A Continuum of Extinction Rates for the Fast Diffusion Equation

We find a continuum of extinction rates for solutions $u(y,\tau)\ge 0$ of the fast diffusion equation $u_\tau=\Delta u^m$ in a subrange of exponents $m\in (0,1)$. The equation is posed in $\ren$ for times up to the extinction time $T>0$. The rates take the form $\|u(\cdot,\tau)\|_\infty\sim (T-\tau)^\theta$ \ for a whole interval of $\theta>0$. These extinction rates depend explicitly on the spatial decay rates of initial data.


Introduction
We consider the Cauchy problem for the fast diffusion equation: u τ = ∆(u m /m), y ∈ R n , τ ∈ (0, T ), u(y, 0) = u 0 (y) ≥ 0, y ∈ R n , (1.1) where m ∈ (0, 1) and T > 0. The factor 1/m is not essential; it is inserted into the equation for normalization so that it can also be written as u τ = ∇ · (u m−1 ∇u). In that way, it is readily seen that the diffusion coefficient c(u) = u m−1 → ∞ as u → 0 if m < 1, hence the name Fast Diffusion Equation (but notice that c(u) → 0 as u → ∞). Furthermore, it is known that for m below a critical exponent m c = (n − 2)/n all solutions with initial data in some convenient space, like L p (R n ) with p = n(1 − m)/2, extinguish in finite time. We will always work in this range, m < m c , and consider solutions which vanish in a finite time. The purpose of this paper is to study the rates of extinction of such solutions. Our main contribution is to provide a continuum of rates of extinction for fixed m. Technical reasons imply that m must be in the range 0 < m < m * = (n − 4)/(n − 2), n ≥ 5, for the construction to work. This restriction may be essential.
Let us review the state of the question from a broader perspective. The description of the asymptotic behaviour of the global in time solutions of (1.1) as τ → ∞ for m ≥ m c is a very active subject, and the study has been extended in recent times to the behaviour near extinction for m < m c , both in bounded domains or in the whole space. In the former case, the rate of decay for bounded solutions is universal, of the form u(·, τ ) ∞ = O((T − τ ) 1/(1−m) ) when m > m s = (n − 2)/(n + 2), cf. [1,6], but the question is more complicated when m ≤ m s .
In the case of the whole space, which is the one of interest here, the book [10] contains a general description of the phenomenon of extinction, where it is explained that not only the occurrence of extinction depends on the size of the initial data, but also that different initial data may give rise to different extinction rates, even for the same extinction time; this may happen for all 0 < m < m c . It is also proved, cf. [10] and quoted references, that the size of the initial data at infinity (the tail of u 0 ) is very important in determining both the extinction time and the decay rates.
Special attention has been given recently to particular classes of data that produce definite estimates. This happens in the case of data with the maximal decay rate compatible with extinction in finite time, which is as |y| → ∞. Note that µ < n for m < m c so these data are not integrable. Thus, the papers [5,2,4] are concerned with the stabilization as τ → T of general solutions towards some special self-similar solutions U D,T known as the generalized Barenblatt solutions, given by the formula where for m < m c we put R(τ ) := (T − τ ) −β , and .
Here T ≥ 0 (extinction time) and D > 0 are free parameters. Note that R depends on T . It has been proved that the corresponding Barenblatt solutions with exponent m > m c play the role of the Gaussian solution of the linear diffusion equation in describing the asymptotic behaviour of a very wide class of nonnegative solutions, i.e., those with initial data in L 1 (R n ), cf. [11]. To some extent, the solutions (1.3) play a similar role for m < m c but their basin of attraction may be much smaller. This is precisely described in [4], with results on the basin of attraction of the family of generalized Barenblatt solutions; it establishes the optimal rates of convergence of the solutions of (1.1) towards a unique attracting limit state in that family. All of these solutions will have a decay rate near extinction of the form u(·, τ ) ∞ = O((T − τ ) nβ ), and it is clear that nβ > 1/(1 − m).
The question that we address here is the following: Can we obtain different decay rates near extinction for bounded data u 0 (y) that behave at infinity in first approximation like the singular solution, i. e., u 0 (y) ∼ A |y| −µ ? We will show that the answer is yes, and actually we will obtain a whole continuum of rates.
and let the initial function u 0 be continuous, bounded, and satisfy the conditions: for some A, c 1 , c 2 > 0, and Then the solution has complete extinction precisely at the time T = (A/k * ) 1−m > 0, and there are positive constants K 1 , K 2 such that for 0 < τ < T we have It is easy to check that under the above assumptions θ covers an interval [θ min , θ max ) with 0 < θ min < θ max = µn/2(n − µ) = nβ. This is the precise range of extinction rates of these solutions, to be compared with the standard extinction rate (T − τ ) nβ of the Barenblatt examples.
As a precedent to this result, the existence of different rates was established in Theorem 7.4 of [10] for all m < m c by means of the construction of self-similar solutions of the form u(y, τ ) = (T − τ ) α f (y (T − τ ) β ). In this way a whole interval (α, ∞) is covered, which extends the scope of our present theorem. However, α (the anomalous exponent) is not explicit, we obtain only one solution for each time-decay rate and the dependence of α on the spatial behavior of the data is not analyzed. Theorem 1.1 clarifies these aspects, explaining the delicate relationship between both limits, |y| → ∞ for u 0 and τ → T for u(y, τ ).
The proof of the theorem needs techniques that are only natural after rescaling the problem. In fact, the rescaled problem allows us to formulate and prove a more precise result about the dependence of the rate on the tail of the data and the convergence of the spatial shapes. We devote the next section to the presentation of the rescaling transformation, the resulting rescaled equation and the asymptotic convergence plus growup result in that context. Sections 3-5 will be concerned with proving the result for the rescaled problem. The last section is devoted to comments and open problems.
Notations. Throughout the rest of the paper and unless mention to the contrary, we keep the conditions n ≥ 5 and m < m * . The exponent m * also plays a big role in the asymptotic results of [2,3,4]. We also keep the above symbols and variables. In particular, µ = 2/(1 − m) so that m < m c means µ < n and m < m * means µ + 2 < n.

