Positivity, local smoothing and Harnack inequalities for very fast diffusion equations

Transcription

1 Positivity, local smoothing and Harnack inequalities for very fast diffusion equations Dedicated to Luis Caffarelli for his ucoming 60 th birthday Matteo Bonforte a, b and Juan Luis Vázquez a, c Abstract We investigate qualitative roerties of local solutions u(t, x) 0 to the fast diffusion equation, t u = (u m )/m with m <, corresonding to general nonnegative initial data. Our main results are quantitative ositivity and boundedness estimates for locally defined solutions in domains of the form 0, T ] R d. They combine into forms of new Harnack inequalities that are tyical of fast diffusion equations. Such results are new for low m in the so-called very fast diffusion range, recisely for all m m c = (d )/d. The boundedness statements are true even for m 0, while the ositivity ones cannot be true in that range. Keywords. Nonlinear evolutions, Fast Diffusion, Harnack Inequalities, Positivity, Smoothing Effects. Mathematics Subject Classification. 35B45, 35B65, 35K55, 35K65. (a) Deartamento de Matemáticas, Universidad Autónoma de Madrid, Camus de Cantoblanco, 8049 Madrid, Sain (b) address: matteo.bonforte@uam.es (c) address: juanluis.vazquez@uam.es 0

2 Introduction We study qualitative and quantitative roerties of solutions u = u(t, x) of the nonlinear diffusion equation t u = (u m u) = (u m /m) (0.) in the whole arameter range < m <, where it is called Fast Diffusion Equation (FDE). We consider local nonnegative weak solutions, defined in an oen cylinder Q of sace-time R R d with d. Note that the factor /m in the last exression is inessential when m > 0 (u to a time rescaling, t = t/m) but becomes essential for m < 0, in order to obtain a arabolic equation; for m = 0 the last exression has to be written as t u = log(u). Assuming the basic existence and uniqueness theory, 4], 3], we are interested in the qualitative roerties of the solutions such as boundedness, ositivity, and Harnack inequalities. For the FDE these roerties deart from the roerties of the linear Heat Equation (case m = ), 34], and even more from the Porous Medium Equation (case m > ), 3]. Moreover, they are still artially understood when m is far from, recisely for m m c where m c = (d )/d is called the first critical fast diffusion exonent. Our goal here is to obtain bounds from above and below for the solutions in that low range of exonents. We look for recise quantitative versions based on local estimates. Such estimates should be of interest in develoing a general theory of this equation in the detail that is already known both for m and for m c < m <. Precedents and roblems The existence and uniqueness of weak solutions of the initial value roblem and other standard initial and boundary value roblems for the FDE, as well as the main qualitative roerties of the solutions (such as the ones already mentioned, or the asymtotic behaviour), are by now well understood when m is close to one, more recisely in the so-called good arameter range: m c < m <. To be secific, when the roblem is osed in the whole sace, weak solutions are uniquely determined by their initial data if u 0 is a locally integrable nonnegative function, or even a locally finite Radon measure. In that case, the solution is C smooth and ositive for all x R d and t > 0, and the initial data are taken in the sense of initial trace, 4], 7], 4]. Solutions are bounded for data u 0 L (R d ) for any, and even for data in the Marcinkiewicz saces M (R d ), >, 3]. They are locally bounded under the very mild restriction that u 0 is Radon measure, even if it is not globally finite. The theory of the FDE has been much less studied until recently in the subcritical fast-diffusion range m < m c, even under the condition m > 0, since essential difficulties have been found in the different chaters of the theory, like existence, uniqueness, and regularity. Note that 0 < m < m c is ossible only if d >. We refer for background to the book 3] that discusses in some detail the range m m c, even for m 0, along with the cases m > m c. Let us give an idea of the difficulties that arise and that we address in our work below: Boundedness. Though weak solutions with data in the saces L (R d ),, exist and are unique for 0 < m <, counterexamles show that for m < m c these weak solutions need not be We will always interret u m /m as log(u) when m = 0. In the whole aer, indicates the gradient oerator, the divergence oerator, and the Lalacian oerator, all of them taken with resect to the sace variables, x R d. With the extra restriction m > 0 if d =, the case < m < 0 and d = being somewhat different, cf. 3].

3 bounded, and as a consequence they are not smooth. The simlest such examle seems to be the searate-variables function (T t)/( m) U(t, x; T, x 0 ) = c x x 0 /( m) (0.) For every m < m c, even m 0, there exists a suitable constant c(m, d) > 0 such that U is a weak solution of the FDE in the cylinder Q = (0, T ) R d, cf. 3], age 80, but obviously the solution never imroves its initial regularity until it extinguishes in finite time. The recise sace regularity is U(, t) L loc (Rd ) for all < c, where the critical integrability exonent is c = d( m)/, which is larger than recisely for m < m c, i.e., in the subcritical range. There is a ositive result concerning boundedness, that is also tied to the exonent c : solutions with initial data in L (R d ) with > c become bounded and C smooth for all ositive times as long as the solution does not disaear. This smoothing effect haens for all if m > m c, for > if m = m c (in the last cases there is no roblem of disaearance). The results are shar, cf. 3]. Extinction in finite time, EFT. The above examle exhibits another tyical feature of the Cauchy roblem for m < m c, namely, the ossible lack of ositivity due to EFT. The occurrence of EFT deends on the tye of roblem we consider. In the case of the Cauchy roblem osed in R d with d 3, Bénilan and Crandall gave in 3] a roof of the extinction in finite time, EFT, of solutions of the FDE in the range 0 < m < m c when u 0 L (R d ) with = c. It is roved in 3] that EFT occurs for the solutions with m < m c for all functions with initial data in the Marcinkiewicz sace M c (R d ), hence in L c (R d ). We recall the EFT does not haen for the Cauchy Problem when m > m c. In the case of the Cauchy-Dirichlet roblem osed in a bounded domain with zero boundary data, EFT haens for all 0 < m <. There is an interesting functional connection: we can show that EFT occurs if we have a global Poincaré and a Sobolev inequality, and this result can be extended to more general settings, such as Riemannian manifolds, as it has been done by the authors in 7]. On the other hand, Bénilan and Crandall s roof for the Cauchy roblem is based only on the Sobolev inequality, but it holds only in the lower range m < m c. Harnack inequalities. Concerning finer regularity roerties, the ossible occurrence of EFT is comatible with the fact that nonnegative bounded solutions are ositive, and consequently C smooth, as long as they are not identically zero, i.e., before extinction. However, the existence of EFT for low m is tied to the breakdown of the standard forms of Harnack inequalities, which are a strong tool in develoing a regularity theory. Obtaining some kind of Harnack inequality is therefore a main research issue for m m c and has been an oen roblem for some years. More secifically, we concentrate on arabolic lower Harnack inequalities of the tye called Aronson-Caffarelli estimates ], and examine their consequences to obtain quantitative forms of ositivity. An extension work has been done in 8] for m c < m < but the method collases for m m c due to the very different roerties of the solutions. As a consequence of our local smoothing effect and of ositivity estimates, we will obtain some intrinsic Harnack inequalities of forward, ellitic or backward tye, which are new in this range. In a recent rerint 8], DiBenedetto, Gianazza and Vesri study the validity of intrinsic Harnack inequalities in the good range m > m c and show, with an exlicit counterexamle, that any kind of Harnack Inequality, intrinsic, ellitic, backward and forward can not hold if m < m c, for a fixed size of the intrinsic cylinder, that is, if we fix the size of the arabolic cylinder a riori in terms of the

