Intermediate asymptotics beyond homogeneity and self-similarity: long time behavior for [formula]

INTERMEDIATE ASYMPTOTICS BEYOND HOMOGENEITY AND SELF-SIMILARITY: LONG TIME BEHAVIOR FOR ut= ∆φ(u) JOS´ E A. CARRILLO, MARCO DI FRANCESCO AND GIUSEPPE TOSCANI Abstract. We investigate the long time asymptotics in L1 +(R) for solutions of general nonlinear diffusion equations ut= ∆φ(u). We describe, for the first time, the intermediate asymptotics for a very large class of non-homogeneous nonlinearities φfor which long time asymptotics cannot be characterized by self-similar solutions. Scaling the solutions by their own second moment (temperature in the kinetic theory language) we obtain a universal asymptotic profile characterized by fixed points of certain maps in probability measures spaces endowed with the euclidean Wasserstein distance d2. In the particular case of φ(u)∼umat first order when u∼0, we also obtain an optimal rate of convergence in L1towards the asymptotic profile identified, in this case, as the Barenblatt self–similar solution corresponding to the exponent m. This second result holds for a larger class of nonlinearities compared to results in the existing literature and is achieved by a variation of the entropy dissipation method in which the nonlinear filtration equation is considered as a perturbation of the porous medium equation. 1. Introduction We consider the Cauchy problem for a general nonlinear diffusion equation, namely    ∂u ∂t = ∆φ(u) u(·,0) = u0. (1) Here, (x, t)∈RN×R+, while the initial datum u0is taken to be nonnegative and belonging in L1(RN). We shall assume throughout this paper that the nonlinearity function φsatisfies the following assumptions: (NL1) φ∈C[0,+∞)∩C1(0,+∞), φ(0) = 0 and φ0(u)>0 for all u > 0. (NL2) ∃C > 0 and m > N−2 Nsuch that φ0(u)≥Cum−1for all u > 0. 1 2 JOS´ E A. CARRILLO, MARCO DI FRANCESCO AND GIUSEPPE TOSCANI (NL3) φ(u) u1−1/N is nondecreasing on u∈(0,∞). Under assumption (NL1), it is well–known that the Cauchy problem (1) is well–posed for any initial datum in L1 +(RN). We refer to [B´en76, V´er79] for the existence and regularity theory for equation (1) with initial data in L1, obtained by means of the Crandall–Ligget formula for nonlinear semigroups. Well-posedness and properties of the solutions were further developed in [BC79, BC81, V´er79]. Solutions are mass-preserving, i.e., the following conservation law holds ZRN u(x, t)dx =ZRN u0(x)dx. Moreover, the unique generalized solution to (1) may not be classical in general. In case of slow diffusion, i.e. φ0(0) = 0, the support of the solution travels with finite speed. This is due to the degeneracy of the parabolic operator as uapproaches zero (see [Kal87, Kne77] and the references therein for the general diffusion equation (1); see also the survey paper by Vazquez [V´az03] and the references therein for the porous medium equation φ(u) = umparticularly [V´az83]). Concerning the evolution of the integral norms, given u0∈Lp(RN), the solution u(·, t) of (1) belongs to Lp(RN) at any time t > 0, and we have ku(·, t)kLp(RN)≤ ku0kLp(RN).(2) Under the extra assumption on the nonlinearity φ(NL2), the equation (1) enjoys an L1–L∞regularizing property. Indeed, it was proved that the solution to (1) with initial datum in L1satisfies the following temporal decay estimate (see [AB79] for the power law case, [V´er79] for the general nonlinear case). Theorem 1.1 (L1–L∞regularizing effect).Let u(x, t)be the solution to the Cauchy problem (1), with u0∈L1 +(RN). Let the nonlinearity φsatisfy assumptions (NL1)-(NL2) above. Then, at any t > 0,u(x, t)∈L∞(RN) and the following estimate hold ku(·, t)kL∞(RN)≤C t−N N(m−1)+2 ku0kL1(RN).