The Aizenman-Bak model for reacting polymers is considered for spatially inhomogeneous situations in which they diffuse in space with a non-degenerate sizedependent coefficient. Both the break-up and the coalescence of polymers are taken into account with fragmentation and coagulation constant kernels. We demonstrate that the entropy-entropy dissipation method applies directly in this inhomogeneous setting giving not only the necessary basic a priori estimates to start the smoothness and size decay analysis in one dimension, but also the exponential convergence towards global equilibria for constant diffusion coefficient in any spatial dimension or for non-degenerate diffusion in dimension one. We finally conclude by showing that solutions in the one dimensional case are immediately smooth in time and space while in size distribution solutions are decaying faster than any polynomial. Up to our knowledge, this is the first result of explicit equilibration rates for spatially inhomogeneous coagulation-fragmentation models.

We analyze the spatial inhomogeneous version of a size-continuous model for reacting polymers or clusters of aggregates: ?t f ? a(y)

x f = Q( f, f ).

Here, f = f (t , x , y) is the concentration of polymers/clusters with length/size y ? 0 at time t ? 0 and point x ? ? Rd , d ? 1. These polymers/clusters diffuse in the environment . This set is assumed to be a smooth bounded domain with normalized volume, i.e., | | = 1. In the one dimensional case, we will set = (0, 1). Equation (1.1) is to be considered with homogeneous Neumann boundary condition ?x f (t , x , y) · ?(x ) = 0 on ? (1.1) with ? the outward unit normal to , so that there is no polymer flux through the physical boundary. We assume the diffusion coefficient a(y) to be non-degenerate in the sense that there exist a?, a? ? R+ such that

0 < a? ? a(y) ? a?.

On the other hand, the reaction term Q( f, f ) of (1.1) models chemical degradationbreak-up or fragmentation- and polymerization -coalescence or coagulation- of polymers/clusters. More precisely, the full collision operator reads as

Q( f, f ) = Qc( f, f ) + Qb( f, f ) = Q+( f, f ) ? Q?( f, f )

= Qc+( f, f ) ? Qc?( f, f ) + Qb+( f, f ) ? Qb?( f, f ) with obvious definitions of the coagulation Qc( f, f ), fragmentation or break-up Qb( f, f ), loss Q?( f, f ) and gain Q+( f, f ) operators which are determined from the four basic terms in (1.4): 1. Coalescence of clusters of size y ? y and y ? y results in clusters of size y: 2. Polymerization of clusters of size y with other clusters of size y produces a loss in its concentration: 3. Break-up of clusters of size y larger than y contributes to create clusters of size y: (1.3) (1.4) (1.5) (1.6) (1.7) (1.8) 0

y Qc+( f, f ) :=

f (t, x , y ? y ) f (t, x , y ) d y .

Qc?( f, f ) := 2 f (t, x , y)

f (t, x , y ) d y .

Qb+( f, f ) := 2 y ?

f (t, x , y ) d y .

Qb?( f, f ) := y f (t, x , y).

0 ? 0 ?

This kind of model finds its application not only in polymers and cluster aggregation in aerosols [S16,S17,AB,Al,Dr] but also in cell physiology [PS], population dynamics [Ok] and astrophysics [Sa]. Here, fragmentation and coagulation kernels are all set up to constants as in the original Aizenman-Bak model [AB]. This will be of paramount importance in the basic a-priori estimates. The conservation of the total number of monomers at time t ? 0 quantified by

N (t, x ) d x , where N (t, x ) := y f (t, x , y) d y is the basic conservation law satisfied by Eq. (1.1) since the reaction term (1.4) satisfies 0 ?

y Q( f, f ) d y d x = 0, and thus, assuming initially a positive total number of monomers, we formally conclude y f (t, x , y) d y d x =

N (t, x ) d x =

N0(x ) d x := N? > 0.

(1.9) Another macroscopic quantity of interest is the number density of polymers, 0 ? M (t, x ) := f (t, x , y) d y, that together with the total number of monomers N (t, x ) satisfies the reaction-diffusion system (1.10) (1.11) (1.12) (1.13) (1.14) (1.16) (1.17) (1.18) ?t N ? ?t M ? x x ? ? 0 0 ya(y) f (t, x , y) d y a(y) f (t, x , y) d y = 0, = N ? M 2, becoming a closed decoupled system in the constant diffusion case (a(y) := a): with obvious notations. for any smooth function ?, the function ? being the primitive of ? (?y ? = ?) such that ?(0) = 0.