4 value of u at the center of the cylinder (t 0, x 0 ). They leave as an oen roblem to find which kind of Harnack Inequalities, if any, are tyical of the very fast diffusion range 0 < m < m c. In this aer we give an answer to this intriguing roblem. Very singular range. Most the literature has avoided the cases m 0, where the diffusivity D(u) = u m is very singular at u = 0. Recently, it has been shown that a large art of the theory of the subcritical range goes over to this very singular range, on the condition of working with solutions that are not too small. See 30] among the older references, then 3], and the books 4], 3] for a more comlete reference. Note that this recovers a subcritical range for dimensions d =,, and also that we can study the interesting log-diffusion roblems where m = 0, cf. 3], 33] and the references. More secifically, there is an extension of the results called smoothing effects, whereby data in L (R d ) with > c imly bounded solutions for all t > 0, and also the extinction in finite time for data in M (R d ), = c. But a very different situation haens for data in L (R d ) with < c, which is called immediate extinction, whereby the solutions obtained as limit of any reasonable aroximation are identically zero for all t > 0. This makes it difficult to think of a general study of ositivity. Immediate extinction haens for the Cauchy-Dirichlet roblem osed in a bounded domain with zero boundary data for all m 0, d. Our study of this range is confined therefore to uer estimates. Comarison with ellitic roblems. Part of the difficulties of the FDE in the lower range of m can be exlained by the intimate relation of the equation with semilinear ellitic theory. This remarkable connection will be briefly exlained in Subsection 4.3. Results and organization Our work focuses on a eculiar feature of the FDE, which is the existence of very strong local estimates. This was resumably first mentioned in the aer by Herrero and Pierre 4], 985, who get solutions in the whole range 0 < m < under the sole condition on the initial data u 0 L loc (Rd ). Much of the subsequent work has been influenced by the local character of the equation. Here, we want ush this idea to its final consequence concerning two different areas: the question boundedness of local solutions, and the question of ositivity of nonnegative solutions, measured quantitatively by so-called lower Harnack inequalities. We will then combine the local uer and lower estimates, into a full form of Harnack inequality. While the boundedness results hold for all m <, ositivity estimates are confined to 0 < m < because of the ossible occurrence of immediate extinction. As we have said, the main interest of our results lies in their alication in the subcritical range, m < m c. They are also new for the critical exonent m = m c. Let us be more secific about the contents of the aer. It is divided into three main arts. (I) The study of ositivity and lower Harnack inequalities, both of local and global tye. The first main contribution of the aer is a arabolic lower Harnack inequality of the Aronson-Caffarelli tye that is resented in Section, along with a detailed comarison with the forms available for other ranges of m. We devote Subsection. to rove the lower estimate, Theorem., for a minimal roblem. This is extended in Subsection. to general solutions. We then show that in the range m c < m < we can further eliminate the resence of the extinction time and recover stronger estimates that are known in that range. Subsection.4 discusses uer bounds for the extinction time T in terms of L norms of the data, which give an alternative tye of lower bound in the range where estimates deending only on L -norms of the data are not true. 3

5 (II) The study of local uer bounds. This takes two forms: the first is the control of the evolution in time of some satial L loc norms, which is erformed in Section.. Then, we get a local in sace-time version of the smoothing effect from L loc into L loc, an imortant regularity result that oens the road to higher regularity and was known for m > m c, and is false in general for m m c. We show in this aer that the estimate holds m < m c, on the condition that must be large enough. We finally obtain the finest local uer estimates, called local smoothing effects, in the form given in Theorem., just by combining the sace-time smoothing effect and the L loc obtained in the first Section.. (III) Parabolic Harnack Inequalities. In Section 3, we combine the local uer and lower estimates obtained in Parts I and II in the form of arabolic Harnack inequalities of forward, backward and ellitic tye, together with an alternative form. To conclude, we sketch a anorama of the obtained local estimates deending on the ranges of m, together with general remarks, some related oen roblem and a short review on related works. A final Aendix contains some useful technical results. Notations. We will work with weak solutions u 0 of the FDE with m <, defined in a cylinder Q = Ω (T 0, T ) for some domain Ω R d and T 0 < T. Usually, we take T 0 = 0, T = T. T can be infinite and Ω can be the whole sace. In view of existing theory we may assume that the solutions are ositive and smooth as long as they do not extinguish identically. We will be mostly interested in the local theory where the sace domain is bounded and the boundary conditions are not taken into account. In deriving local estimates it will be often sufficient to take as sace domain a ball, which we will denote by B = B R (x 0 ) or B = B λr (x 0 ) for some λ >. We will frequently consider the annulus region A R,λ = B λr \ B R. As indicated before, we ut m c = d d, d( m) c =. We have ointed out that c > if and only if m < m c. We will take integrability exonents if m > m c, > c if m m c. Moreover, for c we set which is ositive if and only if > c. ϑ = d( m), (0.3) Part I. Local lower bounds The first art of the aer addresses the question of quantitative estimates of ositivity. The exonent range in this art is 0 < m <, since it is well known that the FDE does not admit solutions of the Dirichlet roblem with zero boundary data when m 0, thus blocking any ossibility of a general local ositivity theory in that range 30, 3]. Our main contribution is a arabolic inequality in the sirit of the one obtained by Aronson and Caffarelli ] in their ath-breaking aer for m >, and the ones roduced by the authors in 8] for m c < m <. The urose of such formulas is giving quantitative information on the ositivity of solutions at later times in terms of information on L norms of u at a former time that we take as t = 0. This is why they are called arabolic lower Harnack formulas. We take 0 < m < and consider a u be a local, nonnegative weak solution of the FDE defined in a cylinder Q = (0, T ) Ω, taking initial data u(0, x) = u 0 (x) in Ω and having finite extinction time T. We make no assumtion on the boundary condition (aart from nonnegativity). For ease of roof we 4

