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We study the nonlinear diffusion equation $ u_t=\Delta\phi(u) $ on generalEuclidean domains, with homogeneous Neumann boundary conditions. We assume that$ \phi^\prime(u) $ is bounded from below by $ |u|^{m_1-1} $ for small $ |u| $and by $ |u|^{m_2-1} $ for large $|u|$, the two exponents $ m_1,m_2 $ beingpossibly different and larger than one. The equality case corresponds to thewell-known porous medium equation. We establish sharp short- and long-time $L^{q_0} $-$ L^\infty $ smoothing estimates: similar issues have widely beeninvestigated in the literature in the last few years, but the Neumann problemwith different powers had not been addressed yet. This work extends someprevious results in many directions.

Smoothing effects for the filtration equation with different powers