Nonlocal porous medium equation: Barenblatt profiles and other weak solutions
aa r X i v : . [ m a t h . A P ] F e b NONLOCAL POROUS MEDIUM EQUATION: BARENBLATT PROFILES AND OTHERWEAK SOLUTIONS
PIOTR BILER, CYRIL IMBERT, AND GRZEGORZ KARCH
Abstract.
A degenerate nonlinear nonlocal evolution equation is considered; it can be understood as a porousmedium equation whose pressure law is nonlinear and nonlocal. We show the existence of sign changing weaksolutions to the corresponding Cauchy problem. Moreover, we construct explicit compactly supported self-similar solutions which generalize Barenblatt profiles — the well-known solutions of the classical porous mediumequation. Introduction
In this work, we study the following degenerate nonlinear nonlocal evolution equation(1.1) ∂ t u = ∇ · (cid:0) | u |∇ α − ( | u | m − u ) (cid:1) , x ∈ R d , t > , where m > ∇ α − denotes the integro-differential operator ∇ ( − ∆) α − , α ∈ (0 , u (0 , x ) = u ( x ) . First, we construct nonnegative self-similar solutions of equation (1.1) which are explicit and compactly sup-ported . They generalize the classical Barenblatt–Kompaneets–Pattle–Zel’dovich solutions of the porous mediumequation, see (1.4) below. Second, we prove the existence of sign changing weak solutions to problem (1.1)–(1.2)for merely integrable initial data, and we prove that these solutions satisfy sharp hypercontractivity L L p estimates. A nonlocal operator.
Equation (1.1) involves a nonlocal operator denoted by ∇ α − which can be definedas the Fourier multiplier whose symbol is iξ | ξ | α − . This notation emphasizes that it is a (pseudo-differential)operator of order α −
1. Recalling the definition of the fractional Laplace operator ( − ∆) α ( v ) = F − ( | ξ | α F v )and the Riesz potential I β = ( − ∆) − β , i.e. Fourier multipliers whose symbols are | ξ | α and | ξ | − β respectively (seefor instance [23, Ch. V]), the fractional gradient ∇ α − can also be written as ∇I − α . Finally, let us emphasizethat the definition of ∇ α − is consistent with the usual gradient: ∇ = ∇ ; the components of ∇ are the Riesztransforms; moreover we have ∇ · ∇ α − = ∇ α · ∇ α = − ( − ∆) α . It is also possible, following the reasoning from Date : August 8, 2018.2000
Mathematics Subject Classification.
Key words and phrases. porous medium equation, nonlocal equation, hypercontractivity, self-similar solutions.The authors wish to thank Jean Dolbeault and R´egis Monneau for the fruitful discussions they had together. The authors weresupported by an EGIDE project (PHC POLONIUM 20078TL, 0185, 2009–2010). The second author was supported by an ANRproject (EVOL). The first and the third authors were supported by the MNSzW grant N201 418839, and the Foundation for PolishScience operated within the Innovative Economy Operational Programme 2007–2013 funded by European Regional DevelopmentFund (Ph.D. Programme: Mathematical Methods in Natural Sciences). [12, Th. 1] to define the fractional gradient via the singular integral formula for smooth and bounded functions v : R d → R (1.3) ∇ α − v ( x ) = C d,α Z (cid:0) v ( x ) − v ( x + z ) (cid:1) z | z | d + α d z with a suitable constant C d,α > Related equations and results.
Our preliminary results for the problem (1.1)–(1.2) have been announcedin [3].First, we would like to shed light on the link between (1.1) and other partial differential equations. Noticethat when α = 2 equation (1.1) coincides with the classical (nonlinear parabolic) porous medium equation(1.4) ∂ t u = ∇ · (cid:0) | u |∇ ( | u | m − u ) (cid:1) = ∇ · (cid:0) ( m − | u | m − ∇ u (cid:1) . For the theory of porous media equations, the interested reader is referred to [27, 28] and references therein. Ofcourse, for m = 2, the Boussinesq equation is recovered.The following nonlinear and nonlocal equation(1.5) ∂ t v + | v x | (cid:18) − ∂ ∂x (cid:19) α v = 0in the one-dimensional case x ∈ R was studied by the first, the third authors and R. Monneau [4]. Such anequation was derived as a model for the dynamics of dislocations in crystals. In [4], the existence, uniquenessand comparison properties of (viscosity) solutions have been proved, and explicit self-similar solutions have beenconstructed. Notice that the function u = v x , where v is a solution to (1.5), solves the one-dimensional case of(1.1) with m = 2. Thus, equation (1.1) is a multidimensional generalization of the one in (1.5).Recently, Caffarelli and V´azquez [5, 7] studied nonnegative weak solutions of (1.1) in the case m = 2 in themultidimensional case. Precisely, they studied the following (nonlocal) porous medium equation in R d (1.6) ∂ t u = ∇ · ( u ∇ p ) , with the nonlocal pressure law p = ( − ∆) − s u , 0 < s <
1, obtained from the density u ≥
0. Notice that,for α = 2 − s ∈ (0 , ∂ t u = ∇ · ( u ∇ α − u ). For sign changing u ’s, our equation (1.1)is a (formally parabolic) extension of equation (1.6) of the structure of (1.5). In [5], Caffarelli and V´azquezconstructed nonnegative weak solutions for (1.6), i.e. for (1.1) with m = 2, with initial data satisfying: u ∈ L ( R d ) ∩ L ∞ ( R d ) and such that 0 ≤ u ( x ) ≤ A e − a | x | for some A, a >
0. Besides the positivity and the masspreservation, the properties of solutions, listed in the next paper [7, p. 4], include the finite speed of propagationproved using the comparison with suitable supersolutions. Further regularity properties of solutions of (1.1)with m = 2 and α ∈ (0 ,
1) are studied in [6].Another nonlocal porous medium equation has been proposed in [10, 11, 29] ∂ t u + ( − ∆) α ( | u | m − u ) = 0 ONLOCAL POROUS MEDIUM EQUATION 3 for α = 1 and α ∈ (0 , L -contraction property, so, they areunique. But self-similar solutions are not compactly supported [29, Th. 1.1].Finally, we recall that the following nonlocal higher order equation, appearing in the modeling of propagationof fractures in rocks, ∂ t u = ∇ · ( u n ∇ ( − ∆) u )(with u ≥ n > n = 1 corresponds to (1.1) with α = 3 and m = 2. Notation.
In this work, Q T denotes (0 , T ) × R d . The usual norm of the Lebesgue space L p ( R d ) is denotedby k · k p for any p ∈ [1 , ∞ ], and H s,p ( R d ) with the norm k · k H s,p is the fractional order Sobolev space, seeSection 3. The Fourier transform F and its inverse transform F − of a function v ∈ L ( R d ) are defined by F v ( ξ ) = (2 π ) − d Z v ( x ) e − ix · ξ d x, F − v ( x ) = (2 π ) − d Z v ( x ) e ix · ξ d ξ. Here, all integrals with no integration limits are over the whole space R d if one integrates with respect to x and over the whole half-line R + = [0 , ∞ ) if the integration is with respect to t . As usual, w + = max { , w } , w − = max { , − w } , so w = w + − w − . Constants (always independent of x and t ) will be denoted by the sameletter C , even if they may vary from line to line. Sometimes we write, e.g. , C = C ( p, q, r ) when we want toemphasize the dependence of C on particular parameters p, q, r , for instance.2. Main results
In this work, we show two main results: we construct explicit self-similar solutions of equation (1.1), as wellas we prove that the initial value problem (1.1)–(1.2) has a global-in-time weak solution which satisfies certainoptimal decay estimates.We first “recall” the appropriate notion of weak solutions for Equation (1.1), see for instance [27, 28].