6 will assume that the solutions are smooth so that the different comutations and comarison results are valid. This assumtion is then eliminated by aroximation, which is justified according to known theory. Theorem. Let 0 < m < and let u be the solution to the FDE under the above assumtions. Let x 0 be a oint in Ω and let d(x 0, Ω) 3R. Then the following inequality holds for all 0 < t < T R d u 0 (x) dx C R /( m) t m + C T m R t m m u m (t, x 0 ). (.) B R (x 0 ) with C and C given ositive constants deending only on d. This imlies that there exists a time t such that for all t (0, t ] u m (t, x 0 ) C R d u 0 L (B R )T m where C > 0 deends only on d; t deends on R and u 0 (x) L (B R ) but not on T. t m m. (.) Simlified version. The deendence on the arameters makes the formula aarently comlicated. But it can be reduced to a simler, equivalent one. Actually, we may assume that x 0 = 0 by translation. Given R > 0 and M = B R (0) u 0(x) dx > 0, we use the rescaling u(t, x) = M ( t R d û τ, x ), τ = R d( m) M m, (.3) R to ass from a solution with mass M in the ball of radius R to a solution û with mass in the ball of radius. So we only need to rove the version with M = R = to get the full version. The scaling is simler for m = m c where τ = M m. Of course, the extinction time has to be rescaled accordingly, T = R d( m) M m T. Imrovements. As stated, estimate (.) alies only to solutions with finite extinction time, and it involves the value of the extinction time T in an exlicit way; both things can make it imractical. However, a simle comarison argument shows that we only need to estimate from below any subsolution. In articular, we may relace the solution under consideration by the solution of the roblem with initial data u 0 (x)χ BR (x 0 )(x), and zero Dirichlet boundary conditions on x B 3R (x 0 ). Let us call this roblem minimal roblem for the given data. The extinction time of the corresonding solution will be called the minimal life time of such domain and data, T m (u 0, B). Clearly, T m (u 0, B) T (u). Corollary. The above ositivity result holds with T (u) relaced by the minimal life time T m (u 0, B), u is defined in Q T, and the estimate alies for 0 < t < T with T = min{t, T m }. This modified result is secially interesting in the range > m > m c where the solutions of the Cauchy Problem do not vanish. On the other hand, it is known that T m is finite if u 0 satisfies some local integrability conditions 6, 3]. 5

7 Comarison with the estimate for the PME and other FDE The PME. Let us write Aronson-Caffarelli s result ] for m > with a similar notation: R d u 0 (x) dx C R /(m ) t m + C R d t d/ u +(d(m )/) (t, x 0 ). (.4) B R (x 0 ) We recall that this formula is valid for all nonnegative weak solutions of the PME defined in the whole sace. The form of the first term in the right-hand side is the same in both results, (.) and (.4). This term lays the role of blocking the ositivity information when it is large relative to the left-hand side integral, and allowing for such information when it is small. The critical time at which we begin to get ositivity information is obtained by making this term a fraction of the left-hand side, i.e., for t c = c(m, d) u 0 m L (B R (x 0 )) R+d(m ). (.5) But since the exonents have just the oosite sign in the above exressions for m > and m <, the consequences are qualitatively very different: the information on ositivity haens for us when t is smaller than t, while for the PME it haens when t is larger. This is in accord with the basic roerties of these equations, which the resent inequalities faithfully reroduce. Rescaling allows to check the inequality only at t = for R =, and in that case we only have to rove that there are constants M 0 = M 0 (n, m) and k = k(m, d) such that for M M 0 u(0, ) k M /(d(m )+). (.6) As to the second term, it is different. We cannot exect to have the A-C term in the range m < m c since then the exonent of u would be negative. In fact, the roof of ] uses conservation of mass that is not valid for the fast diffusion equation in the low m range. The good FDE. The validity of the Aronson-Caffarelli formula was extended by the authors in 8] to local solutions of the FDE in the good exonent range m c < m <, and the already mentioned sign change in the exonents imlies that we get good lower estimates for 0 < t t. Moreover, we can continue these estimates thanks to the fortunate circumstance that we have further differential inequalities, like t u Cu/t in the case of the Cauchy roblem, which allow for a continuation of the lower bounds for t t with otimal decay rates in time. The final form is u(t, x) M R (x 0 ) H(t/t c ), M R (x 0 ) = R d u 0 dx. (.7) B R (x 0 ) The critical time is defined as in (.5); the function H(η) is defined as Kη /( m) for η while H(η) = Kη dϑ for η, with K = K(m, d). Note that for 0 < t < t c the lower bound means u(t, x 0 ) k(m, d)(t/r ) /( m) which is indeendent of the initial mass. Eliminating the time T. A natural question is to try to recover this shar results of the good fast diffusion range via the resent methods. If one wants to do that, one needs uer estimates for the minimal life time, that is uer estimates for the extinction time for the MDP, in terms of the L -norm on the ball B R0. We rove the following result. Theorem.3 Let m c < m <. Then, (i) We have shar uer and lower estimates for the extinction time for the Dirichlet roblem on any ball B R of the form: c u 0 m L (B R/3 ) R d( m) T c u 0 m L (B R ) R d( m). (.8) 6

