Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities

Abstract

The goal of this paper is to state the optimal decay rate for solutions of the nonlinear fast diffusion equation and, in self-similar variables, the optimal convergence rates to Barenblatt self-similar profiles and their generalizations. It relies on the identification of the optimal constants in some related Hardy-Poincaré inequalities and concludes a long series of papers devoted to generalized entropies, functional inequalities, and rates for nonlinear diffusion equations. T he evolution equationwith m ≠ 1 is a simple example of a nonlinear diffusion equation which generalizes the heat equation and appears in a wide number of applications. Solutions differ from the linear case in many respects, notably concerning existence, regularity, and large-time behavior. We consider positive solutions uðτ; yÞ of this equation posed for τ ≥ 0 and y ∈ R d , d ≥ 1. The parameter m can be any real number. The equation makes sense even in the limit case m ¼ 0, where u m ∕m has to be replaced by log u, and is formally parabolic for all m ∈ R. Notice that [1] is degenerate at the level u ¼ 0 when m > 1 and singular when m < 1. We consider the initial-value problem with nonnegative datum uðτ ¼ 0; ·Þ ¼ u 0 ∈ L 1 loc ðdxÞ, where dx denotes Lebesgue's measure on R d . Further assumptions on u 0 are needed and will be specified later.The description of the asymptotic behavior of the solutions of [1] as τ → ∞ is a classical and very active subject. If m ¼ 1, the convergence of solutions of the heat equation with u 0 ∈ L 1 þ ðdxÞ to the Gaussian kernel (up to a mass factor) is a cornerstone of the theory. In the case of Eq. 1 with m > 1, known in the literature as the porous medium equation, the study of asymptotic behavior goes back to ref. On the other hand, when m < m c , a natural extension for the Barenblatt functions can be achieved by considering the same expression [2], but a different form for R, that is,The parameter T now denotes the extinction time, an important feature. The limit case m ¼ m c is covered by RðτÞ ¼ e τ , U D;T ðτ; yÞ ¼ e −dτ ðD þ e −2τ jyj 2 ∕dÞ −d∕2 . See refs. 4 and 5 for more detailed considerations. In this paper, we shall focus our attention on the case m < 1 which has been much less studied. In this regime, [1] is known as the fast diffusion equation. We do not even need to assume m > 0. We shall summarize and extend a series of recent results on the basin of attraction of the family of generalized Barenblatt solutions and establish the optimal rates of convergence of the solutions of [1] toward a unique attracting limit state in that family. Such basin of attraction is different according to m being above or below the value m à ≔ðd − 4Þ∕ðd − 2Þ, and for m ¼ m à the long-time behavior of the solutions has specific features. To state our results, it is more convenient to rescale the flow and rewrite

Nonlinear Diffusion with Fractional Laplacian Operators

“…It has been studied by various authors in recent times in considerable detail, and general accounts are given in [67], [21].…”

Rate of Convergence to Barenblatt Profiles for the Fast Diffusion Equation

“…The exponent m * plays a very important role in the results of [2][3][4]. The proofs of convergence with rates are based on the study of the decay in time of a certain relative entropy and a careful analysis of the linearized problem which leads to certain functional inequalities of the Hardy-Poincaré type.…”

“…We state the initial condition where x and y are related according to (3) with τ = 0. We have taken the precise form of this transformation from [2].…”

“…This fastest rate is the best constant in a Hardy-Poincaré inequality (see [3]). This best constant is also the bottom of the continuous spectrum of the linearization on a suitable weighted space (see [2,3,10]). In the figure below, the gray area indicates the decay exponents that we obtain in Theorem 1 for n = 6.…”

Slow growth of solutions of superfast diffusion equations with unbounded initial data

“…Beyond this, the literature has provided more detailed information on how the asymptotic behaviour near extinction depends on the initial spatial decay, again indicating an important role of self-similar solutions (see e.g. [2], [3], [4], [5], [11], [10], [14], [16], [20]). …”