A regularized Perona-Malik functional: some aspect of the gradient-dynamic

We consider an elliptic regularization of the classical Perona-Malik functional (in 1-D) obtained by adding a small quadratic term depending on the second derivative. After a suitable rescaling we characterize the Gamma-limit for vanishing value of the singular parameter. We then introduce a "slow time" and study the gradient dynamic determined by the Gamma-limit functional. We also discuss some numerical experiments and some relationship between the gradient dynamic corresponding to the regularized functional and the one corresponding to the Gamma-limit.