Quantitative Local and Global A Priori Estimates for Fractional Nonlinear Diffusion Equations

Quantitative Local and Global A Priori Estimates for Fractional Nonlinear Diffusion Equations

Abstract

We establish quantitative estimates for solutions to the fractional nonlinear diffusion equation,  in the whole range of exponents , . The equation is posed in the whole space . We first obtain weighted global integral estimates that allow to establish existence of solutions for classes of large data. In the core of the paper we obtain quantitative pointwise lower estimates of the positivity of the solutions, depending only on the norm of the initial data in a certain ball. The estimates take a different form in three exponent ranges: slow diffusion, good range of fast diffusion, and very fast diffusion. Finally, we show existence and uniqueness of initial traces.


Keywords. Nonlinear diffusion equation, Fractional Laplacian, Weighted global estimates, Existence for large data, Positivity estimates, Initial trace.
Mathematics Subject Classification. 35B45, 35B65, 35K55, 35K65.

1 Introduction

We consider the class of nonnegative weak solutions of the fractional diffusion equation

(1.1)

where , , , and . The precise definition of the fractional Laplacian is given in Appendix 8.1. We are mainly interested in values , since the linear case is rather well known. For we recover the classical porous medium/fast diffusion equation (that will be shortened as PME/FDE respectively), whose theory is well-known, cf. [38]. We will call the case the standard diffusion case. We also assume that we are given initial data

(1.2)

where in principle and . However, larger classes of initial data are sometimes considered, like changing sign solutions or non integrable data. The equation has recently attracted some attention in mathematical analysis. Such an interest has been motivated by its appearance as a model for anomalous diffusion in different applied contexts. For the reader’s convenience we have listed in Appendix II (Section 9) the most relevant sources to the applications that we know of.

We refer to [30, 31] for the basic theory of existence and uniqueness of weak solutions for the Cauchy problem (1.1)(1.2). These papers also describe the main results on boundedness and regularity, they show that nonnegative solutions are indeed positive everywhere, as well as some other basic properties of the nonlinear semigroup generated by the problem. Recently, the existence and properties of Barenblatt solutions for the Cauchy Problem was established in [39]. Related literature is also mentioned in these papers.

The main purpose of the present paper is obtaining quantitative a priori estimates of a local type for the solutions of the problem. Such estimates were obtained for the standard PME by Aronson-Caffarelli [2] and by the authors for the standard FDE [9, 10, 11] . This is not always possible for the present model due to the nonlocal character of the diffusion operator, but then global estimates occur in weighted spaces. The use of suitable weight functions allows to prove crucial -weighted estimates that enter substantially into the derivation of the main results. The results take different forms according to the value of the exponent , a fact that is to be expected since it happens for standard diffusion. The list of bounds is as follows: weighted estimates, half-Harnack parabolic estimates (i. e., quantitative pointwise lower estimates), and tail estimates (i. e., asymptotic spatial behaviour). As a first consequence of these estimates, an existence result for very weak solutions with non-integrable data in some weighted space is obtained. In particular, bounded initial data or data with slow growth at infinity are allowed.

We remind that Harnack inequalities are a standard tool used to develop further theory, see in that respect the work of Di Benedetto for quasilinear equations [19, 20]. Let us quote four examples of application of the line of results of this paper that are already available: the existence and uniqueness of initial traces, which we do in Section 7; understanding the asymptotic behaviour of the fractional KPP equation [35]; dealing with nonlocal symmetrization problems [41]; or uniqueness issues for the fractional PME with variable density [33].

Outline of the paper and main results. First, some preliminary information on case divisions. The case is called the (fractional) porous medium case : contrary to the standard porous medium equation, it does not have the property of finite propagation, an important difference established in [30, 31] . The range of exponents is called the (fractional) fast diffusion equation, and it has special properties when  , which we call the good fast diffusion range. When it is known that some solutions extinguish in finite time, which is a clear manifestation of the change of character of the equation, since solutions of the Cauchy problem exist globally in time and are positive everywhere in if  .

In Section 2, we derive integral bounds in form of weighted estimates, valid for nonnegative solutions of the Cauchy problem in the whole fast diffusion range . Actually, they are valid for the difference of two ordered solutions, the precise statement is given in Theorem 2.2. Contrary to the purely local estimates known in the standard fast diffusion case, cf. [23], the estimates for are valid in weighted -spaces and the weight must decay at infinity with a certain decay rate, not too fast, not too slow. This is again a manifestation of the nonlocal properties of the fractional Laplacian. The estimates will be important as a priori bounds for solutions, or families of solutions, through the rest of the paper.

In Section 3 we use the estimates of Section 2 to construct solutions for initial data that belong to weighted -spaces, in particular for data such that grows less than as , in particular for all bounded data. These solutions can be uniquely identified as minimal solutions in a precise sense and satisfy many of the properties of the known class of bounded and integrable weak solutions.