Definition 2.1 (Weak solutions) . A function u : Q T → R is a weak solution of the problem (1.1)–(1.2) in Q T if u ∈ L ( Q T ), ∇ α − ( | u | m − u ) ∈ L ( Q T ) and | u |∇ α − ( | u | m − u ) ∈ L ( Q T ), and Z Z (cid:0) u∂ t ϕ − | u |∇ α − ( | u | m − u ) · ∇ ϕ (cid:1) d t d x + Z u ( x ) ϕ (0 , x ) d x = 0for all test functions ϕ ∈ C ∞ ( Q T ) ∩ C ( Q T ) such that ϕ has a compact support in the space variable x andvanishes near t = T .The first main result of this work says that there is a family of nonnegative explicit compactly supportedself-similar solutions of (1.1), i.e. nonnegative solutions that are invariant under a suitable scaling. Observethat if u ( t, x ) is a solution of (1.1), then so is L dλ u ( Lt, L λ x ) for each L >
0, where λ = ( d ( m −
1) + α ) − . Thus, PIOTR BILER, CYRIL IMBERT, AND GRZEGORZ KARCH the scale invariant solutions should be of the following form(2.1) u ( t, x ) = 1 t dλ Φ (cid:16) xt λ (cid:17) with λ = 1 d ( m −
1) + α , for some function Φ : R d → R satisfying the following nonlocal “elliptic type” equation(2.2) − λ ∇ · ( y Φ) = ∇ · ( | Φ |∇ α − ( | Φ | m − Φ)) where y = xt λ . Theorem 2.2 (Self-similar solutions) . Let α ∈ (0 , , m > . Consider the function Φ α,m : R d → R defined as (2.3) Φ α,m ( y ) = (cid:16) k α,d (1 − | y | ) α + (cid:17) m − with the constant k α,d = d Γ (cid:0) d (cid:1) ( d ( m −
1) + α )2 α Γ (cid:0) α (cid:1) Γ (cid:0) d + α (cid:1) . Then, the function u : (0 , ∞ ) × R d → R + defined by (2.1) with Φ = Φ α,m is a weak solution of (1.1) in the senseof Definition 2.1 in Q η,T ≡ ( η, T ) × R d for every < η < T < ∞ . Moreover, u ( t, x ) satisfies the equation in thepointwise sense for | x | 6 = t d ( m − α , and is min (cid:8) α m − , (cid:9) -H¨older continuous at the interface | x | = t d ( m − α .Remark . When α = 2 in expression (2.4) below, we recover the classical Barenblatt–Kompaneets–Pattle–Zel’dovich solutions of the porous medium equation (1.4), see for instance [28, 27]. Remark . For each M ∈ (0 , ∞ ) we can find a nonnegative self-similar solution u with prescribed mass M ≡ R u ( t, x ) d x (which is conserved in time) by a suitable scaling of the profile Φ α,m . Indeed, this self-similarsolution is given by the formula(2.4) u ( t, x ) = t − dd ( m − α k α,d (cid:18) R − (cid:12)(cid:12)(cid:12) xt − d ( m − α (cid:12)(cid:12)(cid:12) (cid:19) α + ! m − , where, for each M >
0, there exists a unique
R > R u ( t, x ) d x = M . Remark . Self-similar solutions of equation (1.6) (which is a particular case of equation (1.1)) have beenproved to exist in [7] by studying the following obstacle problem for the fractional Laplacian. For α ∈ (0 ,
2) andΨ( y ) = C − a | y | where a = a ( d, α ) and C >
0, one looks for a function P = P ( y ) with the following properties: P ≥ Ψ , ( − ∆) α P ≥ , and either P = Ψ or ( − ∆) α P = 0 . The novelty of our approach is that we exhibit the explicit self-similar profile Φ α, defined in (2.3) and, conse-quently, the explicit solution of this obstacle problem: P ( y ) = I α (Φ α, ) (cid:0) yR (cid:1) , where I α = ( − ∆) − α is the Rieszpotential and R >
Theorem 2.6 (Existence and decay of L p -norms) . Let α ∈ (0 , and (2.5) ( m > − αd if α ∈ (0 , ,m > − α if α ∈ (1 , . ONLOCAL POROUS MEDIUM EQUATION 5
Given u ∈ L ( R d ) , there exists a global-in-time weak solution u of the Cauchy problem (1.1) – (1.2) . Moreover, Z u ( t, x ) d x = Z u ( x ) d x and (2.6) k u ( t ) k p ≤ C ( d, α, m ) k u k d ( m − /p + αd ( m − α t − dd ( m − α (cid:0) − p (cid:1) for all t > , holds with the constant C ( d, α, m ) independent of p and u .The solution u is nonnegative if the initial condition u is so. If u ∈ L p ( R d ) for some p ∈ [1 , ∞ ] , then k u ( t ) k p ≤ k u k p holds for all t > .Remark . Estimates (2.6) are sharp since the decay in Theorem 2.6 corresponds exactly to that for self-similarsolutions constructed in Theorem 2.2. Moreover, for α = 2, they are similar to those for degenerate partialdifferential equations like the porous medium equation (showing the regularization effect on the L p -norms ofsolutions); see, e.g. , [28], [8, Ch. 2]. Remark . After proving those hypercontractivity estimates in [3], we learned that a similar result is obtainedin [6], however, for a less general model: α ∈ (0 , m = 2, and nonnegative u . Moreover, analogous decayestimates for another fractional porous medium equation of the form ∂ t u + ( − ∆) α ( | u | m − u ) = 0 were provedrecently in [11].Compared with the methods used in [5], we propose an alternative strategy of the proof of the existence ofsolutions. In this paper, we consider approximating solutions u = u δ,ε of the equation ∂ t u = δ ∆ u + ∇ · ( | u |∇ α − G ε ( u )) , considered in the whole space R d , where G ε ( u ) is a sufficiently smooth approximation of u | u | m − , and then wepass to the limit with the parameters ε ց
0, and δ ց
0. Solutions of the approximating equation exist becausethe parabolic regularization term δ ∆ u is strong enough to regularize equation (1.1) when 0 < α <
2, but ofcourse not for α = 2. Our approach resembles the approach to the one-dimensional model achieved in [4, Sec.4 and 5] via viscosity solutions. 3. Preliminaries
In this section, we collect known results that we will used in proofs of the main theorems.
Bessel and hypergeometric functions.
Bessel functions of order ν are denoted by J ν ( z ), and they behavefor small and large values of the (complex) variable z like J ν ( z ) ∼ ν +1) ( z ) ν as z → ,J ν ( z ) ∼ ( π ) cos( z − νπ − π ) z − as | z | → ∞ , PIOTR BILER, CYRIL IMBERT, AND GRZEGORZ KARCH where for functions f , g , the relation f ∼ g means that fg →
1. For the proofs of those properties of J ν , thereader is referred to, e.g. , [30]. The hypergeometric function , denoted by F ( a, b ; c ; z ), is defined for complex numbers a, b, c and z as thesum of the series F ( a, b ; c ; z ) = ∞ X n =0 ( a ) n ( b ) n ( c ) n n ! z n for | z | < , where ( a ) n ≡ Γ( a + n )Γ( a ) , and Γ denotes the Euler Gamma function. This series is absolutely convergent in theopen unit disc and also on the circle | z | = 1 if ℜ ( a + b − c ) < b = − n is a negative integer, F ( a, − n ; c ; z ) is a polynomial function ofdegree n . In particular, we have(3.1) F ( a, − c ; z ) = 1 − ac z. We will also use the following differentiation formula [19, p. 41](3.2) ddz (cid:0) F ( a, b ; c ; z ) (cid:1) = abc F ( a + 1 , b + 1; c + 1; z ) . The Weber–Schafheitlin integral.
If 0 < b < a and if integral (3.3) below is convergent, then the followingidentity holds true(3.3) Z ∞ t − λ J µ ( at ) J ν ( bt ) d t = b ν − λ a λ − ν − Γ( ν + µ − λ +12 )Γ( − ν + µ + λ +12 )Γ(1 + ν ) F (cid:18) ν + µ − λ + 12 , ν − µ − λ + 12 ; ν + 1; b a (cid:19) . According to Watson, [30, pp. 401–403], this result was obtained by Sonine and Schafheitlin. However, it isusually referred to as the
Weber–Schafheitlin discontinuous integral since there occurs a discontinuity for a = b . The Stroock–Varopoulos inequality.
We next recall the the Stroock–Varopoulos inequality , see [18, Theo-rem 2.1 and Condition (1.7)] for a proof.
Proposition 3.1.
For α ∈ (0 , , w ∈ C ∞ c ( R d ) and q > , the following inequality holds true (3.4) Z sgn w | w | q − ( − ∆) α w d x ≥ q − q Z (cid:12)(cid:12)(cid:12) ∇ α (cid:16) sgn w | w | q (cid:17)(cid:12)(cid:12)(cid:12) d x ≥ q − q Z (cid:12)(cid:12)(cid:12) ∇ α | w | q (cid:12)(cid:12)(cid:12) d x. Fractional order Sobolev spaces.
The fractional order Sobolev spaces are defined as H s,p ( R d ) = { v ∈ L p ( R d ) : ∇ s v ∈ L p ( R d ) } = { v ∈ L p ( R d ) : ( I − ∆) s v ∈ L p ( R d ) } , here with p ∈ (1 , ∞ ), supplemented with the usual norm denoted by k · k H s,p , and we refer the reader to thebooks [25, 26] for properties of those spaces. In particular for s = α − α ∈ (1 , H α − ,p ( R d ) ⊂ L ∞ ( R d ) provided p > dα − > . We also recall the fractional integration theorem [23, Ch. V, § I s = ( − ∆) − s satisfies(3.6) kI s u k q ≤ C ( p, q, s ) k u k p for all s ∈ (0 , d ) and p, q ∈ (1 , ∞ ) satisfying q = p − sd . ONLOCAL POROUS MEDIUM EQUATION 7
Some functional inequalities.
We will use the following
Nash inequality (3.7) k v k (cid:0) αd (cid:1) ≤ C N k∇ α v k k v k αd valid for all functions v ∈ L ( R d ), such that ∇ α v ∈ L ( R d ), and with a constant C N = C ( d, α ) >
0. The proofof (3.7) for d = 1 can be found in, e.g. , [16, Lemma 2.2], and this extends easily to the general case d ≥ Gagliardo–Nirenberg type inequality
Lemma 3.2.
For p > and p ≥ m − , the inequality (3.8) k u k ap ≤ C N (cid:13)(cid:13) ∇ α | u | r (cid:13)(cid:13) k u k b holds with (3.9) a = pp − d ( r −
1) + αd , b = a − r = d ( m −
1) + pαd ( p − , r = p + m − . Proof.
This inequality is a consequence of the Nash inequality (3.7) written for v = | u | r , i.e. (3.10) k u k r (cid:0) αd (cid:1) r ≤ C N (cid:13)(cid:13) ∇ α | u | r (cid:13)(cid:13) k u k rαdr , and two H¨older inequalities k u k p ≤ k u k γr k u k − γ with γ = (cid:18) p − r − (cid:19) rp , and k u k r ≤ k u k δp k u k − δ with δ = (cid:18) r − p − (cid:19) pr . Combining the above three inequalities, we get (3.8). (cid:3) Proof of Theorem 2.2
This section is devoted to the study of nonnegative self-similar solutions for (1.1) with m >
1. As explainedabove, this problem reduces to a study of the elliptic-like equation (2.2) which for nonnegative Φ takes the form − λy Φ = Φ ∇ α − (Φ m − ) . Moreover, since we want to construct compactly supported solutions, we are interested in solutions Φ vanishingoutside the unit ball B . This is the reason why we consider the Dirichlet problem − λy = ∇ α − (Φ m − ) in B , Φ = 0 in R d \ B . (4.1)It is well known that, in the case of nonlocal operators (such as ∇ α − ), the homogeneous Dirichlet conditionshould be understood in the form Φ ≡ B , and not only Φ = 0 on the boundary ∂B .The reader is referred to, e.g. , [2] for more explanations.We claim that the proof of Theorem 2.2 reduces to the following key computation. Here, F denotes theclassical hypergeometric function defined in Section 3. PIOTR BILER, CYRIL IMBERT, AND GRZEGORZ KARCH
Lemma 4.1.
