Poincaré inequalities for linearizations of very fast diffusion equations, Nonlinearity 15

In this paper we investigate the largetime asymptotic of linearized very fast diffusion equations with and without potential conÞnements. These equations do not satisfy in general logarithmic Sobolev inequalities, but, as we show by using the BakryEmery reverse approach, in the conÞned 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 conÞnement and algebraic for the pure diffusive case. These results hold for arbitrary algebraically large diffusion speeds, if the solutions have the massconservation property