Section 4 studies the actual positivity of nonnegative solutions via quantitative lower estimates for the good fast diffusion equation. Precise local lower bounds are contained in Theorem 4.1. The behaviour as (so-called tail behaviour) is studied in Section 4.1, and global spatial lower bounds are derived as a consequence in Section 4.2. The merit of the estimates is that they are quantitative and most of the exponents are sharp. The lower estimates of this section can be adapted for the exponent separating both fast diffusion subranges, but only when for some . However, we refrain from doing this particular case in the present paper since the proof uses some other techniques that would lengthen the text.

The very fast diffusion range is studied in Section 5. The weighted estimates of Theorem 2.2 continue to hold, but this does not allow to obtain the same type of quantitative lower bounds since the technique used in the good fast diffusion range does not work anymore. There are two problems: on the one hand the smoothing effect does not hold for general initial data, on the other hand the presence of the extinction phenomenon makes things more complicated, and the extinction time enters directly the estimates of Theorem 5.1. These difficulties have already appeared in the standard FDE, , and were treated in our paper [11]. However the technique used in that paper does not extend to and we present here a technique that is based on the careful use of weight factors, and in the limit gives a simpler proof of the result of [11]. We also study the problem of characterizing the finite time extinction in terms of the initial data; thus, we determine a class of initial data that produces solutions that extinguish in finite time, see Proposition 5.3, as well as a roughly complementary class of initial data for which the solution exists and is positive globally in time, see Corollary 5.2 .

Section 6 is devoted to study similar questions for the porous medium case. Theorem 6.1 establishes local lower bounds of the Aronson-Caffarelli type for all . The question of optimal decay as is an open problem; for selfsimilar solutions it is solved in [39].

In Section 7 we address a different question that complements our previous results, i. e., the question of existence and uniqueness of an initial trace for nonnegative weak solutions defined in a strip . The main results are stated in Theorems 7.2 and 7.4. This result can be combined in the reverse direction with the existence of solutions with initial data a nonnegative Radon measure, Theorem 4.1 of [39].

In Appendix I we collect the definitions of weak, very weak and strong solutions, together with a number of technical results. As already mentioned above, Appendix II discusses applications.

About the linear equation. Our estimates have counterparts for the linear fractional heat equation, case , that are worth commenting. Thus, the lower bound of Section 6 for passes to the limit , and this coincides with the limit of a part of the estimate for obtained in Section 4 for . See Proposition 4.3. Finally, we prove the existence and uniqueness of an initial trace also when , cf. Theorem 7.3. This is an interesting result that is not present in the literature to our knowledge and complements the uniqueness results of [4].

Notations. Throughout the paper, we fix , , , and  , which is positive if  . We will call -Laplacian of the function . This is consistent with the use in the standard case .

2 Weighted estimates in the fast Diffusion range

We will derive weighted estimates which also hold for the standard FDE (i.e., the limit case ). When the equation is nonlocal, therefore we cannot expect purely local estimates to hold. Indeed we will obtain estimates in weighted spaces if the weight satisfies certain decay conditions at infinity. We present first a technical lemma which will be used several times in the rest of the paper.

Lemma 2.1

Let and positive real function that is radially symmetric and decreasing in . Assume also that and that  , for some positive constant and for large enough. Then, for all we have

(2.1)

with positive constants that depend only on and . For the reverse estimate holds from below if :   for all  .

The proof is easy but technical, and is given in Appendix 8.4 for the reader’s convenience. We point out that the large-decay case is what makes the estimate in the fractional Laplacian case very different from the usual Laplacian case. In particular, the -Laplacian of a nonnegative smooth function with compact support is strictly positive outside of the support and has a certain decay at infinity, indeed the minimal decay is obtained for the when is compactly supported, cf. [31]. A suitable particular choice is the function defined for as for and

(2.2)

We are now ready to present the weighted estimates.

Theorem 2.2 (Weighted estimates)

Let be two ordered solutions to the equation (1.1), with . Let where and is as in the previous lemma with for and

Then, for all we have

(2.3)

with that depends only on  .

It is remarkable that the estimate holds for (very) weak solutions, maybe changing sign. Also, it is worth pointing out that the estimate holds both for and for . In the limit we recover the well known local estimates for the standard FDE.

Proof.  Step 1. A differential inequality for the weighted -norm. If is a smooth and sufficiently decaying function we have

Notice that in we have used the fact that is a symmetric operator, while in we have used that , where as mentioned. In we have used Hölder inequality with conjugate exponents and . If the last integral factor is bounded, then we get

Integrating the above differential inequality on with we obtain:

which is (2.3) once we estimate the constant , for a convenient choice of test function.

Step 2. Estimating the constant . Choose  , with as in Lemma 2.1 and  , so that

where it is easy to check that the first integral is bounded, since on  , and when with we know by estimates (2.1) that

(2.4)

therefore is finite whenever  . Note that all the constants depend only on  .      

Remark. The estimate implies the conservation of mass when , by letting . On the other hand, when solutions corresponding to with  , extinguish in finite time  , (see e.g. [31]); the above estimates provide a lower bound for the extinction time in such a case, just by letting and in the above estimates:

(2.5)

Moreover, if the initial datum is such that the limit as of the right-hand side diverges to , then the corresponding solution exists (and is positive) globally in time, as explained in Corollary 5.2 .