For all β ∈ (0 , , β < d , and γ > , we have (4.2) I β (cid:16) (1 − | y | ) γ + (cid:17) = C γ,β,d × F (cid:16) d − β , − γ + β ; d ; | y | (cid:17) for | y | ≤ , ˜ C γ,β,d | y | β − d × F (cid:16) d − β , − β ; d + γ ; | y | (cid:17) for | y | > , with C γ,β,d = 2 − β Γ ( γ +1 ) Γ ( d − β ) Γ ( d ) Γ ( β + γ +1 ) and ˜ C γ,β,d = 2 − β Γ ( γ +1 ) Γ ( d − β ) Γ ( d ) Γ ( d + γ +1 ) . The proof of this lemma is postponed to the end of this section.The following corollary is an immediate consequence of Lemma 4.1 with 2 − β = α = γ , of the property of F formulated in (3.1), and of the identity ( − ∆) α = ( − ∆) I − α . It has an important probabilistic interpretation,and recently, related results and generalizations have been proved in [13]. Corollary 4.2 (Getoor [14, Th. 5.2]) . For all α ∈ (0 , , the identity K α,d ( − ∆) α (1 − | y | ) α + = 1 in B holds true with the constant K α,d = Γ ( d ) α Γ ( α ) Γ ( d + α ) . Before proving Lemma 4.1, we first use it to derive Theorem 2.2.
Proof of Theorem 2.2.
We check that u ( t, x ) = t − dλ Φ α,m ( t − λ x ) is a weak solution of (1.1) in the sense ofDefinition 2.1. First, u ∈ L ( Q T ) if and only if Φ α,m ∈ L ( R d ), which is obviously true. For later use, it isconvenient to introduce the function Φ α ( u ) = (cid:0) − | y | (cid:1) α + , so that k α,d Φ α = Φ m − α,m .The fact that, for all η, T such that 0 < η < T , u ∇ α − ( u m − ) and ∇ α − ( u m − ) are locally integrable in( η, T ) × R d follows from I − α (Φ α ) ∈ H , ( R d ) , which we prove by computing I − α (Φ α ). In order to do so, we first assume that α > − d , and we applyLemma 4.1 with γ = α ∈ (0 ,
2) and β = 2 − α , we use equation (3.1), and we get(4.3) I − α (Φ α )( y ) = ( C α, − α,d (cid:0) − d + α − d | y | (cid:1) if | y | ≤ , ˜ C α, − α,d | y | − ( d + α ) 2 F (cid:16) d + α − , α ; d + α ; | y | (cid:17) if | y | > . The right-hand side of equation (4.3) defines a locally integrable function because a + b − c = ( d + α −
1) + α − d + α < . We then deduce that ∇ α − (Φ α )( y ) = ∇I − α (Φ α )( y ) = − λk α,d y for y ∈ B . Note also that ∇ α − (Φ α ) can be computed outside B thanks to the differentiation formula (3.2).We now remark that Φ α,m ( y ) (cid:0) ∇ α − Φ m − α,m (cid:1) ( y ) = − λy Φ α,m ( y ) for all y ∈ R d , ONLOCAL POROUS MEDIUM EQUATION 9 which is in L ( R d ). Moreover, the following equalities hold true in the sense of distributions in Q T , ∂ t u ( t, x ) = − λt − dλ − ∇ y · ( y Φ α,m )( t − λ x ) , ∇ x · ( u ∇ α − ( | u | m − ))( t, x ) = t − dλ − ∇ y · (Φ α,m ∇ α − Φ m − α,m )( t − λ x ) . This allows us to conclude that u is indeed a weak solution of (1.1) in ( η, T ) × R d for all 0 < η < T < ∞ if α > − d .Assume now that 0 < α ≤ − d , which means that d = 1 and α ≤
1. The critical case α = 1 can be obtainedby passing to the limit as α ց
1; indeed, the constants C α, − α,d ( d + α −
2) and ˜ C α, − α,d ( d + α −
2) appearingin (4.3) simplify thanks to the relation z Γ( z ) = Γ( z + 1). If now α <
1, we can argue as above by analyticcontinuation. The proof is now complete. (cid:3)
Now we turn to the proof of the main technical lemma.
Proof of Lemma 4.1.
We first assume that β ∈ (0 , d ) and that γ > max { , d − β − } , and we then argue bythe analytic continuation with a choice of parameters corresponding each time to F defined and bounded forall | y | ≤ γ ( y ) = (cid:0) − | y | (cid:1) γ + is expressed in terms of Bessel functions, see, e.g. , [23, Ch.IV, Sec. 3] F (Φ γ )( ξ ) = 2 γ Γ (cid:16) γ (cid:17) | ξ | d + γ J d + γ ( | ξ | ) . Since I β is the Fourier multiplier of symbol | ξ | − β , I β (Φ γ ) is the (inverse) Fourier transform of the followingradially symmetric function 2 γ Γ (cid:16) γ (cid:17) | ξ | d + γ + β J d + γ ( | ξ | ) . We recall that by properties of Bessel functions collected in Section 3, we have J ν ( r ) = O ( r ν ) as r → J ν ( r ) = O (cid:16) r − (cid:17) as r → ∞ . We see that the previous function is integrable since β < d and d < γ + 2 β + 1.Thanks to [24, Th. 3.3], we get I β (Φ γ )( y ) = 2 γ Γ (cid:16) γ (cid:17) | y | − d Z ∞ t d + γ + β J d + γ ( t ) t d J d − ( t | y | ) d t = 2 γ Γ (cid:16) γ (cid:17) | y | − d Z ∞ t − ( γ + β ) J d + γ ( t ) J d − ( t | y | ) d t. We obtain (4.2) applying (3.3) with the following choice of parameters: • if | y | ≤
1, we put λ = γ + β , µ = d + γ , ν = d − a = 1 and b = | y | , • if | y | >
1, we put λ = γ + β , µ = d − ν = d + γ , a = | y | and b = 1. (cid:3) A regularized problem
In order to construct weak solutions of (1.1) for general initial data, we first consider the following regularizedproblem(5.1) ∂ t u = δ ∆ u + ∇ · (cid:0) | u |∇ α − ( G ( u )) (cid:1) , u (0 , x ) = u ( x ) , where G : R → R satisfies(5.2) G differentiable and increasing ,G (0) = G ′ (0) = 0 ,G ′ locally Lipschitz continuous . Remark that for m ≥ m = 2, the function G ( u ) = | u | m − u satisfies (5.2). For m ∈ (1 , G = G ε of | u | m − uG ε ( u ) = sgn u (cid:18)(cid:0) u + ε (cid:1) m − − ε m − (cid:19) with ε >
0. The following theorem holds true for a general function G satisfying (5.2). Theorem 5.1 (Existence of solutions to the regularized problem) . Let δ > and assume that G is an arbitraryfunction satisfying (5.2) . Moreover, assume (5.3) u ∈ ( L ( R d ) ∩ L ∞ ( R d ) if α ∈ (0 , ,L ( R d ) ∩ (cid:0) ∩ p>p α H α − ,p ( R d ) (cid:1) if α ∈ (1 , , with p α = dα − > . There exists a unique function u in the space (5.4) u ∈ (cid:26) C (cid:0) [0 , ∞ ) , L ( R d ) ∩ L ∞ ( R d ) (cid:1) if α ∈ (0 , , ∩ p>p α C (cid:0) [0 , ∞ ) , L ( R d ) ∩ H α − ,p ( R d ) (cid:1) if α ∈ (1 , , satisfying problem (5.1) in the usual weak sense (5.5) Z Z (cid:0) u∂ t ϕ − | u |∇ α − ( G ( u )) · ∇ ϕ − δ ∇ u · ∇ ϕ (cid:1) d t d x = 0 for all ϕ ∈ C ∞ c ( Q T ) .Moreover, u ( t, x ) is nonnegative if the initial condition u is so, and for all t > and q ∈ [1 , ∞ ] we have (5.6) Z u ( t, x ) d x = Z u ( x ) d x and k u ( t ) k q ≤ k u k q . Local-in-time existence of mild solutions.Proposition 5.2.
Let p > p α = dα − . There exists T > depending only on u , and a function u in the space (5.7) u ∈ (cid:26) C (cid:0) [0 , T ) , L ( R d ) ∩ L ∞ ( R d ) (cid:1) if α ∈ (0 , , C (cid:0) [0 , T ) , L ( R d ) ∩ H α − ,p ( R d ) (cid:1) if α ∈ (1 , such that (5.8) u ( t ) = e δt ∆ u + Z t ∇ e δ ( t − s )∆ · Ψ( u ( s )) d s with Ψ( u ) = | u |∇ α − G ( u ) , in C [0 , T ] , L ( R d ) ∩ L ∞ ( R d )) where e t ∆ denotes the heat semigroup. ONLOCAL POROUS MEDIUM EQUATION 11
Remark . We identify the heat semigroup e t ∆ and its kernel (4 πt ) − d exp (cid:16) − | x | t (cid:17) . We will use the followingclassical fact(5.9) (cid:13)(cid:13) ∇ β e δt ∆ v (cid:13)(cid:13) p ≤ C ( p, r, β, δ ) t − d ( r − p ) − β k v k r with 1 ≤ r ≤ p ≤ ∞ , and β ∈ [1 , Proof of Proposition 5.2.
We look for a solution u ∈ C ([0 , T ] , X ) as a fixed point of the map T : u e δt ∆ u + Z t ∇ e δ ( t − s )∆ · Ψ( u ( s )) d s, where X is chosen as follows(5.10) X = ( L ( R d ) ∩ L ∞ ( R d ) if α ∈ (0 , ,L ( R d ) ∩ H α − ,p ( R d ) if α ∈ (1 , . The associated norms are k u k + k u k Y with Y = L ∞ ( R d ) and Y = H α − ,p ( R d ), respectively. We show that T has a fixed point by the Banach contraction principle as soon as T = T ( k u k X ) > Lemma 5.4.
