WIAS Preprint No. 2950, (2022)

On a Cahn--Hilliard system with source term and thermal memory


  • Colli, Pierluigi
    ORCID: 0000-0002-7921-5041
  • Gilardi, Gianni
    ORCID: 0000-0002-0651-4307
  • Signori, Andrea
    ORCID: 0000-0001-7025-977X
  • Sprekels, Jürgen
    ORCID: 0009-0000-0618-8604

2020 Mathematics Subject Classification

  • 35K55 35K51


  • Non-isothermal Cahn--Hilliard equation, thermal memory, well-posedness, Cahn--Hilliard equation with source term, Cahn--Hilliard--Oono equation




A nonisothermal phase field system of Cahn--Hilliard type is introduced and analyzed mathematically. The system constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. It couples a Cahn--Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails additional mathematical difficulties because the mass conservation of the order parameter is lost. We provide several mathematical results under general assumptions on the source term and the double-well nonlinearity governing the evolution: existence and continuous dependence results are shown for weak and strong solutions to the corresponding initial-boundary value problem.

Appeared in

  • Nonlinear Analysis, 240 (2024), pp. 113461/1--113461/16 (published online on 14.12.2023), DOI 10.1016/j.na.2023.113461 .

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