Asymptotics of the porous media equation via Sobolev inequalities

Let M be a compact Riemannian manifold without boundary. Consider the porous media equation ˙u= (um), u(0)= u0 ∈ Lq , being the Laplace–Beltrami operator. Then, if q 2 ∨ (m − 1), the associated evolution is Lq − L∞ regularizing at any time t >0 and the bound u(t) ∞ C(u0)/t holds for t <1 for suitable explicit C(u0), . For large t it is shown that, for general initial data, u(t) approaches its time-independent mean with quantitative bounds on the rate of convergence. Similar bounds are valid when the manifold is not compact, but u(t) approaches u ≡ 0 withdif ferent asymptotics. The case of manifolds withboundary and homogeneous Dirichlet, or Neumann, boundary conditions, is treated as well. The proof stems from a new connection between logarithmic Sobolev inequalities and the contractivity properties of the nonlinear evolutions considered, and is therefore applicable to a more abstract setting. © 2005 Elsevier Inc. All rights reserved.