For all T ∈ (0 , , the operator T maps C ([0 , T ] , X ) into itself. Moreover, there exist C > and γ > such that for all u, v ∈ B (0 , R ) ⊂ C ([0 , T ] , X ) , (5.11) kT ( u ) − T ( v ) k C ([0 ,T ] ,X ) ≤ C ( R ) T γ k u − v k C ([0 ,T ] ,X ) , where C ( R ) is a constant which also depends on α, d, ε, m, δ ( and on p if α ∈ (1 , . Indeed, once this lemma is proved, we first derive(5.12) kT ( u ) k C ([0 ,T ] ,X ) ≤ k u k X + RC ( R ) T γ by choosing v = 0 in (5.11) and using estimate (5.9). Now it is enough to choose R = 2 k u k X and T > C ( R ) T γ ≤ in order to ensure that T maps B (0 , R ) into itself, and is a contraction. The case α ∈ (0 , . In order to get estimate (5.11), we first write(5.13) T ( u )( t ) − T ( v )( t ) = Z t ∇ e δ ( t − s )∆ · (Ψ( u ) − Ψ( v )) ( s ) d s, and the difference of Ψ’s is represented as(5.14) Ψ( u ) − Ψ( v ) = ( | u | − | v | ) ∇ α − G ( u ) + | v |∇ α − ( G ( u ) − G ( v )) . Lemma 5.5.
For every α ∈ (0 , and p ∈ (1 , ∞ ) there exists a constant C ( p, α ) > such that for all u ∈ L ∞ ( R d ) ∩ L p ( R d ) the following inequality (5.15) k∇ α − ( G ( u ) − G ( v )) k q ≤ C ( p, α ) sup | z |≤k u k ∞ + k v k ∞ G ′ ( z ) ! k u − v k p holds true with q = p − − αd . Proof.
Since, for α ∈ (0 , ∇ α − = ∇ I − α , where the components of ∇ are the Riesz transforms(which are bounded operators on L p ( R d ) for each p ∈ (1 , ∞ )), we obtain (5.15) from estimate (3.6) as follows k∇ α − ( G ( u ) − G ( v )) k q ≤ kI − α ( G ( u ) − G ( v )) k q ≤ C ( p, α ) k G ( u ) − G ( v ) k p ≤ C k u − v k p . (cid:3) Now, we come back to the proof of (5.11) with X = L ( R d ) ∩ L ∞ ( R d ), First, for all u, v ∈ B (0 , R ) ⊂ X andsome q ∈ (1 , ∞ ), k Ψ( u ) − Ψ( v ) k ≤ C k u − v k q ∗ k∇ α − G ( u ) k q + C k v k q ∗ k∇ α − ( G ( u ) − G ( v )) k q ≤ C k u − v k q ∗ G ′ ( k u k ∞ ) k u k p + C k v k q ∗ G ′ ( k u k ∞ + k v k ∞ ) k u − v k p (5.16) ≤ C ( R ) k u − v k X with C ( R ) = C ( α, d, q, RG ′ (2 R )) and q + q ∗ = 1 and q = p − − αd . We used estimate (5.15) twice to get thesecond line in (5.16), and the inequality k u k r ≤ k u k X which is valid for all r ∈ [1 , ∞ ] to obtain the last one.The estimate of the second norm in X is obtained similarly: for all u, v ∈ B (0 , R ) ⊂ X and some q ∈ (1 , ∞ ),(5.17) k Ψ( u ) − Ψ( v ) k q ≤ k u − v k ∞ k∇ α − G ( u ) k q + k v k ∞ k∇ α − ( G ( u ) − G ( v )) k q ≤ C ( R ) k u − v k X . Now, we apply inequalities (5.9) with β = 1, h p, r i = h , i and h p, r i = h∞ , q i , respectively, and the estimates(5.16), (5.17) yield kT ( u ) − T ( v ) k C ([0 ,T ] ,L ( R d )) ≤ CC ( R ) T k u − v k C [0 ,T ] ,X ) , (5.18) kT ( u ) − T ( v ) k C ([0 ,T ] ,L ∞ ( R d )) ≤ CC ( R ) T − d q ∗ k u − v k C [0 ,T ] ,X ) . (5.19)Combining (5.13), (5.14), (5.18) and (5.19), we thus get (5.11) for α ∈ (0 , γ = − d q ∗ , now with a newconstant C ( R ) = C ( α, d, q ∗ , RG ′ (2 R )), where we have chosen q ∗ > d to ensure d q ∗ < .As far as the continuity of T ( u ) with respect to time is concerned, it is enough to study S ( u ) = Z t ∇ e δ ( t − s )∆ · Ψ( u ( s )) d s. We fix t ∈ [0 , T ] and write for h small enough (and positive if t = 0, negative if t = T ), S ( u ( t + h )) − S ( u ( t )) = Z t + ht ∇ e δ ( t + h − s )∆ · Ψ( u ( s )) d s + Z t ∇ e δ ( t − s )∆ · (e δh ∆ Ψ( u ( s )) − Ψ( u ( s ))) d s. As above, use two key estimates (5.16) and (5.16) together with (5.9) (and the dominated convergence theorem)to conclude the proof of Proposition 5.2 for α ∈ (0 , The case α ∈ (1 , . We argue as before, using (5.13) and (5.14). We need now to state and to prove thecorresponding key technical lemma.
Lemma 5.6.
For α ∈ (1 , , p > dα − , and u ∈ H α − ,p ( R d ) , (5.20) k G ( u ) − G ( v ) k H α − ,p ≤ C ( k u k H α − ,p + k v k H α − ,p ) k u − v k H α − ,p , ONLOCAL POROUS MEDIUM EQUATION 13 where C ( k u k H α − ,p + k v k H α − ,p ) depends on k u k H α − ,p , k v k H α − ,p and on the W , ∞ -norm of G ′ in the interval [0 , k u k ∞ + k v k ∞ ] .Proof. We use the classical identity(5.21) G ( u ) − G ( v ) = K ( u, v )( u − v ) with K ( u, v ) = Z G ′ ( τ u + (1 − τ ) v ) d τ, and the Moser estimate for the product of two functions in H α − ,p ( R d ), see, e.g. , [25, Ch. 2, ineq. (0.22)], toobtain the following inequality(5.22) k G ( u ) − G ( v ) k H α − ,p ≤ k u − v k ∞ k K ( u, v ) k H α − ,p + k u − v k H α − ,p k K ( u, v ) k ∞ . Moreover, we recall [25, Ch. 2, Prop. 4.1] that for every increasing locally Lipschitz function H we have k H ( u ) k H α − ,p ≤ C | H ′ | ( k u k ∞ ) k u k H α − ,p for every p ∈ (1 , ∞ ) and α − ∈ (0 , H = G ′ , we deduce that k K ( u, v ) k H α − ,p ≤ Z k G ′ ( τ u + (1 − τ ) v ) k H α − ,p d τ ≤ C | G ′′ | ( k u k ∞ + k v k ∞ )( k u k H α − ,p + k v k H α − ,p ) . (5.23)Moreover, we have the trivial estimate(5.24) k K ( u, v ) k ∞ ≤ | G ′ | ( k u k ∞ + k v k ∞ ) . Combining (5.22), (5.23), (5.24) and (3.5), we finally complete the proof of inequality (5.20). (cid:3)
Now, we are in a position to obtain the estimate of the first component of the norm of X . From (5.14), weget(5.25) k Ψ( u ) − Ψ( v ) k ≤ C ( R ) R k u − v k p ′ + C (2 R ) k v k p ′ k u − v k X ≤ C ( R ) k u − v k X with C ( R ) = CR ( C ( R ) + C (2 R )). We used inequality (5.20) twice, as well as the fact that k u k r ≤ C k u k X for all r ∈ [1 , ∞ ] and u ∈ X . From (5.25) and (5.9) with β = 1 and h p, r i = h , i , we get inequality (5.18)where C ( R ) is replaced with ˜ C ( R ) = C ( R ).The estimate of the H α − ,p -norm is obtained analogously. First, we have(5.26) k Ψ( u ) − Ψ( v ) k p ≤ k u − v k ∞ k∇ α − G ( u ) k p + k v k ∞ k∇ α − ( G ( u ) − G ( v )) k p ≤ C ( R ) k u − v k X . From inequalities (5.26) and (5.9) with β = 1 and β = α and h p, r i = h p, p i , we get(5.27) kT ( u ) − T ( v ) k C ([0 ,T ] ,H α − ,p ( R d )) ≤ CC ( R )( T + T − α ) k u − v k C [0 ,T ] ,X ) . Finally, combining (5.18) and (5.27), we complete the proof of (5.11) with γ = − α and with some ˜ C ( R ).The time continuity of T ( u ) is proved as in the case α ∈ (0 , (cid:3) Regularity of the solutions.Corollary 5.7 (Regularity of the solutions) . Consider u ∈ L ( R d ) ∩ L ∞ ( R d ) if α ∈ (0 ,
1] ( as before ) , and u ∈ L ( R d ) ∩ (cid:0) ∩ p>p α H α − ,p ( R d ) (cid:1) if α ∈ (1 ,
2) ( a strengthened assumption ) . Then the solution constructed inProposition 5.2 enjoys the following regularity (5.28) u ∈ C (cid:0) (0 , T ) , L p ( R d ) (cid:1) ∩ C (cid:0) (0 , T ) , H ,p ( R d ) (cid:1) for every p ∈ (¯ p α , ∞ ) with ¯ p α = ( dd − (1 − α ) if α ∈ (0 , , dα − if α ∈ (1 , . In particular, u = u ( t, x ) is a weak solution of equation (5.1) , i.e. ∂ t u = δ ∆ u + ∇ · Ψ( u ) with Ψ( u ) = | u |∇ α − G ( u ) in (0 , T ) × R d in the sense of distributions ( cf. equation (5.5)) , and (5.29) Z u ( t, x ) d x = Z u ( x ) d x for all t ∈ (0 , T ) .Proof. If α ∈ (0 , u ∈ C ([0 , T ] , X ), where the space X is defined in (5.10), we obviously have u ∈ L ∞ ((0 , T ) , L ∞ ( R d )). We derive from inequality (3.6) that for all p ∈ (¯ p α , ∞ ), we have ∇ α − G ( u ) ∈ L ∞ ((0 , T ) , L p ( R d )) with ¯ p α = dd − (1 − α ) >
