Asymptotics of the porous media equation via Sobolev inequalities.

Let M be a compact Riemannian manifold without boundary. Consider the porous media equation u_t =\Delta (|u|^(m-1)m), u(0)= u0 ∈ L^q , \Delta being the Laplace–Beltrami operator. Then, for a suitable range of q, the associated evolution is L^q − L^\infty regularizing at any time t >0. 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 with different asymptotics. The case of manifolds with boundary 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.