Special Fast Diffusion with Slow Asymptotics: Entropy Method and Flow on a Riemannian Manifold |
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Authors: | Matteo Bonforte Gabriele Grillo Juan Luis Vázquez |
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Institution: | 1. Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049, Madrid, Spain 2. Dipartimento di Matematica, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129, Torino, Italy
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Abstract: | We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation ${u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)}We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation ut=D(um/m) = div (um-1 ?u){u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)} posed for
x ? \mathbb Rd{x\in\mathbb R^d}, t > 0, with a precise value for the exponent m = (d − 4)/(d − 2). The space dimension is d ≧ 3 so that m < 1, and even m = −1 for d = 3. This case had been left open in the general study (Blanchet et al. in Arch Rat Mech Anal 191:347–385, 2009) since it requires quite different functional analytic methods, due in particular
to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted
here as the heat flow of the Laplace– Beltrami operator of a suitable Riemannian Manifold
(\mathbb Rd,g){(\mathbb R^d,{\bf g})}, with a metric g which is conformal to the standard
\mathbb Rd{\mathbb R^d} metric. Studying the pointwise heat kernel behaviour allows to prove suitable Gagliardo–Nirenberg inequalities associated
with the generator. Such inequalities in turn allow one to study the nonlinear evolution as well, and to determine its asymptotics,
which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear
Fokker–Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known
exponential decay of such representation for all other values of m. |
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