $\L$ ojasiewicz-Simon gradient inequality, J. Funct. Anal., 201 (2003), 572-601. MR 1986700 (2005c:26019)
10.: R. Chill, M.A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations, Nonlinear Anal., 53 (2003), 1017-1039. MR 1978032 (2004d:34103)
11.: E. Feireisl, F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions, J. Dynam. Differential Equations, 12 (2000), 647-673. MR 1800136 (2002g:35116)
12.: E. Feireisl, F. Issard-Roch, H. Petzeltová, A non-smooth version of the $\L$ ojasiewicz-Simon theorem with applications to non-local phase-field systems, J. Differential Equations, 199 (2004), 1-21. MR 2041509 (2005c:35284)
13.: H. Gajewski, On a nonlocal model of non-isothermal phase separation, Adv. Math. Sci. Appl., 12 (2002), 569-586. MR 1943981 (2003k:35103)
14.: H. Gajewski, K. Zacharias, On a nonlocal phase separation model, J. Math. Anal. Appl., 286 (2003), 11-31. MR 2009615 (2004i:35163)
15.: P. Galenko, D. Jou, Diffuse-interface model for rapid phase transformations in nonequilibrium systems, Phys. Rev. E, 71 (2005), 046125(13).
16.: G. Giacomin, J.L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits, J. Statist. Phys., 87 (1997), 37-61. MR 1453735 (98m:82053)
17.: G. Giacomin, J.L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. II. Interface motion, SIAM J. Appl. Math., 58 (1998), 1707-1729 (electronic). MR 1638739 (99m:35249)
18.: C. Giorgi, M. Grasselli and V. Pata, Uniform attractors for a phase field model with memory and quadratic nonlinearity, Indiana Univ. Math. J., 48 (1999), 1395-1445. MR 1757078 (2001h:37160)
19.: M. Grasselli, A. Miranville, V. Pata, S. Zelik, Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials, Math. Nachr., to appear.
20.: M. Grasselli, V. Pata, Existence of a universal attractor for a parabolic-hyperbolic phase-field system, Adv. Math. Sci. Appl., 13 (2003), 443-459. MR 2029927 (2004k:37172)
21.: M. Grasselli, V. Pata, Asymptotic behavior of a parabolic-hyperbolic system, Commun. Pure Appl. Anal., 3 (2004), 849-881. MR 2106302 (2005h:35150)
22.: M. Grasselli, H. Petzeltová, G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwendungen, 25 (2006), 51-72. MR 2216881 (2007b:35159)
23.: M. Grasselli, H. Petzeltová, G. Schimperna, Convergence to stationary solutions for a parabolic-hyperbolic phase-field system, Commun. Pure Appl. Anal., 5 (2006), 827-838. MR 2246010
24.: S.-Z. Huang, P. Takác, Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear Anal., 46 (2001) 675-698. MR 1857152 (2002f:35125)
25.: M.A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon, J. Funct. Anal., 153 (1998), 187-202. MR 1609269 (99c:35101)
26.: P. Krejcí, J. Sprekels, Nonlocal phase-field models for non-isothermal phase transitions and hysteresis, Adv. Math. Sci. Appl., 14 (2004), 593-612. MR 2111831 (2006d:74063)
27.: P. Krejcí, J. Sprekels, Long time behaviour of a singular phase transition model, Discrete Contin. Dyn. Syst., 15 (2006), 1119-1135. MR 2224500
28.: P. Krejcí, E. Rocca, J. Sprekels, Nonlocal temperature-dependent phase-field models for non-isothermal phase transitions, WIAS preprint n. 1006, Berlin (2005).
29.: S. $\L$ ojasiewicz, Une propriété topologiqque des sous-ensembles analytiques réels, in Colloques internationaux du C.N.R.S. 117: Les équations aux dérivées partielles (Paris, 1962), 87-89. Editions du C.N.R.S., Paris, 1963. MR 0160856 (28:4066)
30.: S. $\L$ ojasiewicz, ``Ensembles Semi-analytiques'', notes, I.H.E.S., Bures-sur-Yvette, 1965.
31.: L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Ann. of Math. (2), 118 (1983), 525-571. MR 727703 (85b:58121)
32.: J. Sprekels, S. Zheng, Global existence and asymptotic behaviour for a nonlocal phase-field model for non-isothermal phase transitions, J. Math. Anal. Appl., 279 (2003), 97-110. MR 1970493 (2004c:45015)
33.: X. Wang, Metastability and stability of patterns in a convolution model for phase transitions, J. Differential Equations, 183 (2002), 434-461. MR 1919786 (2003f:35157)
34.: H. Wu, M. Grasselli, S. Zheng, Convergence to equilibrium for a parabolic-hyperbolic phase-field system with Neumann boundary conditions, Math. Models Methods Appl. Sci., 17 (2007), 125-153. MR 2290411
35.: S. Zheng, ``Nonlinear Evolution Equations'', Chapman & Hall/CRC, Boca Raton, Florida, 2004. MR 2088362 (2006a:35001)

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 34D05, 35B40, 35Q99, 80A22

Maurizio Grasselli
Affiliation: Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi, 9, I-20133 Milano, Italy
Email: maurizio.grasselli@polimi.it

Hana Petzeltová
Affiliation: Mathematical Institute AS CR, Zitná, 25, CZ-115 67, Praha 1, Czech Republic
Email: petzelt@math.cas.cz

Giulio Schimperna
Affiliation: Dipartimento di Matematica, Università di Pavia, Via Ferrata, 1, I-27100 Pavia, Italy
Email: giusch04@unipv.it

PII: S0033-569X-07-01070-9
Received by editor(s): May 8, 2006
Posted: July 19, 2007
Additional Notes: This work was partially supported by the Italian PRIN Research Project \textit{Problemi a frontiera libera, transizioni di fase e modelli di isteresi}
The work of the second author was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan no. AV0Z10190503 and by Grant IAA1001190606 of GA AV CR
{The work of the last author was partially supported by the HYKE Research Training Network}
Copyright of article: Copyright 2007, Brown University

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