8 (ii) In that range of m the lower estimates of Theorem. imly the lower Harnack inequalities of 8, 0,, 8], in the form ] t m u(t, x 0 ) c m,d R (.9) for any 0 < t < t and any x B R, where t is given by (.6). This result shows that the form of the lower bounds given in Theorem. is shar, since it allows to obtain shar local lower bounds not only in the good fast diffusion range. And it also alies in the very fast diffusion range, that is the new interesting art of this aer. We are thus led to the question of eliminating all extinction times from the estimate, i.e., relacing T or T m by some information on the initial data, also in the range 0 < m < m c. Theorem.4 Let 0 < m < m c and let u be the solution to the FDE under the above assumtion that u 0 L c loc (Rd ). Let x 0 be a oint in Ω and let d(x 0, Ω) 3R. Then, the following inequality holds for all 0 < t < T R d u 0 L (B R (x 0 )) C R /( m) t m + C3 u 0 L c (B R (x 0 ))R t m m u m (t, x 0 ). (.0) with C and C 3 given ositive constants deending on d. We can also obtain formulas in terms of the norms u 0 L (B R (x 0 ) for all > c, that can be seen below. Obstruction to a simler estimate with L norm The resence of the extinction time T in the lower estimates, or equivalently of some L norm of the initial data, is a drawback in the formulas that is not resent in the original Aronson-Caffarelli estimate for m >, or in the version of the authors for m (m c, ) in the whole sace. But it is a consequence of the bad behaviour of the fast diffusion equation for low values of m, a fact that can be seen in different ways. Thus, we will show here that the local lower estimates cannot deend only on the local L norm of the initial data when 0 < m m c. We do it by means of a counterexamle based on the behaviour of solutions with data that aroximate a Dirac delta. We solve the FDE for smooth and ositive initial data ϕ(x) L (R d ) with integral equal to. We assume that ϕ is radially symmetric, comactly suorted and decreasing with x. We obtain a smooth and ositive solution u(t, x) defined in a cylinder Q T and vanishing identically at some t = T. The scale invariance of the equation imlies that the solution corresonding to data ϕ k (x) = k d ϕ(kx) is u k (x) = k d u(k σ t, kx), σ = d( m) > 0, (.) so that it has extinction time T k = T k σ. As k it is clear that u k (0, t) converges to the Dirac delta. We also observe that T k, so that we lose the revious estimates. On the other hand, we see that losing the estimates is inevitable. If we consider a oint x 0 very close to x = 0 and take a radius R > x 0, then u k (0, x) L (B R (x 0 )) =. However, by continuity of u with resect to the initial data at t = 0, x large, we have u k (t, x 0 ) = k d u(k σ t, kx 0 ) 0 7

9 (note that R d u k (t, x) dx at all times). This means that no lower estimate could be uniformly valid for this sequence. A scaling argument was used by Brezis and Friedman ] to rove that there exist no weak solutions with initial data a Dirac delta.. Positivity for a minimal Dirichlet Problem We will assume that 0 < m < in the study of ositivity (cf. the comment in the Introduction). Since m > 0 we eliminate the factor /m from equation (0.) for simlicity without loss of generality. As a reliminary ste, we first rove ositivity for a roblem osed on a ball of radius R 0, zero boundary data and articular initial data. Since the roblem of getting quantitative ositivity estimates has been successfully studied in 8] in the range m c < m <, the techniques we introduce are mainly aimed at roducing ositivity in the cases 0 < m m c, where revious methods failed. Secifically, we shall consider the following Dirichlet roblem on the ball B R0 R d : t u = (u m ) in Q T,R0 = (0, T ) B R0 u(0, x) = u 0 (x) in B R0, and su(u 0 ) B R u(t, x) = 0 for t > 0 and x B R0, (.) where R 0 > R > 0. We only consider nonnegative data and solutions. The roblem admits a unique solution u C(0, ) : L (B R0 )) for every u 0 L ( B R0 ), 4]. We will refer to this roblem as the minimal Dirichlet roblem, or more briefly, the minimal roblem, because obtaining ositivity for solutions to this roblem imlies in an easy way local ositivity for any other roblem, thanks to the comarison rincile. The solution vanishes in finite time; let T > 0 be the finite extinction time, shortly FET. Later on we would like to eliminate the deendence of the results on T and make the estimates deend only on the initial data, see Section.4. Our goal is to obtain ositivity with a quantitative estimate for this minimal roblem. Our most novel idea consists in assing the information on the initial data via the flux of the solution on the boundary of the ball B R into an averaged ositivity result outside the ball for times that are not too small, more recisely on the annulus A 0 := B R0 \ B R. This roerty can be interreted as the exansion of ositivity outside a ball in which the initial datum has nonzero mean. It is in some sense it is analogous to the exansion of ositivity already introduced by DiBenedetto et al., see e.g. 9, 8, ] for the uer m-range. The exansion of ositivity turns out to be a key tool in roving lower Harnack also in our case. Once we have roved that ositivity sreads out from a ball, then for suitable ositive times the mean value of the solution on an annulus is ositive. We then fill the hole in the middle using Aleksandrov s Reflection Princile, cf. the Aendix and 8]. In this way we arrive at the ositivity result in the inner ball for any ositive time... Flux and transfer of ositivity We start the roof of the ositivity results for the minimal roblem by a result on mass transfer to an outside annulus based on the flux across an internal boundary. We recall that R 0 > R and A 0 := B R0 (x 0 ) \ B R (x 0 ). In order to simlify the final formulas, we write λ = R 0 /R > (we take for instance R 0 = 3R). 8

10 Lemma.5 (Flux Lemma) If u is a ositive smooth solution of the Minimal Problem (.) in Q T with extinction time T > 0. Then, the following estimate holds true k 0 (R 0 R) B R0 u(s, x) dx for any 0 s T, and any 0 < R < R 0, and for a suitable constant k 0 = k 0 (d). T s A 0 u m dx dt, (.3) Proof. We shall use a C test function ϕ(x) that is suorted in the ball B R0 and takes the value in B R. It is clear that we can choose ϕ such that there exist a constant k 0 > 0 deending only on d such that ϕ(x) k 0 (R 0 R), (.4) Let 0 s < t T. We comute t t t t t u ϕ dx dt = (u m )ϕ dx dt = u m ϕ dx dt + ν (u m )ϕdσ dt s A 0 s A 0 s A 0 s B R t + ν (u m )ϕ u m ν ϕ ] t dσ dt u m ν ϕdσ dt. s B R0 s B R We remark that the last three integrals vanish since ϕ and u vanishes identically near the boundary B R0, and ν ϕ 0 on B R. We also have t s t u ϕ dx dt = A 0 u(t, )ϕ dx A 0 u(s, )ϕ dx A 0 Hence, u(t, )ϕ dx u(s, )ϕ dx = A 0 A 0 t s A 0 u m ϕ dx dt + t s B R ν (u m )ϕdσ dt. We will use this equality with t = T, T = T (u 0 ) being the finite extinction time for the solution to Problem (.), so that we obtain A 0 u(s, )ϕ dx = T s A 0 u m ϕ dx dt + T s B R ν (u m )ϕdσ dt, (.5) where ν is the exterior normal to B R, which is the oosite of ν which is the exterior normal to the inner boundary of A 0, so that ν (u m ) = ν (u m ). On the other hand, a simle calculation shows that t (u(t, x) u(s, x)) dx = t u dx dt = B R s B R Letting t = T, with T as above, we obtain t s B R (u m ) dx dt = t s B R ν (u m )dσ dt. T u(s, x) dx = ν (u m )dσ dt. (.6) B R s B R 9

