NONLOCAL PHASE-FIELD SYSTEMS WITH GENERAL POTENTIALS

作者：Grasselli Maurizio^*; Schimperna Giulio

来源：Discrete and Continuous Dynamical Systems, 2013, 33(11-12): 5089-5106.

DOI：10.3934/dcds.2013.33.5089

• Phase-field models
• regular and singular potentials
• nonlocal operators
• well-posedness
• uniform regularization properties
• finite dimensional global attractors

We consider a phase-field model of Caginalp type where the free energy depends on the order parameter chi in a nonlocal way. Therefore, the resulting system consists of the energy balance equation coupled with a nonlinear and nonlocal ODE for chi. Such system has been analyzed by several authors, in particular when the configuration potential is a smooth double-well function. More recently, the first author has established the existence of a finite-dimensional global attractor in the case of a potential defined on (-1, 1) and singular at the endpoints. Here we examine both the case of regular potentials as well as the case of physically more relevant singular potentials (e.g., logarithmic). We prove well-posedness results and the eventual global boundedness of solutions uniformly with respect to the initial data. In addition, we show that the separation property holds in the case of singular potentials. Thanks to these results, we are able to demonstrate the existence of a finite-dimensional global attractor in the present cases as well.

• 出版日期2013-12