Local Boundedness of Weak Solutions to the Diffusive Wave Approximation of the Shallow Water Equations

In this paper we prove that weak solutions to the Diffusive Wave Approximation of the Shallow Water equations $$\partial_t u - \nabla\cdot ((u-z)^\alpha|\nabla u|^{\gamma-1}\nabla u) = f$$ are locally bounded. Here, $u$ describes the height of the water, $z$ is a given function that represents the land elevation and $f$ is a source term accounting for evaporation, infiltration or rainfall.