Smoothing effects for the filtration equation with different powers

TL;DR

This paper investigates the smoothing effects of a nonlinear diffusion equation with variable powers under Neumann boundary conditions, extending existing results to cases with different growth exponents.

Contribution

It establishes sharp smoothing estimates for the filtration equation with different powers, addressing a gap in the literature for Neumann problems.

Findings

Derived sharp short-time $L^{q_0}$-$L^$ smoothing estimates.

Extended previous results to equations with different growth exponents.

Addressed the Neumann boundary condition case for the first time with variable powers.

Abstract

We study the nonlinear diffusion equation $ u_t=\Delta\phi(u) $ on general Euclidean 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 $ 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^\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.