Existence of Solutions to a Cahn–Hilliard Type Equation with a Logarithmic Nonlinear Term

  • Alain MiranvilleEmail author


Our aim in this paper is to prove the existence of solutions to a Cahn–Hilliard type equation with a proliferation term and a logarithmic nonlinear term. Such an equation was proposed in view of biological applications. The main difficulty comes from the fact that we no longer have the conservation of the spatial average of the order parameter, contrary to the original Cahn–Hilliard equation. This makes the derivation of uniform (with respect to the regularization parameter) estimates on the solutions to approximated problems delicate, as blow up in finite time may occur.


Cahn–Hilliard equation proliferation term logarithmic nonlinear term existence blow up 

Mathematics Subject Classification

35K55 35B45 



The author wishes to thank an anonymous referee for several useful comments.


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Authors and Affiliations

  1. 1.School of Mathematical SciencesXiamen UniversityXiamenChina
  2. 2.Laboratoire de Mathématiques et Applications UMR CNRS 7348, Equipe DACTIM-MISUniversité de PoitiersChasseneuil Futuroscope CedexFrance

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