(3) Assumption (NL3) implies that the entropy functional associated to equation (1) is displacement convex [McC97] and thus, the flow map of the nonlinear diffusion (1) is a non-expansive contraction in time with respect to the euclidean Wasserstein distance d2in probability measures [Ott01, Agu02, CMV04]. The first result in this work proves the existence of a universal asymptotic profile for general nonlinearities satisfiying assumptions (NL1)-(NL3) LONG TIME BEHAVIOR FOR ut= ∆φ(u) 3 above. Let us define the “temperature” of a solution uof (1) as its second moment, i.e., θu(t) = ZRN |x|2 2u(x, t)dx. This definition is natural from a kinetic theory point of view, see for instance [CIP94], and is nothing else than the variance of the density u(x, t) from a probabilistic point of view. Time-dependent rescalings have been used to translate the Cauchy problem for homogeneous nonlinearities φ(u) = umonto nonlinear Fokker- Planck type equations [CT00, Ott01, DdP02]. These changes of variables translate particular self-similar Barenblatt-Pattle solutions onto stationary states for the corresponding nonlinear Fokker-Planck equations. In fact, it is quite easy to check that the change of variables is defined in terms of the explicit value of the temperature of the expected asymptotic profile, i.e., the Barenblatt-Pattle self-similar solution. This fact has been noted recently in [Tos04]. We propose to study the behavior of the solution when we perform a nonlinear time-dependent rescaling depending on the own temperature of the solution. More precisely, given a solution of (1) we will study the long time behavior of θu(t)N/2u(θu(t)1/2x, t).(4) Similar scalings have been used in the analysis of homogeneous cooling states in granular media equations (see for instance [BCP97, Tos03] and the references therein). The nonlinear time dependent scaling (4) can be also seen as the projection of the solution u(·, t) onto the manifold of probability measures with unit temperature (see also [CG03]) M=½µ∈ P2(RN),1 2ZRN |x|2dµ(x) = 1¾. In order to have the above scaling (4) well defined for all positive times t, in addition we must require the following natural assumption on the nonlinearity φ, namely (FT) For any solution u(x, t) to (1), the following relation holds, Z|x|2u(x, 0)dx < ∞,⇒Z|x|2u(x, t)dx < ∞for all t > 0 We postpone to section 3 the discussion of providing reasonable sufficient conditions on φsuch that condition (FT) is satisfied. Let us briefly point out that for the slow diffusion case is clearly satisfied while for the fast diffusion case the control of moments is delicate even in the power-law case [CV03], and thus, this case has to be analyzed carefully. We focus our next section on proving the main result of this work: 4 JOS´ E A. CARRILLO, MARCO DI FRANCESCO AND GIUSEPPE TOSCANI Theorem 1.2 (Asymptotic profile for general nonlinear diffusions). Given φverifying the hypotheses (NL1)–(NL3)–(FT), there exists t∗>0 and a one parameter curve of probability densities v∞ t, with unit temperature defined for t≥t∗such that, for any solution of (1) with initial data (1 + |x|2)u0∈L1 +(RN)of unit mass and temperature, d2³θu(t)N/2u(θu(t)1/2·, t), v∞ t´−→ 0as t→ ∞. Moreover, the asymptotic profile v∞ tis characterized as the unique fixed point of the renormalized flow map S(t) S(t)u0:= θu(t)N/2u(θu(t)1/2·, t), where u(·, t)is the solution to (1) with initial datum u0. The main ingredient of previous theorem is the proof of a contraction property for the maps S(t), t≥t∗>0, obtained as compositions of the flow map for the nonlinear diffusion equation ut= ∆φ(u) and the projection of the solution onto the unit second moment manifold M. The asymptotic profile v∞ tis nothing else that the Barenblatt-Prattle solution at the time in which it has unit temperature in the case of the homogeneous nonlinearities φ(u) = um[Tos04]. Indeed, thanks to our point of view we can generalize the classical notions of self–similarity and source–type solution by means of the idea of invariance of the solution orbit after projection onto the subset M. The contraction property makes use of the L1−L∞regularizing effect to control the behavior of the temperature of the solution. Let us mention that the existence of ‘caloric’ self–similar profiles for general diffusion equations has been the subject of [vDP77] in a context of solutions attaining different end–states at infinity. Moreover, the asymptotic behavior of nonlinear diffusion equations has been studied in [BP85] in a bounded domain context. Even if far from our purposes, we also mention [BV04] which contains very recent development in general nonlinear diffusion equations applied to image contour enhancement. As far as we are concerned, our theorem is the first result concerning L1asymptotic profiles for such general nonlinearities. We can obtain more detailed information of the asymptotics in specific cases. Let us consider the particular case of asymptotically homogeneous nonlinearities, i.e., let us assume that the nonlinearity φsatisfies the additional assumptions (NL4) φ(u) = umψ(u) for some m > max(N−1 N,N N+2 ), with ψsatisfying: (P1) ∃limu→0+ψ(u) = l∈(0,+∞) (for simplicity we assume l= 1). (P2) ψ∈C[0,+∞)∩C1(0,+∞). (P3) ψ0(u) = O(uk) as u→0+, for some k > −1. LONG TIME BEHAVIOR FOR ut= ∆φ(u) 5 (NL5) h(·)∈L1 loc([0,+∞)). where his given by h(u) = Zu 1 φ0(η) ηdη. (5) In this case, we can use a variant of the entropy method for nonlinear homogeneous equations [CT00, Ott01, DdP02] based on the L1−L∞regularizing effect and on the generalized Log-Sobolev inequalities [CJM+01]. More precisely, we show in section 4: Theorem 1.3 (Asymptotically homogeneous nonlinearities). Let u(x, t)be the solution to the Cauchy problem (1), with nonlinearity φsatisfying hypotheses (NL1)-(NL5)-(FT) above and with initial datum u0≥0such that ZRN£um 0(x) + |x|2u0(x) + F(u0(x))¤dx < +∞ with F(u) = uh(u)−φ(u). Let u∞(x, t)be the Barenblatt self–similar function such that u∞has the same mass as u0. Then, the following estimate holds for all t≥0 ku(t)−u∞(t)kL1(RN)≤C(t+ 1)−λδ α, where C,λ,δand αare explicitly given in section 4. Let us remark that this particular case was first treated in [Kam75, Kam76] obtaining L∞decay and L1convergence without rate to the corresponding Barenblatt-Pattle self-similar solution. Recently, another variant of the entropy method was proposed in [BDE02] but the range of nonlinearities in there has some stricter hypotheses that we can avoid by the perturbation argument proposed in section 3 reminiscent of arguments done for convection-diffusion equations in [CF04]. 