Asymptotics of the Porous Media Equation via Sobolev Inequalities

Let M be a compact Riemannian manifold without boundary. Consider the porous media equation u̇ = 4(um), u(0) = u0 ∈ Lq , 4 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 (1) holds for t < 1 for suitable explicit C(u0), γ. 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.