11 Joining equalities (.5) and (.6) we get u(s, x) dx = B R0 u(s, x) dx + B R u(s, x) dx = A 0 T s A 0 u m ϕ dx dt We conclude by using estimates (.4) for ϕ: for any 0 < R < R 0 we then get u(s, x) dx u(s, x) dx = B R B R0 The roof is comlete. T s A 0 u m ϕ dx dt k0 T (R 0 R) u m dx dt. (.7) s A 0 Remark. Lower Bound on the Extinction Time. As a first consequence of this Lemma we can easily obtain useful lower estimates for the FET: T k 0 (R 0 R) B u(s, x) dx u m dx dt (T s) Vol(A 0 ) u m dx (s, x) R0 s A 0 A 0 Vol(A 0 ) ] m dx (T s) Vol(A 0 ) u(s, x) B R0 Vol(A 0 ) (T s) Vol(A 0 ) m B R0 u(s, x) dx ] m where in the first ste we have used the mean value theorem for the time integral (see details in Ste of next section), with s (s, T ), in the second ste the Hölder inequality, and in the third ste we used the contractivity of the global L (B R0 )-norm. Letting then s = 0 gives the desired lower bound, once we notice that su(u 0 ) B R ] m k 0 (R 0 R) B R u 0 dx Vol ( ) T. (.8) A 0.. Pointwise lower estimate for initial times We have just shown how ositivity of the initial datum roagates on the annulus in the weak form of a ositive sace-time mean value. We will now see that this is sufficient to fill the hole inside the annulus. As in the study of the exonent range m c < m < erformed in 8], the estimate uses a critical time that is defined in terms of the initial norms. In the resent case it is given by t := k 0 (R 0 R) B R u 0 dx Vol ( ) A 0 ] m (.9) where k 0 as in the Flux Lemma.5. Note in assing that the ositivity result that follows, formula (.5), imlies that this quantity is less than T. Obtaining the lower bound needs several stes. 0

12 Ste. Time Integrals. Hölder s inequality, together with the fact that the global L ( ) B R0 -norm decreases, gives u(t, x) m dx Vol ( ] ) m m A 0 u(t, x) dx Vol ( ) m A 0 u(t, x) dx A 0 A 0 B R0 Vol ( ] m ) m A 0 u(0, x) dx = Vol ( ] ) m m A 0 u 0 dx B R0 B R ] m since su(u 0 ) B R. For any 0 s t we then have t s A 0 u(τ, x) m dxdτ Vol ( A 0 ) m ] m u 0 dx (t s) B R We use this estimate together with estimate (.3) to get T t T k 0 (R 0 R) B u 0 (x) dx u m dx dt = u m dx dt + u m dx dt R 0 A 0 0 A 0 t A 0 Vol ( ) m T m A 0 u 0 dx] t + u B m dx dt R A 0 t (.0) In view of the definition of t we can eliminate one term and get k (R 0 R) B R u 0 (x) dx T t A 0 u m dx dt. (.) with k = k 0 /. In articular, this means that the left-hand side remains strictly ositive. Ste. We introduce the function Y (t) = u m (t, x) dx, A 0 and aly the mean value theorem -for the time integral- to rove that there exists t t, T ] such that T t Y (t) dt = (T t ) Y ( t ). In other words, T t u m (t, x) dx dt = ( ) T t A 0 A 0 u m( t, x ) dx. Using now the estimate obtained in the revious ste, we conclude that there exists t t, T ), such that T k (R 0 R) B u 0 (x) dx u m (t, x) dx dt = ( ) T t u m( t, x ) dx, R A 0 A 0 and this imlies that for some t t, T ] we have k (R 0 R) T BR u 0 (x) dx k (R 0 R) ( T t ) t u 0 (x) dx u m( t, x ) dx. (.) B R A 0

13 Ste 3. Aleksandrov Princile. Positivity at the critical time. We can now use the Aleksandrov Princile to deduce ositivity at x 0 from inequality (.). In fact, u m( t, x ) dx Vol(A 0 ) u m (t, x 0 ) (.3) A 0 where x 0 R d is the center of the ball B R0, since we know from Aleksandrov Princile, that u(t, x) u(t, x 0 ) for any x A 0 and any t > 0 (see Aendix for details). Joining inequality (.) and (.3) we obtained that there exists a t t, T ) such that k (R 0 R) u 0 (x) dx u m (t, x 0 ) (.4) Vol(A 0 ) T B R Ste 4. Positivity backward in time. The last ste consists in obtaining a lower estimate when 0 t t. This argument is based on Bénilan-Crandall s differential estimate, cf. 4] : t u(t, x) u(t, x) ( m)t that is valid for all nonnegative solutions of this initial and boundary value roblem. It easily imlies that the function: u(t, x)t m is non-increasing in time, thus for any t (0, t ] we have that u(t, x) t m t m u(t, x) t m T m u(t, x) since t T. It is now sufficient to aly inequality (.4) to the l.h.s. in the above inequality to get k (R 0 R) u 0 (x) dx u m (t, x 0 ) t m m m T m u m (t, x 0 ) (.5) Vol(A 0 ) T B R This is the inequality we were looking for. Ste 5. In order to simlify the final formulas, it is convenient to make a choice for the ratio λ = R 0 /R > (for instance R 0 = 3R). The formula for t becomes t = c 0 R d( m) u 0 m L (B R ) (.6) and c 0 > 0 deends only on d and λ. We have roved the following ositivity result. Theorem.6 Let 0 < m < and let u be the solution to the Minimal Problem (.) and let T = T (u 0 ) be the MET. Then T t, and the following inequality holds true for all t (0, t ] where c > 0 deends only on d. u m (t, x 0 ) c R d u 0 L (B R )T m For the articular time t = t we get ( ) m u(t, x 0 ) c (R /T ) /( m) u(0, x) dx. B R u(0, x) dx B R t m m. (.7)