Let us consider the (free-energy) entropy functional associated to any positive density f as 0 ? H ( f )(t, x ) = ( f ln f ? f ) d y, and the relative entropy H ( f |g) = H ( f ) ? H (g) of two states f and g not necessarily with the same L1y -norm. Then, the entropy formally dissipates as d dt

H ( f ) d x = ? a(y) |?x f |2 d y d x ? f ? ? f (y ) ? f (y) f (y ) (?(y) + ?(y ) ? ?(y )) d y d y (1.15) for any smooth function ?(y), where y = y + y and the dependence on (t, x ) of the density function has been dropped for notational convenience. An alternative weak formulation that can be useful in several arguments below is obtained integrating by parts in the Qb+ part giving < Q( f, f ), ? >= ? 2 f (y) f (y )?(y ) d y d y

Global existence and uniqueness of classical solutions has been studied in [Am, AW] for some particular cases, namely, for constant diffusion coefficient or dimension one with additional restrictions for the coagulation and fragmentation kernel not including the AB model. The initial boundary-value problem to (1.1)?(1.2) was then analyzed in [LM02-1], for much more general coagulation and fragmentation kernels including the AB model (1.5) ? (1.8), proving the global existence of weak solutions satisfying the entropy dissipation inequality 0 t H ( f (t )) d x +

DH ( f (s)) ds ?

H ( f0) d x for all t ? 0.

The equilibrium states for which the entropy dissipation vanishes are better understood after applying a remarkable inequality proven in [AB, Props. 4.2 and 4.3]. A modified version of this inequality (reviewed in Sect. 2) reads: ? ? 0 0 f ? f f ln f f f d y d y ? M H ( f | f?N ,N ) + 2(M ? ?N )2.

(1.19) Herein, f?N ,N denotes a distinguished, exponential-in-size distribution with the very moments M = ?N and N : y f?N ,N (t , x , y) = e? ?N .

These distributions f?N ,N appear as analogues to the so-called intermediate or local equilibria in the study of inhomogeneous kinetic equation (e.g. [DV01, CCG, FNS, DV05, FMS, NS]). Finally, the conservation of mass (1.9) identifies (at least formally) the global equilibrium f? with constant moments M ?2 = N = N?: y f? = e? ?N? .

The analogy to intermediate equilibria carries over to the following additivity of relative entropies:

H ( f | f?) = H ( f | f?N ,N ) + H ( f?N ,N | f?).

It is worth pointing out that even if f?N ,N and f? do not have the same L 1y ?norm, its global relative entropy (1.20) (1.21) H ( f?N ,N | f?) d x = 2

N d x ? ?

N d x ? 0 is a nonnegative quantity, as easily checked via Jensens?s inequality. In [LM02-1], it is proved that f? attracts all global weak solutions in L 1( × (0, ?)) of (1.1)-(1.2) but no time decay rate is obtained. This result is the analogue to convergence results along subsequences for the classical Boltzmann equation in [De].

Other existence and uniqueness results for inhomogeneous coagulation-fragmentation models were given in [CD] and the references therein. Finally, let us mention that the conservation law (1.9) is known not to hold for certain coagulation-fragmentation kernels, phenomena known as gelation [ELMP], and the convergence or not towards typical self-similar profiles for the pure coagulation models is a related issue; we refer to [Le,LM05,MP]. We refer finally to [LM02-1,LM03,LM04] for an extensive list of related literature.

Let us now discuss some works on the study of the long time asymptotics for related models. Qualitative results concerning a discrete version of a coagulation-fragmentation system, the Becker-Döring system, have been obtained in [CP,LW,LM02-2] and the references therein. We emphasize that global explicit decay estimates towards equilibrium were obtained for the Becker-Döring system without diffusion in [JN] by entropy-entropy dissipation methods. Other techniques have recently been developed for inhomogeneous kinetic equations. We refer to [MN] for a spectral approach and to [V06] for a general description of the concept of hypocoercivity. Note that the presence of diffusion instead of advection makes it possible in our present context, not to use the concept of hypocoercivity.