3 Existence of solutions in weighted -spaces

Theorem 3.1

Let and let , where is as in Theorem 2.2 with decay at infinity , . Then there exists a very weak solution to equation (1.1) on , in the sense that

This solution is continuous in the weighted space,  .

Proof. Let be as in Theorem 2.2 with the decay at infinity . Let be a non-decreasing sequence of initial data , converging monotonically to  , i. e., such that as . Consider the unique solutions of equation (1.1) with initial data . By the comparison results of [31] we know that they form a monotone sequence. The weighted estimates (2.3) show that the sequence is bounded in uniformly in  . By the monotone convergence theorem in , we know that the solutions converge monotonically as to a function . Indeed, the weighted estimates (2.3) show that when then

(3.1)

At this point we need to show that the function constructed as above is a very weak solution to equation (1.1) on  , more precisely we have to show that for all we have

(3.2)

By the results of [31] we know that each is a bounded strong solutions, since the initial data  , therefore for all we have

(3.3)

Now, for any we easily have that

since is compactly supported and we already know that in . On the other hand, for any we have that

since and

where we have used Hölder inequality with conjugate exponents and , and we notice that

since is compactly supported, therefore by Lemma 2.1 we know that , and the quotient

is integrable when that is when . In the last step we already know that when is as above, i.e. as in Theorem 2.2. Therefore we can let in (3.3) and obtain (3.2) .

For the solutions constructed above, the weighted estimates (2.3) show that when imply

(3.4)

which gives the continuity in  . Therefore, the initial trace of this solution is given by       

Remark. The solutions constructed above only need to be integrable with respect to the weight , which has a tail of order less than . Therefore, we have proved existence of solutions corresponding to initial data that can grow at infinity as for any  . Note that for the linear case this exponent is optimal in view of the representation of solutions in terms of the fundamental solution, but this does not seem to be the case for .

Theorem 3.2 (Uniqueness)

The solution constructed in Theorem 3.1 by approximation from below is unique. We call it the minimal solution. In this class of solutions the standard comparison result holds, and also the estimates of Theorem 2.2 .

Proof.  We keep the notations of the proof of Theorem 3.1. Assume that there exist another sequence which is monotonically non-decreasing and converges monotonically to  . By the same considerations as in the proof of Theorem 3.1, we can show that there exists a solution . We want to show that , where is the solution constructed in the same way from the sequence . We will prove equality by proving that and then that . To prove that we use the estimates

(3.5)

which hold for any , see Theorem 6.2 of [31] for a proof. Letting we get that

since by construction. Therefore also for , so that in the limit we obtain  . The inequality can be obtained simply by switching the roles of and  . The validity of estimates of Theorem 2.2 is guaranteed by the above limiting process. The comparison holds by taking the limits in inequality (3.5), as it has been done for -solutions in [31].      

4 Good fast diffusion range

The first result of the section will be the existence of local lower bounds. In the proof we will use Lemma 8.6, which is a simple optimization lemma that we state in Appendix 8.5 . We recall that and which is positive for  .

Figure 1: Black: Lower bounds in the two time ranges. Blue: Upper bounds (smoothing effects), which has the same behaviour when  .
Theorem 4.1 (Local lower bounds)

Let , and let , where is as in Theorem 2.2 with decay at infinity , . Let be a very weak solution to Equation (1.1) corresponding to the initial datum . Then there exists a time

(4.1)

such that

(4.2)

and

(4.3)

The positive constants depend only on and . See Fig. 1.

Remarks. (i) The lower estimate for small times is an absolute bound in the sense that it does not depend on the initial data (though does depend).
(ii) We obtain the following expressions for and and :

(4.4)

where is the constant in the -weighted estimates of Proposition 2.2 that depends on , with , and is the constant in the smoothing effects (4.6), cf. Theorem 2.2 of [31].
(iii) We can always choose  , since .

Proof. The proof is divided in several steps.

Step 1. Reduction. By the comparison principle that it is sufficient to prove lower bounds for solutions to the following reduced problem:

(4.5)

where  ,  , and  . We only assume that  , which implies that since and also that  . It is not restrictive to assume that the ball is centered at the origin.

Step 2. Smoothing effects. In [31] there are the global smoothing effects which provide global upper bounds for solutions to the Cauchy problem 1.1 . We apply such smoothing effects to solutions to our reduced Problem 4.5 to get

(4.6)

where and the constant only depends on  .

Step 3. Aleksandrov principle. We recall Theorem 11.2 of [39], we have that

Therefore one has that

(4.7)

Step 4. Lower estimates for the -norm on an annulus. We combine the -weighted estimates of Theorem 2.2 with the smoothing effects of Step 2: estimates (2.3) read in this context

(4.8)

we have chosen and with as in Lemma 2.1 (with the explicit form given in formula (2.2)) , so that on and for with  , and we recall that depends only on  .

(4.9)

We first estimate (I), to this end we observe that if we choose we have that