1. This implies ∇ · Ψ( u ) ∈ L q (cid:0) (0 , T ) , H − ,p ( R d ) (cid:1) for all q ∈ (1 , ∞ ) and p ∈ (¯ p α , ∞ ). Thus, the maximal regularity of mild solutions for the nonhomogeneousheat equation [17] gives us ∇ u ∈ L q ((0 , T ) , L p ( R d )) for every q ∈ (1 , ∞ ) and p ∈ (¯ p α , ∞ ). Consequently, ∇ · (cid:0) | u |∇ α − G ( u ) (cid:1) = f + f with(5.30) f = ∇| u | · ∇ α − G ( u ) , f = −| u | ( − ∆) α G ( u ) . First, we remark that f ∈ L q ((0 , T ) , L p ( R d )) for every q ∈ (1 , ∞ ) and p ∈ (¯ p α , ∞ ). Second, we notice that G ( u ) ∈ L q ((0 , T ) , H ,p ( R d )), hence ( − ∆) α G ( u ) ∈ L q ((0 , T ) , H − α,p ( R d )) ⊂ L q ((0 , T ) , L p ( R d )). Hence, we alsohave f ∈ L q ((0 , T ) , L p ( R d )) for every q ∈ (1 , ∞ ) and p ∈ (¯ p α , ∞ ). Using again the maximal regularity result,we obtain that ∂ t u ∈ L q ((0 , T ) , L p ( R d ))for every q ∈ (1 , ∞ ) and p ∈ (¯ p α , ∞ ). Thus, using the following representation in L p ( R d ) for all 0 < s < tu ( t ) − u ( s ) = Z ts ∂ t u ( τ ) d τ, and the H¨older inequality we obtain for every p ∈ (¯ p α , ∞ ) k u ( t ) − u ( s ) k p ≤ Z ts k ∂ t u ( τ ) k p d τ ≤ (cid:18)Z ts k ∂ t u ( τ ) k qp d τ (cid:19) q ( t − s ) q ∗ , ONLOCAL POROUS MEDIUM EQUATION 15 i.e. u ∈ C ,β ((0 , T ) , L p ( R d )) for all p ∈ (¯ p α , ∞ ) and β ∈ (0 , u ) ∈ C ,β ((0 , T ) , L p ( R d )) for every p ∈ (¯ p α , ∞ ). Now, the classical theory of linear parabolic equations,see, e.g. , [20, Ch. 4, Theorem 3.5], implies that ∂ t u ∈ C ,β ((0 , T ) , L p ( R d )) and u ∈ C ,β ((0 , T ) , H ,p ( R d )) whichis the desired regularity result.If α ∈ (1 , u ∈ L q ((0 , T ) , H α − ,p ( R d )) for all q ∈ (1 , ∞ ) and p ∈ ( p α , ∞ ), where p α = ¯ p α = d/ ( α − ∇ α − G ( u ) ∈ L q ((0 , T ) , L p ( R d ))for all p ∈ ( p α , ∞ ) and q ∈ (1 , ∞ ). In particular, Ψ( u ) = | u |∇ α − G ( u ) ∈ L q ((0 , T ) , L p ( R d )). Hence, themaximal regularity gives us(5.32) u ∈ L q ((0 , T ) , H ,p ( R d ))for all p ∈ ( p α , ∞ ) and q ∈ (1 , ∞ ). We now write once again ∇ · Ψ( u ) = f + f with f = sgn u ∇ u · ∇ α − G ( u ) , f = | u |∇ α − ( G ′ ( u ) ∇ u ) . In view of (5.31) and (5.32), we have f ∈ L q ((0 , T ) , L p ( R d )) . We now claim that f ∈ L q ((0 , T ) , H − − α ) ,p ( R d )) . Indeed, G ′ ( u ) ∇ u ∈ L q ((0 , T ) , L p ( R d )) . This implies that ∇ α − ( G ′ ( u ) ∇ u ) ∈ L q ((0 , T ) , H − − α ) ,p ( R d ))which, in turn, implies the claim. Hence, ∇ · Ψ( u ) ∈ L q ((0 , T ) , H − − α ) ,p ( R d ))for all p ∈ ( p α , ∞ ) and q ∈ (1 , ∞ ). Then, the maximal regularity implies that u ∈ L q ((0 , T ) , H − α ) ,p ( R d ))for all for all p ∈ ( p α , ∞ ) and q ∈ (1 , ∞ ), thus we see that the space regularity of u is improved. More generally,the same argument shows that if u ∈ L q ((0 , T ) , H β,p ( R d )) , with β ≤ α, then u ∈ L q ((0 , T ) , H β +(2 − α ) ,p ( R d )) . Now choose the least integer k ≥ β k = 1 + k (2 − α ) > α , and notice that β k <
2. Then f ∈ L q ((0 , T ) , H β k − α,p ( R d )) ⊂ L q ((0 , T ) , L p ( R d )) . Then the maximal regularity implies that ∂ t u ∈ L q ((0 , T ) , L p ( R d )), which implies, as was in the case α ∈ (0 , u ∈ C ,β ((0 , T ) , L p ( R d )) for all p ∈ (1 , ∞ ) and β ∈ (0 , L q ((0 , T ) , Y ) extends readily to spaces of the form C ,β ((0 , T ) , Y ). This yields the desired regularity result inthe case α ∈ (1 , ∂ t u = δ ∆ u + ∇ · f with f ∈ C ,β ((0 , T ) , L p ( R d )) are in fact weak solutions, i.e. they satisfy the equation in the sense of distributions.Under these regularity properties, the proof of the mass conservation property (5.29) is completely standard. (cid:3) Convexity inequalities.
First, we show a simple but useful technical result involving monotone functions andthe fractional Laplacian.
Lemma 5.8.
Let α ∈ (0 , . Assume that g, h ∈ C [0 , ∞ ) are strictly increasing functions. Then, for everynonnegative v ∈ C ∞ c ( R d ) we have Z h ( v )( − ∆) α g ( v ) d x ≥ . Proof.
Notice that for α = 2 this lemma is obviously true, which one checks integrating by parts. Now, let α ∈ (0 , − ∆) α C = 0 for every constant C ∈ R , we can assume that g (0) = 0. In the same way, wecan assume that h (0) = 0, because R ( − ∆) α w d x = 0 for every w ∈ C ∞ c ( R d ). Defining w = g ( v ), it suffices toshow that Z h ( g − ( w ))( − ∆) α w d x ≥ w ∈ C ∞ c ( R d ) such that w ≥
0. To do it, notice that f ∈ C [0 , ∞ ) defined via the relation f ( s ) = R s h ( g − ( τ )) d τ for s ≥ f ′′ ( s ) ≥ − ∆) α g ( v ) ≤ g ′ ( v )( − ∆) α v, (see, e.g. , [9], [12, Lemma 1]) we obtain R h ( g − ( w ))( − ∆) α w d x ≥ R ( − ∆) α f ( w ) d x = 0 . (cid:3) Next, we formulate a crucial technical tool used in the derivation of various integral estimates for solutionsof the regularized problem (5.1).
Proposition 5.9 (Convexity inequalities) . Consider a C function ϕ : R → R + such that, for all r ∈ R , r = 0 , ϕ ′′ ( r ) > , and (5.34) ϕ ( r ) + | ϕ ′ ( r ) | + ϕ ′′ ( r ) ≤ C ( | r | M + | r | M ) for some constant C > and M , M ∈ [1 , ∞ ) . Then for all ≤ s < t ≤ T , the function u given byProposition 5.2 satisfies (5.35) Z ϕ ( u ( t, x )) d x + Z ts Z ψ ( u ( τ, x ))( − ∆) α G ( u ( τ, x )) d x d τ + δ Z ts Z ϕ ′′ ( u ) |∇ u | d x d τ ≤ Z ϕ ( u ( s, x )) d x, where ψ ( r ) = | r | ϕ ′ ( r ) − ϕ ( r ) sgn r . The proof of Proposition 5.9 is more or less classical, and we recall it in Appendix for the sake of completeness.
ONLOCAL POROUS MEDIUM EQUATION 17
Remark . Remark that ψ ′ ( r ) = | r | ϕ ′′ ( r ) >
0, hence the function ψ is increasing. Since G is also increasing,the result stated in Lemma 5.8 can be applied to show that the first dissipation term Z ts Z ψ ( u ( τ, x ))( − ∆) α G ( u ( τ, x )) d x d τ is nonnegative. The fact that this quantity is finite is a part of the result stated in Proposition 5.9. Moreover,[18, Theorem 2.2] implies that g ϕ ( G ( u )) ∈ L ((0 , T ) , H α , ( R d )) for a function g ϕ constructed from ϕ , see [18]for the detailed presentation. The special case ϕ ( r ) = | r | p is treated below. Remark . The convexity of ϕ also implies that the second dissipative term in (5.35) is nonnegative. Hence,Proposition 5.9 implies that R ϕ ( u ( t, x )) d x decreases along the flow of the regularized equation (5.1). Corollary 5.12 (Estimates of the L p -norms) . For all p ∈ (1 , ∞ ) and < s < t , (5.36) Z | u ( t ) | p d x + ( p − Z ts Z | u | p − u ( − ∆) α G ( u ) d x d τ + δp ( p − Z ts Z | u | p − |∇ u | d τ ≤ Z | u ( s ) | p d x. In particular, for p ≥ ¯ p α (see Corollary 5.7), (5.37) dd t Z | u | p d x ≤ − ( p − Z | u | p − u ( − ∆) α G ( u ) d x − δp ( p − Z | u | p − |∇ u | . Thus, for all p ∈ [1 , ∞ ] , the norm k u ( t ) k p decreases as t increases.Proof. If p ≥
3, we can apply Proposition 5.9 with 0 ≤ s < t ≤ T , ϕ ( r ) = | r | p ; indeed, in this case, ϕ is a C -function and satisfies the growth assumption (5.34) with h M , M i = h p, p − i . Next, since u ∈C ((0 , T ) , L p ( R d )), we obtain (5.37) from the inequality in Proposition 5.9 by a direct computation. We leavethe details to the reader.If p ∈ (1 , η >
0, the function ϕ η such that ϕ η (0) = ϕ ′ η (0) = 0 and ϕ ′′ η ( r ) = p ( p − r + η ) p − − η p − ) . In particular, the function ϕ η satisfies the assumptions of Proposition 5.9, hence, we have Z ϕ η ( u ( t, x )) d x + Z ts Z ψ η ( u ( s, x ))( − ∆) α G ( u ( s, x )) d x d s + δ Z ts Z ϕ ′′ η ( u ( s, x )) |∇ u ( s, x ) | d x d s ≤ Z ϕ η ( u ( s, x )) d x, with ψ ′ η ( r ) = | r | ϕ ′′ η ( r ), ψ η (0) = 0. Letting now η → k u ( t ) k p ≤ k u k p for all t > p ∈ (1 , ∞ ). By computing the limits as p → ∞ and p →
1, the bounds k u ( t ) k ≤ k u k and k u ( t ) k ∞ ≤ k u k ∞ are also obtained. (cid:3) We can now complete the proof of Theorem 5.1.