14 ..3 Estimate of Aronson-Caffarelli tye for Very Fast Diffusion It is interesting to resent the above result in the form that has been used by Aronson and Caffarelli in their work ]. We have to argue as follows: we have arrived at the following alternative either t < t or Writing the exression of t, we either have or R d R d k (λ ) ω d (λ d )R d T m B R u 0 (x) dx C R /( m) t m, C = u 0 (x) dx t m u m (t, x 0 ) B R ω d (λ d ) m m k (λ ) m u 0 (x) dx C T m R t m m u m (t, x 0 ), C = ω d (λ d ) B R k (λ ) We now sum u the two exressions to get R d u 0 (x) dx C R /( m) t m + C T m R t m m u m (t, x 0 ). (.8) B R with C and C given constants deending on d and λ >.,. Proof of Theorem. and Corollary. The revious results will now be used to rove uniform ositivity on balls for any local solution of the FDE roblem. We roceed as follows: let u be a ositive and continuous weak solution of the FDE defined in Q = (0, T ) Ω taking initial data u(0, x) = u 0 (x) in Ω. We make no assumtion on the boundary condition (aart from nonnegativity). Let us select a oint x 0 Ω R d. Select two radii R 0 3R > 0 so that B R0 (x 0 ) Ω, that is R 0 dist(x 0, Ω). In the case Ω = R d there is obviously no restriction on R 0. Let u D be the solution to the corresonding MDP, as defined in (.). It has extinction time T m. By arabolic comarison, it is then clear that u D u in Q = (0, T ) B R0 (x 0 ), where T = min{t, T m }, hence we can easily extend the ositivity results for the MDP obtained in the revious section to any other local weak solution, either in the form given by Theorem.6, or in the Aronson-Caffarelli form (.8). This concludes the roof of Corollary., and as a consequence we get Theorem...3 Lower estimates indeendent of the extinction time for m c < m <. Proof of Theorem.3 The roof is divided in some short stes. Here, m c < m <. Reduction. Let u R (t, x) be the solution to the homogeneous Dirichlet roblem on the ball B R, corresonding to the initial datum u 0 L (B R ) and having extinction time T (R, u 0 ). The rescaled solution u(t, x) = M ( t R d û τ, x x ) 0, τ = R d( m) M m, M = u 0 (x) dx (.9) R B R 3

15 allows us to ass from a solution with mass M defined in the ball of radius R centered at x 0 to a solution û with mass in the ball of radius centered at 0. The extinction times have to be rescaled accordingly, T (u) = R d( m) M m T, where T = T (û). Therefore, we will work with rescaled roblems and solutions. Barenblatt solutions. We now consider the solution B of the Dirichlet roblem osed on B, and corresonding to the Dirac mass as initial data, B(0, ) = δ 0, and zero boundary data, that we call Barenblatt solution. First we recall that by aroximation with L -data, or by comarison with the solutions of the Cauchy roblem, it is easy to check that the smoothing effect alies and B(t, x) c m,d t dϑ, for any (t, x) 0, + ) B. (.30) Moreover, it is known that this is the solution that extinguishes at the later time among all nonnegative solutions with the same mass of the initial data and same boundary data. Such comarison is a consequence of the concentration-comarison and symmetrization arguments develoed in detail in 3, 3]. Thus, we need to rove that the Barenblatt solution extinguishes in finite time T. The roof is based on the fact that it is bounded for t t 0 > 0. Solution by searation of variables. Consider now the solution U r (t, x) = S(x)(T t) /( m) of the Dirichlet roblem on B r, r >. It corresonds to the initial datum U r (0, x) = S(x)T /( m), and extinguishes at a time T, to be chosen later. Here, S is the solution to the stationary ellitic Dirichlet roblem S m + /( m)s = 0 on B R0, and therefore it can be chosen radially symmetric, S(x) = S( x ). It will also be nonincreasing in r = x. By standard regularity theory S(x) m can be bounded from above an below by the distance to the boundary. The arameter T, extinction time of U r can be chosen at will. To fix it, we ick a time t 0 > 0 and define the T through the relation S()(T t 0 ) m = cm,d t dϑ 0. (.3) Comaring the two solutions. We now consider the homogeneous Dirichlet roblem on t 0, T ] B r, and we comare the Barenblatt solution B with the solution U r constructed above in the cylinder Q = B (0) t 0, T ). At the initial time t 0 we know by construction that B(t 0, x) U r (t 0, x). The comarison of the boundary data is immediate. By well-known arabolic comarison results, this imlies that B(t, x) U r (t, x), on t 0, T ] B r, and hence the extinction times satisfy T T = c m,d S() t dϑ 0 ] m + t 0. (.3) We only need to choose a t 0 (0, T ) to obtain an exression for the uer bound of T that deends only on m and d (the reader may choose to otimize the exression for T with resect to t 0 ). Conclusion. As a consequence of the above uer bound, we know that any solution to the Dirichlet roblem on the unitary ball and with unitary initial mass extinguish at a time T T τ(m, d). Rescaling back to the original variables we have roved that any solution u R to the Dirichlet roblem on the ball B R and with initial mass M = B R u 0 dx extinguish at a time T R that can be bounded above by T (R, u 0 ) τ m,d R d( m) u 0 m L (B R ). The lower bounds come from the fact that t T and is given by (.6). 4