2. Intermediate asymptotics for general nonlinearities Throughout this section the finite temperature hypothesis (FT) is assumed, further discussion about sufficient conditions ensuring (FT) is postponed to section 3. We start by performing the following nonlinear time dependent scaling u(x, t) = θu(t)−N/2v(y, t), y =θu(t)−1/2x. (6) Let us remark that the above scaling (6) has the advantage of fixing the temperature or variance of the solution at any time t > 0 to be one, namely ZRN |y|2 2v(y, t)dy = 1. 6 JOS´ E A. CARRILLO, MARCO DI FRANCESCO AND GIUSEPPE TOSCANI Let us then analyze the time evolution of the the temperature θu(t) and show that it diverges as t→ ∞ with a precise rate. Lemma 2.1. Assume the nonlinearity φsatisfies (NL1)–(NL2) and (FT), then θu(t)≥1 4t1 N(m−1)+2 .(7) Proof. We have, for an arbitrary R(t), θu(t) = ZRN |x|2 2u(x, t)dx =Z|x|≥R(t) |x|2 2u(x, t)dx +Z|x|≤R(t) |x|2 2u(x, t)dx ≥R(t)2 2Z|x|≥R(t) u(x, t)dx =R(t)2 2"1−Z|x|≤R(t) u(x, t)dx# ≥R(t)2 2£1−CNku(t)kL∞R(t)N¤, where CNis the volume of the unit sphere in RN. Taking into account the smoothing effect in Theorem 1.1, we have ku(t)kL∞≤C0t−N N(m−1)+2 ,(8) where the constant C0depends on the mass of the initial datum, and thus, by choosing R(t) = 1 2C0 t1 N(m−1)+2 , we obtain the desired below estimate (7). ¦ We then recall the definition of euclidean Wasserstein distance between any two probability densities u1and u2, d2(u1, u2)2= inf u2=T]u2ZRN |x−T(x)|2u1(x)dx, where u2=T]u2if and only if ZRN ψ(x)u2(x)dx =ZRN ZRN ψ(T(x))u1(x)dx, for all ψ∈L1(RN, u2dx), with T:RN−→ RNis any measurable map. Previous definition can be extended properly to space of probability measures with bounded second moment P2(RN) (see for instance [Vil03]). The flow map of the nonlinear difussion equation (1) enjoys the following property with respect to the euclidean Wasserstein distance. This property is hidden in the arguments in [Ott01, Agu02] and fully proved in a much more general setting in [CMV04]. LONG TIME BEHAVIOR FOR ut= ∆φ(u) 7 Theorem 2.2. Let u1(x, t)and u2(x, t)be solutions to (1) with the nonlinearity φverifying (NL1)–(NL3)–(FT) and with initial data u1,0and u2,0 of unit mass and finite temperature, then d2(u1(t), u2(t)) ≤d2(u1,0, u2,0),for all t≥0.(9) Given u1(x, t) and u2(x, t) the solutions to (1) with the nonlinearity φ verifying (NL1) and (NL3) and with initial data u1,0and u2,0of unit mass and finite temperature, we set, for all t≥0, θ1(t) = ZRN |x|2 2u1(x, t)dx, θ2(t) = ZRN |x|2 2u2(x, t)dx and θ(t) = min {θ1(t), θ2(t)}. We define ui(t), i= 1,2 at time t≥0 as follows. Suppose for simplicity that, for t≥0, θ(t) = θ1(t). Then, u1(x, t) = u1(x, t)u2(x, t) = µθ2(t) θ1(t)¶d/2 u2õθ2(t) θ1(t)¶1/2 x, t! (of course, a symmetric definition holds if θ(t) = θ2(t)). It is easily seen that u1and u2have the same temperature θ(t) = θ1(t) at any time t≥0. Lemma 2.3. d2(u1(t), u2(t)) ≤d2(u1(t), u2(t)). Proof. Let t≥0 be fixed. Suppose θ1(t) = θ(t). Set θ2(t) θ1(t)= 1 + a, for some a≥0. Throughout this lemma we use the notation u1(x, t) = u1(x, t)−→ u(x, t) u2(x, t)−→ v(x, t) u2(x, t) = (1 + a)−N/2v((1 + a)−1/2x, t)−→ va(x, t). Then, the proof of the lemma comes from the following claim, d2(u, v) = inf a≥0d2(u, va).(10) To prove (10), we have by definition of d2, d2(u, va)2= inf T]u=vaZRN |x−T(x)|2u(x)dx. For a nonnegative a, let us denote the dilation da(x) = (1 + a)1/2x. We claim the following assertion, ©T:RN→RN, T]u=vaª=©da◦T, T :RN→RN, T]u=vª.