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A non-local, phenomenological continuum equation describing surface growth in unstable systems with anomalous scaling is presented. The roughness of unstable surface is first studied considering different origin of surface diffusion processes (temperature, flux, diffusion anisotropy). Then, the non-locality is assumed to come from self-pinning of the interface at slowly growing deep grooves. Our description is applied to the Lai-Das Sarma-Tamborenea (LDV) model, whose exponents are computed analytically for the first time. The continuum equation is then solved in a generalized form and compared to the results of a two-dimensional atomistic model for deposition on vicinal crystal surfaces. The atomistic model exhibits kinetic roughening due to limited interface diffusion and unstable growth leading to a self-organised pattern with deep grooves, then to mounding originated from Ehrlich-Schwoebel effect in step edges. It is found that the surface height-height correlation function shows multiple scaling behavior in the instable growth regime, such as super-roughening, anomalous scaling and multiscaling. These critical phenomena are discussed in detail in many different growth regimes and we find that their scaling exponents computed numerically agree very well with the continuum equation description. |