Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



A nonlocal phase-field system with inertial term

Authors: Maurizio Grasselli, Hana Petzeltová and Giulio Schimperna
Journal: Quart. Appl. Math. 65 (2007), 451-469
MSC (2000): Primary 34D05, 35B40, 35Q99, 80A22
DOI: https://doi.org/10.1090/S0033-569X-07-01070-9
Published electronically: July 19, 2007
MathSciNet review: 2354882
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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 $ (-1,1)$. 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 \Lojasiewicz-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.

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Additional Information

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, Žitná, 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

DOI: https://doi.org/10.1090/S0033-569X-07-01070-9
Received by editor(s): May 8, 2006
Published electronically: July 19, 2007
Additional Notes: This work was partially supported by the Italian PRIN Research Project 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 ČR
The work of the last author was partially supported by the HYKE Research Training Network
Article copyright: © Copyright 2007 Brown University

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