Smoothing effects for the filtration equation with different powers

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