(11) 8 JOS´ E A. CARRILLO, MARCO DI FRANCESCO AND GIUSEPPE TOSCANI To prove (11), let T]u=v. Then, we claim that Ta(x) = (1 + a)1/2T(x) is such that Ta ]u=va. To see this, let ψ∈L1(RN, vadx), we have (we skip the dependence on time twhich is fixed) ZRN ψ(x)va(x)dx =ZRN ψ(x)(1 + a)−d/2v((1 + a)−1/2x)dx =ZRN ψ(x(1 + a)1/2)v(x)dx =ZRN ψ((1 + a)1/2T(x))u(x)dx. By inverting the roles of vand va, we conclude that the assertion (11) is true. We go back to the proof of (10). From (11) we have d2(u, va)2= inf T]u=vZRN¯¯¯x−(1 + a)1/2T(x)¯¯¯ 2u(x)dx = inf T]u=vfT(a),(12) with fT(a) = ZRN |x|2u(x)dx + (1 + a)ZRN |T(x)|2u(x)dx −2(1 + a)1/2ZRN x·T(x)u(x)dx. Computing directly the first derivative of fT(a) w.r.t. ayields f0 T(a) = ZRN |T(x)|2u(x)−(1 + a)−1/2ZRN x·T(x)u(x)dx ≥ZRN |T(x)|2u(x)dx −(1 + a)−1/2[2θ(t)]1/2·ZRN |T(x)|2u(x)dx¸1/2 and since uand vhave the same (finite) temperature, we have by definition of push–forward ZRN |T(x)|2u(x)dx =ZRN |x|2v(x)dx =ZRN |x|2u(x)dx = 2θ(t). Therefore, f0 T(a)≥0 for all a≥0. Hence, the minimum is attained at a= 0, which proves (10) and the lemma. ¦ Let us consider again any two solutions u1and u2of (1). By performing the time dependent scaling (6), with vi,i= 1,2, denoting the two rescaled solutions, it is easily seen that vi(y, t) = θ(t)N/2ui(θ(t)1/2y, t), i = 1,2, where again θ(t) is the minimum between the two temperatures θ1and θ2. Hence, by a well known scaling property of the Wasserstein distance (see LONG TIME BEHAVIOR FOR ut= ∆φ(u) 9 [Vil03]), we obtain d2 2(v1(t), v2(t)) = θ(t)−1d2 2(u1(t), u2(t)).(13) Our first result is a straightforward consequence of collecting lemmas 2.3 and 2.1 and identity (13). Theorem 2.4. Let u1(x, t)and u2(x, t)be solutions to (1) with the nonlinearity φverifying (NL1)–(NL3)–(FT) and with initial data u1,0and u2,0 of unit mass and finite temperature, Let v1and v2be their projections onto the ‘unit second moment sphere’ in P2(RN)as in (6). Then, for all t≥0, d2 2(v1(t), v2(t)) ≤θ(t)−1d2 2(u1,0, u2,0)→0as t→ ∞.(14) Our intention now is to recover the existence of some fixed points from the result in Theorem 2.4. To perform this task, we consider the complete metric space of probability measures M=½µ∈ P2(RN),1 2ZRN |x|2dµ(x) = 1¾, endowed with the 2–Wasserstein distance d2(see [Vil03]). Let us define a ‘renormalized’ flow map S(t) : M −→ M as follows. For u0belonging to the subspace of absolutely continuous with respect to Lebesgue measures M ∩ L1(RN), we consider the unique solution u(x, t) to equation (1) with initial datum u0and its projection v(y, t) given by the scaling (6). We then set S(t)u0:= v(t) by definition. Theorem 2.4 and Lemma 2.1 implies that the map S(t) : M ∩ L1(RN)−→ M ∩ L1(RN) verifies d2 2(S(t)(u1,0), S(t)(u2,0)) ≤4t−1 N(m−1)+2 d2 2(u1,0, u2,0) (15) We now trivially extend the map S(t) to the whole space Mby using property (15) of S(t). Consider µ∈ M. There exists a sequence un∈ M ∩ L1(RN) such that d2(un, µ)→0 as n→ ∞. It suffices to consider a normalized regularization via standard mollifiers ρn vn=ρn∗µ. We then perform the mass preserving scaling un(x) = αnvn(αnx), where αnis the temperature of vn. Since the sequence unis tight and converges in the sense of measures to µ, then αn→1 (see [Vil03]). Moreover, it is easily seen than unconverges also to µin the sense of measures. Therefore, we conclude that d2(un, µ)→0. Since the map S(t) verifies (15), then the sequence {S(t)un}nis a Cauchy sequence in M. By completeness of M,S(t)un→ν. We define ν=S(t)µ. It is easy to see based again on (15) that this definition is independent on the choice of the sequence un. 16 JOS´ E A. CARRILLO, MARCO DI FRANCESCO AND GIUSEPPE TOSCANI (a) Let v(y, s)be the solution to (24) with initial datum u0∈L1 +(RN). Then, for all p∈[1,+∞)the following estimate holds for some fixed constant C > 0 kv(·, s)kLp(RN)≤CeN(1−1 p)s£λ¡es λ−1¢¤−λN(1−1 p)ku0kL1(RN),(26) while, for p= +∞we have kv(·, s)kL∞(RN)≤CeNs £λ¡es λ−1¢¤−λN ku0kL1(RN).