This talk will be concerned with the equation $$u_t + \mathrm{div} f (u) + (-\Delta)^{\alpha/2}\varphi(u) = 0,$$ where $\alpha\in(0, 2)$ and $\varphi$ is a nondecreasing nonlinearity. It will focus on continuous dependence estimates with respect to the nonlinearities and the fractional power $\alpha$. The results are optimal and robust as the fractional Laplacian approaches the classical one, thus giving a new proof of the know estimates for classical degenerate parabolic problems. Joint work with Simone Cifani and Espen R. Jakobsen (Norwegian University of Science and Technology, Trondheim, Norway).
