Conditions for correctness of anisotropic diffusion equations in filtering theory (Q1386963)

In their paper [J. Scale-space and edge-detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell. 12, No. 7 (1990)] \textit{P. Perona} and \textit{J. Malik} suggest to use a nonlinear modification of the heat-equation to enhance noisy images. More specifically, the idea is to make the heat conductivity coefficient dependent on the first derivatives of the unknown solution \(u=u(x,t)\) (where \(x\in \mathbb{R}^n)\) in such a way that it tends to zero when its gradient grows: \[ {\partial u\over \partial t}= \nabla\bigl( c(\nabla u) \nabla u \bigr) \] and the positive conductivity coefficient \(c:\mathbb{R}^n\to\mathbb{R}\) tends to zero as \(\| \nabla u \|\) increases. The rationale for doing is that diffusion is stopped (or at least greatly slowed down) at places where large gradients are present (typically around edges) whereas it can proceed unhampered whenever gradients are more typical. The purpose of this note is to establish the conditions under which this initial boundary value problem is well-posed. The main result of this short note is a sufficient condition of a correct initial boundary value problem for nonlinear diffusion equation when the heat conductivity coefficient depends on the gradient of unknown variables.