16 The lower Harnack inequality. Inequality (.9) follows by lugging the uer bound (.8) into the lower bound (.7). The reader should notice that the roerties that we have used are tyical of the good fast diffusion range, m c < m <, and cannot be extended to the very fast diffusion range, m < m c..4 Lower estimates indeendent of the extinction for 0 < m < m c. The resence of the minimal extinction time T m = T in the formula for the lower Harnack inequality resonds to an essential characteristic of the roblem. Actually, lower estimates in terms of only L norms cannot be true for m m c as we have shown at the beginning of this section: there is no ositive lower bound at a time t 0 > 0 and a oint x 0 that deends only on t 0, R and the mass of u 0 in B R (x 0 ). Similar examles can be constructed if u 0 L loc (Rd ) with < c, and we can be found in 3], Chaters 5 and 7. Fortunately, controlling the local (or global) L norm gives a control on the MET T, and in this way we get valid lower estimate without T, as we exlain next..4. Estimates in terms of the L c norm In 3] Bénilan and Crandall rove that for any 0 s t T, and for any m < m c u(t) m c u(s) m c K c (t s), with K c = 8 d( m) ] S (d ) ( m), (.33) where S in the constant of the Sobolev inequality f S f (.34) and the above estimate holds for any solution with initial datum u 0 L c. We also stress on the fact that the constant K c is universal in the sense that it only deends on m and d. As a consequence of (.33), we have the following universal uer bound for the extinction time T (u 0 ) K c u 0 m c. (.35) We remark that while for lower bounds on FET we only need local information on the initial datum, uer estimates for the FET require global information. Fortunately, in the minimal roblem that we are considering, global and local are equivalent since u 0 (x) = 0 for x x 0 R. Proof. We sketch here the roof for the reader s convenience. It is well known that the time derivative of the global L norm of the solution u(t) of the MDP roblem under consideration is given by d 4( ) u +m dt u(t) = ( + m ) dx 4( ) ] S ( + m ) u (+m ) dx = 4( ) (.36) S ( + m ) u +m (+m ), where in the last ste we used the Sobolev inequality (.34) alied to the function f = u (+m )/, where = d/(d ) and S is the Sobolev constant. 5

17 Notice that if m > m c, then c <, so that the global L -norm increases, and this originates the so called Backward Effect, see e.g. 3]. This exlains our assumtion m < m c. Moreover ( c + m = c ), d so that (.36) becomes ( c + m ) = c, d dt u(t) c c 8 d( m) ] S (d ) ( m) 4 c ( c ) S ( c + m ) = 8 d( m) ] S (d ) ( m) ( d) u(t) c c integrating the differential inequality gives the bound (.36) for any 0 s t. Letting s = 0 and t = T (u 0 ) in (.36) finally gives (.35). Alication to Theorem.4. Using this bound, we can now formulate the second version of the lower Harnack estimate, reflected in the theorem. The roof is immediate when u 0 is as in the minimal roblem, since in that case local and global norm is the same. Comarison as done in Subsection., allows to ass to the general solutions. Notice that in this way we use a local L c norm, not the global one! Remark. When we have not only the Sobolev inequality, but also the Poincaré, we can rove similar estimates for any m (0, ). This haens for instance for roblems osed in bounded domains, or for the minimal Dirichlet roblem. > 0,.4. Estimates in terms of other L norms Proosition.7 Let m <, α, R > 0 and let u be the solution to the Dirichlet roblem u t = m (um ) in (0, T ) B αr u(0, x) = u 0 (x) in B αr, and su(u 0 ) B R u(t, x) = 0 for t > 0 and x B αr with u 0 L ( B αr ), with > max{c, } = max {d( m)/, }. Then the following estimate hold for any 0 s t, where u(t) m u(s) m K (t s). (.37) ( ) ( m)( ) K = 4 ] P αr ] c + m S c > 0 and where S is the Sobolev constant of R d and P is the Poincaré constant on the unit ball. Proof. First consider, for any f W, ( ) 0 BαR, the Sobolev and Poincaré inequalities: f S f, and f P α R f, 6

18 where = d/(d ), and where the constants S, the otimal Sobolev constant on R d, and P, the Poincaré constant on the unit ball, only deend on the dimension d. By combining them through the Hölder inequality, we then get for any q (, ) f q f ϑ f ϑ P αr ] ϑ S ϑ f. Let now f = u +m, q := + m and ϑ = d( m) = c. We remark that q < if and only if > c, while q > if and only if q <. We obtain then ] u m P αr ] ( ) c c S u +m := K 0 u +m (.38) The derivative of the global L -norm then satisfies d 4( ) u +m dt u(t) = ] + m )K 0 4( ] u m (.39) + m where in the last ste we used (.38). Integrating the differential inequality over s, t] 0, T ], gives u(t) m u(s) ( ) m 4( m)( ) ] P αr ] c + m ] S c (t s). Uer Bounds on the Extinction Time. The above estimates (.37), rove that any solution of the Dirichlet roblem extinguish in finite time, and this is not surrising, but they also rovide an Uer Bound for the extinction time T, indeed letting s = 0 and t = T, we obtain T K u 0 m = ] + m P αr ] 4( m)( ) ( ) c S c u 0 m Notice that in the limit c we recover the revious result (.33). Summing u, the above result roves that a global Sobolev and Poincaré inequality rovides that the solution extinguish in finite time T and an gives a quantitative uer bound for T. Remarks. These results can be extended to different domains or manifolds in a straightforward way, the only imortant thing is to have global Sobolev and Poincaré inequalities, as already studied by the authors in 7], in the case of Riemannian manifolds with nonositive curvature Using this bound, we can now formulate a version of the lower Harnack estimate similar to Theorem.4. We leave the easy details to the reader. Part II. Local uer bounds In the second art of this work we turn our attention to the question of uer estimates for solutions with data in some L loc,, and obtain quantitative forms of the bounds that are shar in various resects. The range of alication is all m <, even m 0. We assume moreover that d 3, which is 7