The rescaled flow
As we have just said, it is very convenient to rescale the flow and rewrite (1.1) in self-similar variables by introducing the time-dependent change of variables with R as above, and the rescaled function v(x, t) := R(τ ) n u(y, τ ).
In these new variables, the generalized Barenblatt functions U D,T (y, τ ) are transformed into generalized Barenblatt profiles V D (x), which are stationary: If u is a solution to (1.1), then v solves the rescaled fast diffusion equation which is a nonlinear Fokker-Planck equation (NLFP). We put as initial condition v 0 (x) := R(0) −n u 0 (y), where x and y are related according to (2.1) with τ = 0, x = cy. Roughly speaking, v 0 is a rescaling of u 0 depending only on T . We have taken the precise form of this transformation from [4]. Note also that the factors 1/m and µ in equation (2.4) can be eliminated by manipulating the change of variables, but then the expression of the Barenblatt solutions would contain new constants. Thus, in our scaling the singular solution becomes

Main result for the NLFP equation
In the following sections we consider the v-equation (2.4) with initial data given by a bounded function 0 ≤ v 0 ≤ V 0 , and such that the difference V 0 − v 0 has a tail controlled by a power rate. This is our detailed result about asymptotic behaviour of the solution whose initial data v 0 (x) are perturbations of the steady state V 0 (x).
Theorem 2.1 Assume that n and m are as in (1.5). Suppose that v 0 is continuous, bounded and nonnegative, and fulfils where l is as in (1.6) and c 1 , c 2 > 0. Assume also that v 0 (x) ≤ |x| −µ for all x = 0. Let v denote the solution of (2.4). Then: (ii) For each r 0 > one can find C 1 , C 2 > 0 such that for t ≥ 1 and |x| ≥ r 0 the following holds Let us comment on the contents and scope of the result.
1. First of all, it states the two main aspects of the convergence of the solution v(·, t) towards the singular steady state V 0 : (2.8) establishes the uniform convergence of v(·, t) towards V 0 in the complement of a ball centered at the origin, with a precise rate that depends explicitly on the tail decay exponent l. On the other hand, estimate (2.7) gives the exact rate of growth of the solutions as t → ∞ to account for the approach to the singular value V 0 (0) = +∞.
2. An important feature of the result is the existence of a continuum of grow-up rates for v(·, t) ∞ , and a corresponding continuum of stabilization rates of v(·, t) towards V 0 in the outer region. Note furthermore that as l approaches the lower value µ + 2, the rates go to zero. This limit case is on the other hand easier and does not produce any convergence, since we can consider the example of the generalized Barenblatt solutions V D given in (2.3). Indeed, they satisfy 0 < V D < V 0 and Since they are stationary, no convergence to V 0 holds in this case.
3. The conditions on l imply that the perturbation V 0 − v 0 is never integrable, contrary to the usual assumptions made in variational methods. Let us now examine the maximal grow-up rate that we have achieved. Note first that γ(µ + 2) = γ(n) = 0. The maximum of γ in (2.7) is attained at l = L, and This is lower than the maximal growth rate of any bounded solution that is given by the growth of the spatially homogeneous solutionṽ(t) = ce µnt . We conjecture that γ(L) is the largest exponent that can be achieved by the solutions under the conditions of the theorem, even if we allow l to be larger than L. The bound from below follows immediately from the lower bound in (2.7), but to obtain the corresponding bound from above is still an open problem.

