A nonlocal phase-field system with inertial term

Grasselli, Maurizio; Petzeltová, Hana; Schimperna, Giulio

doi:10.1090/S0033-569X-07-01070-9

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 Łojasiewicz-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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