19 the interesting case also for the lower estimates, in order to avoid technical comlications which break the flow of the roofs and results, but we remark that the qualitative fact, the existence of local uer bounds, is also true for d =,. As a reliminary for the main result, we devote Section. to establish the conservation of the local L integrability of the solutions and the control of the evolution of the local L norm for suitable. Let u = u(t, x) be a nonnegative weak solution of the FDE for m < defined in a sace-time cylinder Q = (0, T ] B R0 for some R 0, T > 0. This is the form of the estimate we get: ( m)/ ( m)/ u(t, x) dx] u(s, x) dx] + K (t s), B R (x 0 ) B R0 (x 0 ) for any R 0 > R and 0 s t < T. It is valid for all m < if, > m. The deendence of K on R and R 0 is exlicitly given in Theorem.3 below. The estimate extends Herrero-Pierre s well-known estimate to > and is valid for m 0. The main result of this art is the local uer bound that alies for the same tye of solution and initial data, under different restrictions on. Here is the recise formulation. Theorem. Let if m > m c or > c if m m c. Let u be a local weak solution to the FDE in the cylinder (0, T ) Ω (0, + ) R d. Then there are ositive constants C, C such that we have u(t, x 0 ) C t dϑ ] ϑ t u 0 (x) dx + C B R0 (x 0 ) where R 0 dist(x 0, Ω) and the constants C i deend on m, d and. R 0 ] m. (.) We recall that ϑ = /( d( m)) = /( c ). Note that the constants C i do not deend on the radii, but only on m, d and. An exlicit formula for them is given at the end of the roof, but we oint out that such values need not be otimal. The result is roved in Sections. and.3. A similar smoothing effect result has been roved for the first time by Herrero and Pierre in 4] in the good fast diffusion range m c < m < using =, but it is new in the range m m c where HP s result cannot hold in view of solutions like (0.). HP s technique relies on stronger differential estimates that do not hold in the subcritical fast diffusion case or on the local setting; our imression is that their techniques can not be adated to the very fast diffusion range. Related estimates for > are due to DiBenedetto and Kwong, 9], and Daskalooulos and Kenig, 4], but as far as we know no results cover the very fast diffusion range. Finally, note that the smoothing effect L loc into L loc is false for exonents < c as has been demonstrated in 3]. In fact, that monograh studies the existence of the so-called backward smoothing effects that go from L (R d ) into L (R d ) for < c. The local bound in (.) is exressed as the sum of two indeendent terms, one due to initial data, the other one due to effects near the boundary. The estimate is otimal in the following senses: (i) The first term resonds to the influence of the initial data and has the exact form that has been demonstrated to be exact for solutions that are defined in the whole sace and have initial data in L (R d ), see 3, Chaters 3,5]. By exact we mean that the integral is the same (but extended to the whole sace) and the exonents are the same, only the constant C may differ. We can then recover the global smoothing effect on R d, just by letting R so that the second term disaears; as mentioned 8

20 above the constant C is not the otimal one: the best constant for the global smoothing effect on R d has been calculated by one of the authors in 3]. (ii) The last term accounts for the influence of the boundary data and is secial to fast diffusion in the sense that it does include any information on the recise boundary data, thus allowing for the so-called large solutions that take on the value u = + on B R. The term has the exact form rescribed by the exlicit singular solutions (0.). This last term has the meaning of an absolute bound for all solutions with zero or bounded initial data; thus it can also be interreted as a universal bound for the influence of any boundary effects. In alications it is interreted as an absolute daming of all external influences.. Evolution of Local L -norms A basic question in the existence theory is obtaining a riori estimates of the solutions in terms of the data measured in some aroriate norm. The eculiar feature of the FDE is the local nature of the estimates. A fundamental result in this direction is the local L loc -L loc estimate due to Herrero and Pierre (which is valid for m > 0): Lemma. Let u, v C ( 0, + ) ; L loc (Rd ) ) be weak solutions of t u = (u m /m), 0 < m <. Let R > 0, R 0 = λr with λ >, and x 0 R d be such that B = B R0 (x 0 ) R d. Let moreover v u a.e. Then, the following inequalities hold true: ] ] m ] ] m u(t, x) v(t, x) dx u(s, x) v(s, x) dx + K R,R0, t s, (.) B R B for any t, s 0, where K R,R0, = and the constant c > 0 deends only on m, d. c ( R0 R ) Vol (B R 0 \ B R ) ( m) > 0 (.3) This result was roven in Pro. 3. of 4] and has been generalized to the case of fast diffusion on a Riemannian manifold by the authors in 7]. Our goal here is to extend such a result into and L loc -L loc estimate for suitable >. This estimate has two merits: first, it is valid for all < m < ; second, it is needed for some values > c for the roof of boundedness estimates. Theorem.3 Let u C ( (0, T ) ; L loc (Ω)) be a nonnegative weak solution of t u = (u m /m), (.4) and assume that u(t, ) L loc (Ω) for some, > m, and for all 0 < t < T. Here, Ω is a domain in R d that contains the ball B = B R0 (x 0 ). Then, the following inequality holds true: ( m)/ ( m)/ u(t, x) dx] u(s, x) dx] + K R,R0, (t s), (.5) B R (x 0 ) B R0 (x 0 ) 9

21 for any 0 s t < T, where K R,R0, = c m,d ( R0 R ) Vol (B R 0 \ B R ) ( m)/ > 0, (.6) and the constant c m,d > 0 deends only on m, d. Remarks. (i) The result imlies for those values of that whenever u(s, ) L loc (Ω) for some s > 0, then u(t, ) L loc (Ω) for all t > s. Note that the deendence of the local L norm is again exressed as the sum of two indeendent terms, one due to the initial data, the other one due to effects near the boundary. (ii) Note that the times t and s must be ordered in this result, a condition that is not required in Lemma.. (iii) The last term may blow u as we aroach the boundary of Ω (where no information on the data is used). Indeed, the constant can be written in the form K R,R0, = c m,d R( c)/ 0 F (R/R 0 ), F (s) = ( sd ) ( m)/ ( s). Now, if x 0 Ω we may take R 0 = d(x 0, Ω) and R = R 0 ( ε). In that case the constant in the last term behaves as ε 0 in the form K R,R0, R ( c)/ 0 ε β, β = ( m)/ = ( + m )/. (iv) The constant blows u in the limit m, and this is erfectly coherent, since a similar estimate is false for the Heat Equation. Moreover, the constant K R,R0,, blows u when, thus it does not rovide L local stability, while it rovides local L stability, for > c. Proof of Theorem.3. (i) Let u 0 and take a test function ψ Cc (Ω) and ψ 0. We can comute d ψ u dx = ψ u t u dx = ( ψ u ) ( ) u m dx dt Ω Ω Ω m ] = ψ (u +m u) dx + ( ) ψ u +m 3 u dx Ω Ω = ψ (u +m 4( ) ] ) dx + + m Ω ( + m ) ψ u +m dx (.7) Ω = ( ψ) u +m 4 ( ) dx + m Ω ( + m ) ψ u +m dx Ω ψ u +m dx. + m Ω This comutation holds true for any, and any m R, when one relaces, in the limit m 0, the quantity (u m )/m with log u; we also have to relace u +m /( + m ) by log(u) if + m = 0. Of course, when + m 0 the last term may be infinite, since it contains u +m so we make the assumtion > m. 0