Poincare Inequalities for linearizations of very fast diffusion equations

CARRILLO, JOSE A.; LEDERMAN, CLAUDIA; MARKOWICH, PETER A.; TOSCANI, GIUSEPPE

NONLINEARITY

Institute of Physics Publishing

Lugar: Bristol, UK; Año: 2002 vol. 15 p. 565 - 580

0951-7715

In this paper we investigate the large--time asymptotics of linearized very fast diffusion equations with and without potential confinements. These equations do not satisfy in general logarithmic Sobolev inequalities, but, as we show by using the ``Bakry--Emery reverse approach´´, in the confined case they have a positive spectral gap at the eigenvalue zero. We present estimates for this spectral gap and draw conclusions on the time decay of the solution, which we show to be exponential for the problem with confinement and algebraic for the pure diffusive case.  These results hold for arbitrary algebraically large diffusion speeds, if the solutions have the mass--conservation property.