4.
As m → m * we have µ + 2 → n and the interval (µ + 2, L] shrinks to the empty set while the admissible values of the exponents γ and λ go to zero.

5.
Results similar as in Theorem 2.1 were obtained for the standard Fujita equation in [7,8,9].
6. Finally, we apply the results of Theorem 2.1 (i) to prove Theorem 1.1. Notice that under the assumptions of Theorem 1.1, if we take the prescribed value of T then v 0 satisfies the hypotheses of Theorem 2.1, so that the solution v is global in time and stabilizes to V 0 ; this means that the extinction time of u is precisely T . The extinction rate of u is obtained by rewriting the bounds in (2.7). Recall that and T − τ = T e −2(n−µ)t . The conclusion follows.

Auxiliary results for the rescaled problem
After the previous transformation, in the radially symmetric case we end up with the problem An important role is played by the quadratic equation is positive if µ + 2 < l < n. The roots α − and α + of (3.2) are given by 4) and the following way to rewrite α ± indicates why the value l = µ + 2(n − µ) plays an important role in the sequel (cf. Section 5).
The following two lemmata apply to parameters n, m and κ more general than required in (1.5) and (3.3).
as well as at such points. Thus, in view of the identity m + 2 µ = 1 and, equivalently, µm if (r, t) ∈ S. As rξ r = ξ, we finally have and therefore obtain from (3.7) and (3.8) that which after a straightforward rearrangement yields (3.5).
where A is defined by (3.6).
Proof. We directly compute and thereby immediately obtain (3.9).

Lower bound
Once we have the preparatory material, we proceed next to establish the lower bound for the solutions mentioned in Theorem 2.1. This is the content of Proposition 4.2. Section 5 will contain the proof of the corresponding upper bound, Proposition 5.3. and such that ξ α ψ(ξ) → a as ξ → ∞.
Given r 0 > 0, this easily yields (4.9) upon an obvious choice of C 2 .

Upper bound and proof of Theorem 2.1
Lemma 5.1 Assume (1.5), and let l ∈ (µ + 2, n) and α − be as defined in (3.4) with κ given by (3.3). Then there exist β > α − and C β > 0 with the following property: Suppose that A > 0 and B > 0 are such that

1)
and let Then the function ψ out defined by
As we have said, Propositions 4.2 and 5.3 together imply Theorem 2.1.
6 Comments and open problems , is an interesting related problem. The difference with the above analysis is that the v-profile is regular, so no grow-up is expected if l > µ + 2.
Since the behaviour of V D at infinity is similar to the singular one, V 0 , and V D is still stationary, we also expect a continuum of convergence rates depending on l from a certain range. In this case we have to mention that for l > n there is a variational theory developed in the recent papers [2,3,4] that proves convergence with rate using the techniques of entropies, linearization and functional inequalities.

5.
Our methods are not variational and our solutions do not belong to the usual spaces of that theory, like spaces of finite relative energy or finite relative mass.