ENTROPY ESTIMATES AND LARGE-TIME BEHAVIOR OF SOLUTIONS TO A FOURTH-ORDER NONLINEAR DEGENERATE EQUATION

The main aim of this paper is to give some theoretical analysis for a nonlinear fourth-order degenerate equation related to image processing, with homogeneous Neumann and no-flux boundary conditions. The asymptotic behavior of solutions to the equation is discussed in the paper using the entropy dissipation method. We firstly derive some entropy estimates via the algebraic approach, and present a polynomial decay of an entropy. As a result, we prove solutions of the equation eventually converge to equilibrium, which can help us understand the model in greater depth. To the best of our knowledge, this is the first result on the long-time behavior of the fourth-order model in image processing.