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

摘要

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.