Proof of Theorem 5.1.
In view of Proposition 5.2 and Corollary 5.7, it remains to prove that solutions arenonnegative if initial data are so, and that solutions are global in time.
The positivity property is derived immediately in a usual way from the conservation of mass property (5.6)and the monotonicity of the L -norm. Indeed, (5.6) yields Z u ( T, x ) d x = Z u + ( T, x ) d x − Z u − ( T, x ) d x = Z (( u ) + − ( u ) − )( x ) d x and Z | u ( T, x ) | d x = Z u + ( T, x ) d x + Z u − ( T, x ) d x ≤ Z (( u ) + + ( u ) − )( x ) d x (here, as usual, u + = max { , u } and u − = max { , − u } ). These inequalities imply R u − ( T, x ) d x ≤ R ( u ) − ( x ) d x ,and, in particular, the assumption ( u ) − = 0 gives us u ≥ α ∈ (0 , k u k + k u k ∞ , and this norm of the solution does not increase. Hence, we extend u = u ( t, · ) to thewhole half-line [0 , ∞ ), step-by-step.For α ∈ (1 , v = 0 yield k u ( t ) k H α − ,p ≤ k u k H α − ,p + C ( k u k p ∗ ) Z t ( t − s ) − α k u ( s ) k H α − ,p d s. Due to the singular Gronwall lemma, see, e.g. , [20, Ch. 5, Lemma 6.7], we deduce that the norm k u ( t ) k H α − ,p cannot explode in finite time. This shows that local-in-time solutions of the regularized equation (5.1) can bealso continued to global-in-time ones. (cid:3) Hypercontractivity and compactness estimates
Hypercontractivity estimates.
We now turn to prove certain L L p estimates for solutions of problem(5.1). Theorem 6.1 ( L p -decay of solutions to the regularized problem) . Let u = u ( t, x ) be a solution to the regularizedproblem (5.1) constructed in Theorem 5.1. There exists a constant C = C ( d, α, m ) > such that for all ε > , δ > and p ∈ [1 , ∞ ] , (6.1) k u ( t ) k p ≤ C k u k d ( m − /p + αd ( m − α t − dd ( m − α (cid:0) − p (cid:1) for all t > .Proof. We first remark that it is enough to prove the decay estimate (6.1) for large p ’s, since the general resultfollows by the interpolation of the L p -norms combined with the estimate k u ( t ) k ≤ k u k from Corollary 5.12.This is the reason why we will now prove (6.1) for p ≥ max { m − , , ¯ p α } = p m (see Corollary 5.7 for a definitionof ¯ p α ).We also remark that we can assume that M = k u k = 1 by rescaling the solution u in the followingway. First, we consider the function e u ( t ) = M u (cid:0) tM m − (cid:1) which satisfies equation (5.1) with suitably rescaledparameters: ˜ δ = δM m − and (if applicable) ˜ ε = εM . Scaling back, we recover the desired inequality (6.1). ONLOCAL POROUS MEDIUM EQUATION 19
We first prove the result when Corollary 5.12 holds true with G ( r ) = | r | m − r and in the differentialform (5.37). This is the case when m = 2 or m ≥
3. In the case m ∈ (1 , G . We will see below how to pass to the limit asthe regularization parameter ε goes to 0 and get (5.36) with G ( r ) = | r | m − r . For expository reasons, we preferto present the proof when we indeed have a differential inequality, and then to explain how to adapt it if onlyan integral version of it is available.The proof on Theorem 6.1 in the cases m ≥ m = 2 is split into two steps: first, we show inequalities(6.1) with non-optimal constants C which blow up for p = ∞ ; then, we improve those constants by an iterationmethod. Decay estimates with optimal exponents and nonoptimal constants.
Our computation consists in getting thefollowing differential inequality for p ∈ ( p m , ∞ ). Lemma 6.2.
There exists a constant K = K ( p, m ) > independent of ε > and δ > such that K and K − are bounded as p → ∞ and (6.2) dd t k u k pp ≤ − K k u k ap , with a defined in (3.9) .Proof. We get from Corollary 5.12dd t Z | u | p d x ≤ − ( p − Z u | u | p − ( − ∆) α ( u | u | m − ) d x (6.3) ≤ − p ( p − m − p + m − (cid:13)(cid:13)(cid:13) ∇ α (cid:16) | u | p + m − (cid:17)(cid:13)(cid:13)(cid:13) , after applying the Stroock–Varopoulos inequality (3.4) with w = u | u | m − and q = pm − + 1. We use next theGagliardo–Nirenberg inequality from Lemma 3.2 combined with k u ( t ) k ≤ k u k = 1 to estimate the right-handside of the above inequality. Thus, we get the differential inequality (6.2) with the constant(6.4) K = 4( m − p ( p − C N ( p + m − . (cid:3) With the differential inequality (6.2) in hand, a direct computation shows that every nonnegative solution ofthe inequality dd t f ( t ) ≤ − Kf ( t ) ap has to satisfy the algebraic decay f ( t ) ≤ (cid:18) K (cid:18) ap − (cid:19) t (cid:19) − ap − . We recognize (6.1) for p > p m with the constant C p = (cid:16) K (cid:16) ap − (cid:17)(cid:17) − a − p . Here, let us notice that the constants C p = C ( d, α, m, p ) which are obtained at this stage of the proof blow up as p → ∞ . Thus, we cannot get the L ∞ -bound directly in this way. Recurrence step.
To improve the constant C p and to handle the limit case p = ∞ , we apply a variation on theMoser–Alikakos method of estimating the L p -norms with p = 2 n recursively, see, e.g. , [1] and [16, Lemma 3.1].The starting point is the already obtained estimate for p = 2 k ≥ m − k . Lemma 6.3.
For each n ≥ k , the following estimate (6.5) k u ( t ) k n ≤ κ n t − µ n for all t > , holds true with µ n = − − nαd + m − and with a positive κ n satisfying the recursive estimate (6.6) κ n +1 ≤ (cid:20) n (cid:0) αd + m − n (cid:1) µ n + 1 K n (cid:0) αd + m − n (cid:1) κ n (cid:0) αd + m − n (cid:1) n (cid:21) − n − αd + m − n , where K n is given by formula (6.4) with p = 2 n +1 . Thus, having this estimate we see that lim sup n →∞ κ n < ∞ (irrespective of the value of κ k at the beginningof the recurrence), essentially since P ∞ n = k n − n < ∞ , see Appendix B for details. Recall that the constants K = K n in the preliminary estimates have been such that K n and K − n were bounded uniformly when p =2 n → ∞ . Proof of Lemma 6.3.
We combine (6.3) with the Nash inequality (3.10) and two H¨older inequalities with 2 n +1 Interpolation between p = 2 n and p = 2 n +1 and the passage to the limit p → ∞ finish the proof of thehypercontractivity estimates in the cases m ≥ m = 2.To deal with the case m ∈ (1 , g ( t ) = t and g ( t ) = Ct ν for some well chosen positive constants ν, C , successively. ONLOCAL POROUS MEDIUM EQUATION 21 Lemma 6.4. Consider functions f : [0 , T ] → (0 , ∞ ) nonincreasing, and g : [0 , T ] → (0 , ∞ ) increasing, smooth,and g (0) = 0 . Assume that for a.e. t ∈ (0 , T ) , s < t , f ( t ) + K Z ts f ( τ ) γ +1 g ′ ( τ ) d τ ≤ f ( s ) . Then for a.e. t ∈ [0 , T ] , we have f ( t ) ≤ ( Kγg ( t )) − γ . The proof of this lemma is given in Appendix C for the readers’ convenience. Now, the proof of Theorem 6.1is complete. (cid:3) Compactness estimates. Now, we prove estimates which will allow us to pass to the limit as ε → δ → 0, successively, in the regularized problem (5.1). We are going to use the fact that the approximatingfunctions G = G ε are γ -H¨older continuous with γ = min { m − , } on the interval (cid:2) , k u k ∞ (cid:3) , uniformly in ε ∈ [0 , Lemma 6.5. Assume that G is γ -H¨older continuous with γ = min { m − , } . For every α ∈ (0 , and m > − αd ∈ (1 , , we have (6.8) k∇ α − G ( u ) k q ≤ C k u k γγp with q ≥ ( m − − (1 − α ) /d ) − and p = (1 /q + (1 − α ) /d ) − . Proof. Here, it suffices to combine the estimate on the Riesz potential with k G ( u ) k p ≤ C k u k γγp . The choice of q (or equivalently, the restriction on m ) ensures that γp ≥ (cid:3) The compactness of a sequence of solutions to the regularized problem (5.1) in the case α ∈ (1 , 2) is aconsequence of the following estimate. Lemma 6.6. Consider a γ -H¨older continuous function G : R → R . Then, for every α ∈ (1 , such that (2 − γ ) α < , we have k∇ α − G ( u ) k p ≤ C k u k H α , with α − γ − dp = α − d . Remark . Note that the assumptions of Lemma 6.6 ensure that p > Proof of Lemma 6.6. We use successively the fact that H s,p ( R d ) = F sp, ( R d ) [22, p.14], the characterization ofTriebel-Lizorkin spaces with difference quotients [22, p.41], and finally known embedding theorems for Besovand Triebel-Lizorkin spaces [22, p.31] in order to derive k∇ α − G ( u ) k p ≤ C k G ( u ) k F α − p, ≤ C [ G ] γ k u k F α − γp, γ ≤ C k u k H α , , where [ G ] γ := sup r = s | G ( r ) − G ( s ) | / | r − s | γ and p > (cid:3) Lemma 6.8. For α ∈ (1 , and (3 − m ) α < , there exists p ∈ (1 , ∞ ) such that (6.9) k∇ α − G ( u ) k p ≤ C k u k H , ( R d ) where C depends on the γ -H¨older seminorm of G in (0 , k u k ∞ ) ( with γ = min { m − , } ) .Proof. Apply Lemma 6.6 with γ = m − (cid:3) Lemma 6.9. If α ∈ (1 , and m > α , there exists p ∈ (1 , ∞ ) and r ∈ (1 , ∞ ) ( and r > m − such that (6.10) k∇ α − | u | m − sgn u k p ≤ C k| u | r + m − sgn u k H α , ( R d ) . Proof. Consider v = | u | r + m − sgn u . Then estimate (6.10) is equivalent to the following one k∇ α − | v | m − r + m − sgn v k p ≤ C k v k H α , ( R d ) . We then apply Lemma 6.6 with γ = m − r + m − and get the desired result if γ > − α ), or equivalently, r < m − α − . Here, in order to find r ∈ (cid:0) , m − α − (cid:1) , the condition m > α is needed. The proof is now complete. (cid:3) Proof of Theorem 2.6 This section is devoted to the proof of our main result on the existence of solutions to problem (1.1)–(1.2)satisfying decay estimates (2.6). Proof of Theorem 2.6. We first consider very regular initial data, i.e. we assume that u satisfies (5.3). Then,this condition is relaxed by considering initial data that are merely integrable.The proof proceeds in three steps: passage to the limit with the parameter of the regularization of thenonlinearity, then with the parameter of the parabolic regularization, and finally — stability with respect toinitial data. Passage to the limit as ε → . Consider u satisfying (5.3). From Theorem 5.1, we have a sequence ofsolutions u ε of (5.1) for G ε defined in such a way that there exists C > ε ∈ (0 , G ε ] γ := sup r = s | G ε ( r ) − G ε ( s ) || r − s | γ ≤ C where γ = min { m − , } . Thanks to Corollary 5.12, there exists also a constant C > δ > ε ∈ (0 , k u ε k L ∞ ((0 ,T ) × R d ) ≤ C, k u ε k L ((0 ,T ) ,H , ( R d )) ≤ C. Hence, we can construct a sequence ( ε n ) n such that u n ⇀ u in L ((0 , T ) , H , ( R d )) ONLOCAL POROUS MEDIUM EQUATION 23 where u n denotes u ε n . Moreover, for all R > 0, the embedding H , ( B R ) ⊂ L ( B R ) is dense and compact.Using (7.1) we also obtain lim meas( E ) → ,E ⊂ [0 ,T ] Z E Z B R | u n ( t, x ) | d t d x = 0 . Hence, we infer from [21] (which contains an optimal result on the compactness for Hilbert space valued vectorfunctions) that, up to a subsequence, for every R > u n → u in L ((0 , T ) × B R ) . Moreover, passing again to a subsequence if necessary, we can assume that for every R > ∇ u n ⇀ ∇ u in L ((0 , T ) × B R ) , u n → u for a.e. ( t, x ) ∈ Q T , which imply for all ϕ ∈ C ∞ c ( ¯ Q T ), Z Z Q T u n ∂ t ϕ d t d x → Z Z Q T u∂ t ϕ d t d x, Z Z Q T ∇ u n · ∇ ϕ d t d x → Z Z Q T ∇ u · ∇ ϕ d t d x. As far as the nonlinear term in equation (5.1) is concerned, we have | u n | → | u | for a.e. ( t, x ) ∈ Q T . Hence, the dominated convergence theorem combined with (7.1) imply that | u n |∇ ϕ → | u |∇ ϕ in L ((0 , T ) , L q ′ ( R d ))for all q ′ ∈ (1 , ∞ ). Lemmas 6.5 and 6.8 imply that(7.3) ∇ α − G ε n ( u n ) is bounded in L q ( Q T )for some q ∈ (1 , ∞ ).We deduce from (7.3) that we can extract a weakly converging subsequence of ( ∇ α − G ε n ( u n )) n ⊂ L q ( Q T )which limit is denoted by L . Since G ε n ( u n ) → | u | m − sgn u a.e. in Q T , we have ∇ α − G ε n ( u n ) ⇀ L in L q ( Q T ) , where L = ∇ α − ( | u | m − sgn u ) in D ′ ( Q T ) . In particular, ∇ α − ( | u | m − sgn u ) ∈ L q ( Q T ) holds, and Z Z Q T ∇ α − G ε n ( u n ) · | u n |∇ ϕ d t d x → Z Z Q T ∇ α − ( | u | m − sgn u ) · | u |∇ ϕ d t d x as n → ∞ . Hence, we obtain a weak solution of (5.1) with G ( r ) = | r | m − sgn r for u satisfying (5.3). By the Fatou lemma, we derive from Corollary 5.12 and Proposition 3.1 the following estimates: for a.e. t > s > q ∈ [1 , ∞ ], p ∈ (1 , ∞ ), uniformly in δ ∈ (0 , k u ( t ) k q ≤ k u k q , (7.4) k u ( t ) k pp + 4 p ( p − m − p + m − Z Z ( s,t ) × R d (cid:12)(cid:12)(cid:12) ∇ α (cid:16) sgn u | u | p + m − (cid:17)(cid:12)(cid:12)(cid:12) d s d x ≤ k u ( s ) k pp , (7.5) k u ( t ) k q ≤ C k u k d ( m − /q + αd ( m − α t − dd ( m − α (cid:0) − q (cid:1) , (7.6) 2 δ Z Z Q t |∇ u | d s d x ≤ k u k . (7.7)Remark that now we can derive the hypercontractivity estimates in the case m ∈ (1 , 3) from inequality (7.5).We also know that the solution u is nonnegative if u is so. Passage to the limit as δ → . Let u = u δ denote the solution constructed above. We consider v δ = | u | m sgn u. Using (7.5) with s = 0 and p = m + 1 > 1, we get that, for all T > v δ is bounded in L ((0 , T ) , H α , ( R d )) . Hence, there exists a subsequence δ n → v n ⇀ v in L ((0 , T ) , H α , ( R d )) , where v n denotes v δ n . Moreover, inequality (7.4) with q = ∞ implies that for R > meas( E ) → ,E ⊂ [0 ,T ] Z E Z B R | v n ( t, x ) | d t d x = 0 . Hence, we can use [21] once more, and conclude that for all R > v n → v in L ((0 , T ) × B R ) . Moreover, passing to a subsequence if necessary, we can assume that u n → u for a.e. ( t, x ) ∈ Q T , where u n denotes u δ n . Now Lemmas 6.5 and 6.9 imply that(7.8) ∇ α − ( | u n | m − sgn u n ) is bounded in L q ( Q T )for some q > 1. Hence, we can pass to the limit in the nonlinear term in the weak formulation of equation (5.1).To complete this proof, notice that inequality (7.7) implies that δ n ∇ u n → L ( Q T ) . In particular, δ n Z Z Q T ∇ u n · ∇ ϕ d t d x → n → ∞ . We thus conclude that u is a weak solution of (1.1)–(1.2). ONLOCAL POROUS MEDIUM EQUATION 25 The conservation of mass, the positivity property, the monotonicity of L p -norms, and the hypercontractivityestimates follow from (7.4)–(7.7) in a standard way. Stability with respect initial conditions. Assume now that u ∈ L ( R d ) ∩ L ∞ ( R d ). Consider an approx-imating sequence u n satisfying (5.3). Then, the sequences of L p -norms are bounded and we can pass to thelimit as we did already when letting δ → 0. We also recover the expected properties of the weak solution.Assume finally that u is merely integrable. Then T n ( u ) ∈ L ( R d ) ∩ L ∞ ( R d ) where T n ( r ) = min { max { r, − n } , n } . Hence, there exists a weak solution u n of (1.1). In view of the hypercontractivity estimates, we can extracta converging subsequence in ( η, T ) × R d , with arbitrary η > 0, to a function u η in the following sense u n → u η in L (( η, T ) × B (0 , R )) . Moreover, the L -norm of u n is bounded in (0 , T ) × R d . Then we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Z (0 ,η ) × R d u n ∂ t ϕ d t d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cη, Z Z ( η,T ) × R d u n ∂ t ϕ → Z Z ( η,T ) × R d u η ∂ t ϕ Z Z ( η,T ) × R d ∇ α − ( | u n | m − sgn u n ) · ∇ ϕ d t d x → Z Z ( η,T ) × R d ∇ α − ( | u η | m − sgn u η ) · ∇ ϕ d t d x as n → ∞ for each η > 0. We can now conclude the proof of Theorem 2.6 through a diagonal procedure. (cid:3) Appendix A. Proof of Proposition 5.9 Proof of Proposition 5.9. We fix an arbitrary p ≥ p α = dα − . First, we remark that Z Z ( ∂ t u ) φ d t d x + Z | u |∇ α − G ( u ) · ∇ φ d t d x + δ Z Z ∇ u · ∇ φ d t d x = 0for all φ compactly supported in time and in L ((0 , T ) , H ,p ′ ( R d )). Recall that we have ∂ t u ∈ C [0 , T ] , L p ( R d ))for p ≥ p α = dα − and Ψ( u ) = | u |∇ α − G ( u ) ∈ L ∞ ((0 , T ) , L p ( R d )). To justify the previous equality, it is enoughto mollify the function φ in time and space, and remark that the mollified function φ η satisfies (for η smallenough) φ η (0 , x ) = φ η ( T, x ) = 0, uφ η ∈ C ((0 , T ) , L ( R d )) and Z Z u ( ∂ t φ η ) d t d x = − Z Z ( ∂ t u ) φ η d t d x. Letting η → u ) yields the desired result.Next, consider φ ( τ, x ) = ϕ ′ ( u ( τ, x ))Θ η ( τ ), where Θ η is truncation function in time of [ s, t ]. Remark that φ ∈ L ((0 , T ) , H ,p ′ ( R d )); indeed, ϕ ′ ( u ) ∈ L ∞ ((0 , T ) , L p ′ ( R d )), and we also have for all w ∈ ( H ,p ∩ L ∩ L ∞ )( R d ) k ϕ ′′ ( w ) ∇ w k p ′ ≤ k∇ w k p k ϕ ′′ ( w ) k q with q = 1 − p . As far as Θ η is concerned, we choose it such that Θ η ( s ) = 0 and Θ ′ η ( τ ) = ρ η ( τ − s − η ) − ρ η ( τ − t + η )where ρ η is an even mollifier supported in [ − η, η ]. Now we can write Z Z ( ∂ t u ) ϕ ′ ( u )Θ η d τ d x = Z Z ∂ t ( ϕ ( u ))Θ η d τ d x = Z Z ϕ ( u )( ρ η ( τ − t + η ) − ρ η ( τ − s − η )) d τ d x. Hence, Z Z ( ∂ t u ) ϕ ′ ( u )Θ η d τ d x → Z ϕ ( u ( t, x )) d x − Z ϕ ( u ( s, x )) d x. Moreover, | u |∇ φ ( u ) = | u | ϕ ′′ ( u ) ∇ u = ∇ ( ψ ( u )) in L p ( R d )thanks to the Stampacchia theorem ( u ∈ L ∞ ( R d ) hence ϕ ′ and ψ locally Lipschitz is enough). Hence Z ts Z | u |∇ α − G ( u ) · ∇ φ ( u ) d τ d x = Z ts Z ∇ α − G ( u ) · ∇ ( ψ ( u )) d x d τ. It remains to prove that(A.1) Z ts Z ψ ( u )( − ∆) α ( G ( u )) d τ d x ≤ Z ts Z ∇ α − G ( u ) · ∇ ( ψ ( u )) d x d τ < ∞ . The last inequality comes from the computations we made above. As far as the second inequality is concerned,we use (1.3) to write Z ts Z ∇ α − G ( u ) · ∇ ( ψ ( u )) d x d τ = lim η → Z ts Z Z ( G ( u )( τ, x ) − G ( u )( τ, y )) F η ( y − x ) · ∇ ( ψ ( u ( τ, x ))) d x d y d τ where F η is defined as follows F η ( z ) = C d,α zη d + α + | z | d + α . Through an integration by parts, we now get Z ts Z Z ( G ( u )( τ, x ) − G ( u )( τ, y )) F η ( y − x ) · ∇ ( ψ ( u ( τ, x ))) d x d y d τ = − Z ts Z Z ∇ G ( u )( τ, x ) · F η ( y − x ) ψ ( u ( τ, x )) d x d y d τ + Z ts Z Z ( u ( τ, x ) − u ( τ, y ))( ∇ · F η )( y − x ) ψ ( u ( τ, x )) d x d y d τ. The first term on the right-hand side equals 0 since F η is odd. Moreover, we have ∇ · F η ( z ) = C d,α η d + α + ( d + α + 1) | z | d + α ( η d + α + | z | d + α ) . Hence, the Fatou lemma yields (A.1). The proof of the proposition is now complete. (cid:3) Appendix B. From (6.6) to the boundedness of κ n Consider l n = log κ n and write (6.6) as follows l n +1 ≤ a n + b n l n ONLOCAL POROUS MEDIUM EQUATION 27 with a n = 12 n +1 (cid:0) αd + m − n (cid:1) log n (cid:0) αd + m − n (cid:1) + 1 K n (cid:0) αd + m − n (cid:1) ! ; b n = 1 − m − αd n +1 + 2( m − . Remark next that a n ≤ C n n and b n ≤ − C n . In particular X n ≥ k a n < ∞ and Y n ≥ k b n < ∞ Using the fact that b n ≤ 1, we get l n ≤ X n ≥ k a n + Y n ≥ k b n a k . Hence, l n does not blow up, and neither does κ n . Appendix C. Proof of Lemma 6.4 Proof of Lemma 6.4. First, we remark that we can reduce to the case g ( t ) = t and K = 1 through a change ofvariables.The proof is simple if f is smooth. If f is not, extend f by 0 to R and consider a mollifier ρ ε . Then writefor t < t , f ( t − s ) + K Z t − s −∞ f γ +1 ( τ ) d τ ≤ f ( t − s ) . Now integrate against ρ ε ( s ) and use the Jensen inequality to get f ε ( t ) + K Z t −∞ f γ +1 ε ( τ ) d τ ≤ f ε ( t ) . We are now reduced to the case f = f ε , which is smooth. Passing to the limit, the proof is now complete. (cid:3) References [1] N. D. Alikakos , An application of the invariance principle to reaction-diffusion equations , J. Differential Equations, 33 (1979),pp. 201–225.[2] G. Barles, E. Chasseigne, and C. Imbert , On the Dirichlet problem for second-order elliptic integro-differential equations ,Indiana Univ. Math. J., 57 (2008), pp. 213–246.[3] P. Biler, C. Imbert, and G. Karch , Barenblatt profiles for a nonlocal porous medium equation. , C. R., Math., Acad. Sci.Paris, 349 (2011), pp. 641–645.[4] P. Biler, G. Karch, and R. Monneau , Nonlinear diffusion of dislocation density and self-similar solutions , Comm. Math.Phys., 294 (2010), pp. 145–168.[5] L. Caffarelli and J. L. V´azquez , Nonlinear porous medium flow with fractional potential pressure , Arch. Ration. Mech.Anal., 202 (2011), pp. 537–565.[6] L. A. Caffarelli, F. Soria, and J. L. V´azquez , Regularity of solutions of the fractional porous medium flow . PreprintarXiv:1201.6048, 2012.[7] L. A. Caffarelli and J. L. V´azquez , Asymptotic behaviour of a porous medium equation with fractional diffusion , DiscreteContin. Dyn. Syst., 29 (2011), pp. 1393–1404.[8] J. A. Carrillo, A. J¨ungel, P. A. Markowich, G. Toscani, and A. Unterreiter , Entropy dissipation methods for degen-erate parabolic problems and generalized Sobolev inequalities , Monatsh. Math., 133 (2001), pp. 1–82.[9] A. C´ordoba and D. C´ordoba , A maximum principle applied to quasi-geostrophic equations , Comm. Math. Phys., 249 (2004),pp. 511–528.[10] A. de Pablo, F. Quir´os, A. Rodr´ıguez, and J. L. V´azquez , A fractional porous medium equation , Adv. Math., 226 (2011),pp. 1378–1409.[11] , A general fractional porous medium equation , Comm. Pure Applied Mathematics, 65 (2012), pp. 1242–1284. [12] J. Droniou and C. Imbert , Fractal first order partial differential equations , Arch. Ration. Mech. Anal., 182 (2006), pp. 299–331.[13] B. Dyda , Fractional calculus for power functions and eigenvalues of the fractional laplacian , Fractional Calculus and AppliedAnalysis, 15 (2012), pp. 535–555.[14] R. K. Getoor , First passage times for symmetric stable processes in space , Trans. Amer. Math. Soc., 101 (1961), pp. 75–90.[15] C. Imbert and A. Mellet , Existence of solutions for a higher order non-local equation appearing in crack dynamics , Non-linearity, 24 (2011), pp. 3487–3514.[16] G. Karch, C. Miao, and X. Xu , On convergence of solutions of fractal Burgers equation toward rarefaction waves , SIAM J.Math. Anal., 39 (2008), pp. 1536–1549.[17] O. Ladyzhenskaya, V. Solonnikov, and N. Ural’tseva , Linear and quasi-linear equations of parabolic type. Translatedfrom the Russian by S. Smith. , Translations of Mathematical Monographs. 23. Providence, RI: American Mathematical Society(AMS). XI, 648 p. , 1968.[18] V. A. Liskevich and Y. A. Semenov , Some problems on Markov semigroups , in Schr¨odinger operators, Markov semigroups,wavelet analysis, operator algebras, vol. 11 of Math. Top., Akademie Verlag, Berlin, 1996, pp. 163–217.[19] W. Magnus, F. Oberhettinger, and R. P. Soni , Formulas and theorems for the special functions of mathematical physics ,Third enlarged edition. Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., NewYork, 1966.[20] A. Pazy , Semigroups of linear operators and applications to partial differential equations , vol. 44 of Applied MathematicalSciences, Springer-Verlag, New York, 1983.[21] J. M. Rakotoson and R. Temam , An optimal compactness theorem and application to elliptic-parabolic systems , Appl. Math.Lett., 14 (2001), pp. 303–306.[22] T. Runst and W. Sickel , Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations ,vol. 3 of de Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter & Co., Berlin, 1996.[23] E. M. Stein , Singular integrals and differentiability properties of functions , Princeton Mathematical Series, No. 30, PrincetonUniversity Press, Princeton, N.J., 1970.[24] E. M. Stein and G. Weiss , Introduction to Fourier analysis on Euclidean spaces , Princeton University Press, Princeton,N.J., 1971. Princeton Mathematical Series, No. 32.[25] M. E. Taylor , Tools for PDE , vol. 81 of Mathematical Surveys and Monographs, American Mathematical Society, Providence,RI, 2000. Pseudodifferential operators, paradifferential operators, and layer potentials.[26] , Partial differential equations. III: Nonlinear equations. 2nd ed. , Applied Mathematical Sciences 117. New York, NY:Springer. xxii, 715 p. , 2011.[27] J. L. V´azquez , Smoothing and decay estimates for nonlinear diffusion equations , vol. 33 of Oxford Lecture Series in Mathe-matics and its Applications, Oxford University Press, Oxford, 2006. Equations of porous medium type.[28] , The porous medium equation , Oxford Mathematical Monographs, The Clarendon Press Oxford University Press,Oxford, 2007. Mathematical theory.[29] J. L. V´azquez , Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type .Preprint arXiv:1205.6332v1, 2012.[30] G. N. Watson , A treatise on the theory of Bessel functions , Cambridge Mathematical Library, Cambridge University Press,Cambridge, 1995. Reprint of the second (1944) edition. P. Biler: Instytut Matematyczny, Uniwersytet Wroc lawski, pl. Grunwaldzki 2/4, 50-384 Wroc law, Poland E-mail address : [email protected] C. Imbert: CNRS, UMR 8050, Universit´e Paris-Est Cr´eteil, 61 av. du G´en´eral de Gaulle, 94010 Cr´eteil, cedex,France E-mail address : [email protected] URL : G. Karch: Instytut Matematyczny, Uniwersytet Wroc lawski, pl. Grunwaldzki 2/4, 50-384 Wroc law, Poland E-mail address : [email protected] URL ::