We discuss in this paper equations describing processes involving non-linear and higher-order diffusion. We focus on a particular case (ut = 2λ2(uux)x + λ2uxxxx), which is put into analogy with the KdV equation. A balance of nonlinearity and higher-order diffusion enables the existence of self-similar solutions, describing diffusive shocks. These shocks are continuous solutions with a discontinuous higher-order derivative at the shock front. We argue that they play a role analogous to the soliton solutions in the dispersive case. We also discuss several physical instances where such equations are relevant.