(27) Hence, for any fixed s0>0and for all p∈[0,+∞], we have sup s≥s0 kv(·, s)kLp(RN)≤C(s0)ku0kL1(RN),(28) where C(s0)doesn’t depend on p. (b) Let v(y, s)be the solution to (24) with initial datum u0∈Lp(RN). Then, the following local stability property holds at any s≥0 kv(·, s)kLp(RN)≤eN(1−1 p)sku0kLp(RN).(29) A note about the notation. In the sequel we denote by Ca generic positive constant. Sometimes we shall indicate its dependence on some parameters by means of the expressions C(. . .) or C.... 4.1. Statement of the problem and result. Let vbe the solution to the Cauchy problem    ∂v ∂s =∇ · (yv) + eNms∆φ(e−Nsv) v(y, 0) = u0(y). (30) Due to the conservation of the mass, we can set ZRN u0(y)dy =ZRN u(y, s)dy =M. We expect the solution v(y, s) to behave like the rescaled Barenblatt similarity function v∞(y) =      µCM−λ|y|2 2¶1 m−1 + if m6= 1 CMe−|y|2 2if m= 1, (31) as s→+∞, where the constant CMis chosen in such a way that v∞(y) has mass M, and λdepends only on mand on the space dimension N. We LONG TIME BEHAVIOR FOR ut= ∆φ(u) 17 emphasize that v∞is not a solution to equation (30). Let us then define our entropy functional H(v) =                1 m−1ZRN v(y)mdy +1 2ZRN |y|2v(y)dy if m6= 1 ZRN v(y) log v(y)dy +1 2ZRN |y|2v(y)dy if m= 1 1 1−mZRN (mv(y)−v(y)m)dy +1 2ZRN |y|2v(y)dy if m < 1. (32) Here mis the exponent describing the behavior of the nonlinearity at zero, given by condition (NL4). We recall that the convex functional H(v) attains its minimum over L1 +(RN), under the constraint RRNv= constant, at the state v∞with mass M(see [CT00]). The relative entropy is defined, as usual, by H(v|v∞) = H(v)−H(v∞). The above functional H(v|v∞) is related to a Dirichlet–type integral, the so– called entropy production or generalized Fisher information, by means of the following generalized Log-Sobolev–type inequality (see [CT00, CJM+01]). Theorem 4.2. Let v∈L1 +(RN)such that RRNv(y)dy =M, let v∞be the ground state defined in (31) with mass M. Then, the following inequality holds, H(v)−H(v∞)≤1 2I(v|v∞),(33) where I(v|v∞) = ZRN v¯¯¯¯ ∇µm m−1vm−1+|y|2 2¶¯¯¯¯ 2 dy, where mis the same exponent as in the definition (32) of the entropy functional . We also recall the generalized Csisz´ar–Kullback inequality, which provides an upper bound of the L1norm of the difference between any positive density vand the ground state v∞having the same mass as v, in terms of their relative entropy. More precisely, we have Theorem 4.3. Let v∈L1 +(RN), with RRNv(y)dy =RRNv∞(y)dy. Then, the following holds kv−v∞kα L1(RN)≤C[H(v)−H(v∞)],(34) where α=(2if m≤2 mif m≥2,(35) 18 JOS´ E A. CARRILLO, MARCO DI FRANCESCO AND GIUSEPPE TOSCANI where mis again the exponent in the definition (32) of the entropy functional. In the sequel we shall also need the generalized entropy functional (see [BDE02]) E(v, s) = emNs ZRN F(e−Nsv)dy +1 2ZRN |y|2vdy, (36) where F(u) = Zu 0 h(θ)dθ, (37) and his the enthalpy defined in (5). The function Fis well–defined thanks to condition (NL5). We observe that the following identity holds F(u) = uh(u)−φ(u). In what follows, we shall assume the solution to enjoy enough regularity in order to be treated as a classical solution; the rigorous justification of our result then directly follows by an approximation argument used in [Ott01], subsequently generalized for equations of the form (24) in [CJM+01]. Let us restate the main result of this section already announced as Theorem 1.3. Theorem 4.4. Let v(y, s)be the solution to the Cauchy problem (30). Let the initial datum v0∈L1 +(RN)be such that ZRN F(v0(y))dy +1 2ZRN |y|2v0(y)dy +ZRN v0(y)mdy < +∞,(38) with Fgiven by (37). Then, the relative entropy functional H(v(s)|v∞) satisfies H(v(s)|v∞)≤Ce−δs,for all s≥0,(39) where δ= min{2, N(k+ 1)},kis the exponent in the structural condition (NL4), and C > 0is a constant depending on the mass of v0and on the bounded quantity in (38). The Csisz´ar Kullback inequality (34) above and the time dependent scaling (23), then, provides the rate of convergence in L1for the solution to the original problem (1) towards the Barenblatt self–similar functions u∞(x, t) = ³C−λ|x|2t−2 N(m−1)+2 ´1 m−1 +.(40) LONG TIME BEHAVIOR FOR ut= ∆φ(u) 19 Corollary 4.5. Let u(x, t)be the solution to the Cauchy problem (1), with initial datum u0≥0such that ZRN£um 0(x) + |x|2u0(x) + F(u0(x))¤dx < +∞. Let u∞(x, t)be the Barenblatt self–similar function given by (40) with the constant Csuch that Ru∞=Ru0. Then, the following estimate holds for all t≥0 ku(t)−u∞(t)kL1(RN)≤C(t+ 1)−λδ α,(41) where λis defined in (23) and αis defined by (35). Remark 4.6. The rate of convergence in (41) has been already obtained in [BDE02]. However, our result includes more general nonlinearity functions. In particular, we don’t need to require the technical condition (m−1)uh(u)− mf(u)≤0, which is not satisfied (for instance) by the nonlinearity function φ(u) = u2+P(u),where P(u) is a suitable standard regularization of the function p(u) = (−u3if 0 ≤u≤1/2 −1/8 if u≥1/2. Moreover, our approach seems to be much simpler than that in [BDE02], since we make use the usual nonlinear version of the Log–Sobolev inequality, and we use directly the Csiszar–Kullback inequality (34) in order to get a rate of convergence in L1. 4.2. Proof of Theorem 4.4. In order to prove theorem 4.4, we perform an estimate for large times s, which is obtained basically by means of the regularizing effect (27) and of the Log-Sobolev–type inequality (33). Then we use the generalized entropy functional (36)in order to control the evolution of the entropy in a finite time interval. Proposition 4.7. Under the assumptions (NL1)–(NL5) on the nonlinearity φ, there exist an s0>0and a positive constant C(s0)depending on s0such that, if H(v(s0)) <+∞, then we have H(v(s)|v∞)≤C(s0)e−min{2,N(k+1)}s(42) for all s≥s0. Proof. We recall that the equation (30) can be written in the alternative way (25). Then, integration by parts yields d dsH(v(s)) =ZRN ·mvm−1 m−1+|y|2 2¸∇ · ·v∇µ|y|2 2+eN(m−1)sh(e−Nsv)¶¸dy =− ZRN v∇µmvm−1 m−1+|y|2 2¶· ∇µ|y|2 2+eN(m−1)sh(e−Nsv)¶dy. 20 JOS´ E A. CARRILLO, MARCO DI FRANCESCO AND GIUSEPPE TOSCANI Hence, we employ the structural condition (NL4) to obtain d dsH(v(s)) = =−ZRN ψ(e−Nsv)¯¯¯¯ ∇µm m−1vm−1−|y|2 2¶¯¯¯¯ 2 dy + −me−Ns ZRN v2m−2ψ0(e−Nsv)|∇v|2dy + −e−Ns ZRN vmψ0(e−Nsv)·y· ∇ µm m−1vm−1¶¸dy =− ZRN vµψ(e−Nsv)+ 1 mψ0(e−Nsv)e−Nsv ¶¯¯¯¯ ∇µmvm−1 m−1−|y|2 2¶¯¯¯¯ 2 dy + +eNms ZRN¡e−Nsv¢mψ0(e−Nsv)£y· ∇(e−Nsv)¤dy + +1 me−Ns ZRN v2ψ0(e−Nsv)|y|2dy = 3 X j=1 Ij.(43) We first compute the term I2. We observe that, from the hypothesis (P3) on the nonlinearity, the function g(u) = umψ0(u) is is summable over any interval [0, L), L > 0. Hence, the primitive G(u) = Zu 0 θmψ0(θ)dθ is well defined on [0,+∞). As a consequence of that, and after integration by parts, I2may be written as follows I2=−NeNms ZRN G(e−Nsv)dy. Again from the structural hypothesis (P3), it follows easily that G(u) is a (m+k+1)–H¨older function on a neighborhood of u= 0. This fact, together with estimate (27), yields I2≤C(M, s0)e−N(k+1)sZRN vm+k+1dy ≤C(M, s0)kvkk+1 L∞(RN)e−N(k+1)sZRN vmdy ≤C(M, s0)e−N(k+1)sH(v(s)), LONG TIME BEHAVIOR FOR ut= ∆φ(u) 21 for some s0>0 (chosen in order to have e−Nsvsmall) and for any s≥s0. In a very similar way, we estimate I3 I3≤C(M, s0)e−N(k+1) ZRN v2+k|y|2≤C(M, s0)e−N(k+1)H(v(s)), for s0large and s≥s0. The integral term I1may be written as follows. I1=−ZRN α(s)v¯¯¯¯ ∇µm m−1vm−1+|y|2 2¶¯¯¯¯ 2 dy, where α(s) = ψ(e−Nsv) + 1 mψ0(e−Nsv)e−Nsv. By means of the structural condition (NL2) and the first order Taylor expansion, we have α(s) = 1 + e−Nsvψ0(e−Nsη) + 1 me−Nsvψ0(e−Nsv), η ∈[0, v(s)]. We employ once again the regularizing effect (27) to get α(s)≥1−C(M, s0)e−N(k+1)s, s ≥s0, and we can choose s0large enough in order to have α(s)≥0 for any s≥s0. Hence, we put the above estimates into (43) and we use the Sobolev–type inequality (33) to recover d dsH(v(s)|v∞)≤ −2³1−C(M, s0)e−N(k+1)s´H(v(s)|v∞)+ +C(M, s0)e−N(k+1)sH(v(s)) ≤ −2³1−C(M, s0)e−N(k+1)s´H(v(s)|v∞)+ +C(M, s0)e−N(k+1)sH(v(s0)). Finally, we use the variation of constants formula and obtain the desired estimate (42). ¤ Next, we perform a local–in–time estimate of the entropy, in order to control the constant C(s0) in the inequality (42). Lemma 4.8. Suppose that the initial datum u0satisfies (38). Then, the following inequality holds at any s≥0, E(v(s)) ≤e(m−1)sE(v0),(44) where E(u)is defined in (36). In particular, the entropy H(v(s)) is uniformly bounded on any finite time interval [0, s0]. 22 JOS´ E A. CARRILLO, MARCO DI FRANCESCO AND GIUSEPPE TOSCANI Proof. We calculate the evolution in time of the functional E(v(s), s) defined in (36) (see [BDE02]). After integration by parts, we get d dsE(v(s)) = −ZRN v¯¯¯y+e(m−1)Ns∇h(e−Nsv)¯¯¯ 2dy+ +emNs ZRN£(m−1)ve−Nsh(e−Nsv)−mφ(e−Nsv)¤dy ≤emNs(m−1) ZF(e−Nsv)dy ≤(m−1)E(v(s)), which proves (44). The last assertion comes directly from (29) in case m > 1. In case m≤1, we observe that the absolute entropy H(v) is bounded from below (we are in the finite entropy equilibrium range, see section 3). Moreover, H(v)≤Rv, and therefore the absolute entropy is uniformly bounded. ¤ Remark 4.9. The entropy dissipation method has been successfully used in [CF04] in order to prove convergence towards Barenblatt solutions for diffusion dominated convection–diffusion equations. By means of the same approach in the present paper, one can generalized the results in [CF04] to a generalized convection–diffusion model where the power law in the diffusion term is replaced by a more general φsatisfying the hypothesis above. Acknowledgements: The authors acknowledge support from the European IHP network HYKE “Hyperbolic and Kinetic Equations: Asymptotics, Numerics, Applications” HPRN-CT-2002-00282. JAC acknowledges partial support from DGI-MCYT/FEDER project BFM2002-01710 and CIRIT 2001SGR/00172. GT acknowledges support from the Italian MIUR project “Mathematical Problems of Kinetic Theories”. MDF acknowledges partial support from the Wittgenstein 2000 Award of Peter A. Markowich and from the Italian MIUR–COFIN 2002 project ‘Equazioni Iperboliche e Paraboliche Nonlineari’ . We thank P. Bechouche, M. P. Gualdani, P. A. Markowich and R. J. McCann for stimulating discussions. 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