In this work we prove exponential decay towards equilibrium with explicit rates and constants. Our key result, Lemma 2 in Sect. 2, establishes a functional inequality between entropy and entropy dissipation provided lower and upper bounds on the moment M and (1.3). We are able to apply this functional inequality to solutions of (1.1)-(1.2) in the next two situations.

In the special case of size-independent diffusion coefficients a(y) = a, we show, as the first application of Lemma 2, the exponential decay towards equilibrium in all space dimensions d ? 1 by exploiting the closed system (1.13)?(1.14) for N and M in Sect. 2.

In the case of general diffusion coefficients a(y) satisfying (1.3) we prove in Sect. 3 a-priori estimates in the one-dimensional case d = 1, which entail an entropy-entropy dissipation estimate with a constant sufficient to conclude exponential decay via a suitable Gronwall argument (see Sect. 4). These two cases are summarized in the following theorem: Theorem 1. Let be a smooth bounded connected open set of Rd , d ? 1 and assume a constant diffusion coefficient a(y) = a > 0 or let be the interval (0, 1) and consider a diffusion coefficient satisfying (1.3). Let us also assume that f0 = 0 is a nonnegative initial datum such that (1 + y + ln f0) f0 ? L1((0, 1) × (0, ?)). In the case a(y) = a > 0 assume further that initial moments M0(x ) and N0(x ) are L?( )-functions.

Then, the global weak solutions f (t, x , y) of (1.1)?(1.2) decay exponentially to the global equilibrium state (1.20) with explicitly computable constants C1, C2 and rate ?, both in global relative entropy: and in the L 1x,y sense:

H ( f (t )| f?) d x ? C1

H ( f0| f?) d x f (t, ·, ·) ? f? L1x,y ? C2

H ( f0| f?) d x e?? t , e? ?2 t (1.22) (1.23) for all t ? 0, where f? is defined by (1.20) and N? > 0 is determined by the conservation of mass (1.9).

In the one dimensional case, it is further possible to interpolate the exponential decay in a ?weak? norm like L1 with polynomially growing bounds in ?strong? norms like (weighted) L1y (Hx1) in order to get an exponential decay in a ?medium? norm like L1y (L ?x). Thus, the decay toward equilibrium can be extended to these stronger norms.

Proposition 1. Under the assumptions of Theorem 1 for the case d = 1, for all t? > 0 and q ? 0, there are explicitly computable constants C3, ? > 0 such that whenever t ? t?, (1 + y)q f (t, ·, y) ? f?(y) L?x d y ? C3 e?? t . (1.24)

A bootstrap argument in the spirit of the proof of Proposition 1 allows to replace the L ?x norm by any Sobolev norms in (1.24).

g(y)g(y ) ln g(y + y ) d y d y ? g(y) d y g(y ) ln g(y ) d y 2. Entropy-Entropy Dissipation Estimate Please note that in this section we will systematically use the shortcuts: M = f (x , y) d yd x ,

N = y f (x , y) d yd x .

We start by reminding the reader of the following functional inequality: Lemma 1 ([AB, Prop. 4.3]). Let g := g(y) be a function of L1+((0, ?)) with finite entropy g ln g ? L1((0, ?)), then ? ? 0 0 ? ? 0 0 while for the term 0 ? 0 ? 0 ? ?

. f f ? ? 0 0 ? 0 ? 0 ? ? f f ln f d y d y ,

This inequality allows to show the dissipation inequality (1.19). Following the original paper [AB] or the survey [LM04], one finds that ( f ? f f ) ln d y d y ? M H ( f | f?N ,N ) + (M ?

?N )2 g(y) d y

(2.1) + M 2

N N N M 2 ln M 2 + 1 ? M 2 .

(2.2) In fact, after expanding the left-hand side of (2.2), one applies Lemma 1 to the term one uses Jensen?s inequality for the convex function x ln x and further that ? ? 0 0 f f f f f ln f f f

d y d y ? ? 0 0

f d y d y = N .

Then, after directly calculating the remaining terms one obtains (2.2), as in [LM04], and moreover the inequality (1.19) when applying the elementary inequality x ln(x )+1?x ? (1 ? ?x )2 for x ? 0 to the last term on the right-hand side of (2.2).

For the subsequent large-time analysis, we will rather study the relative entropy with respect to the global equilibrium, which dissipates according to (1.18) and (1.19) as d dt

We introduce a lemma enabling to estimate the entropy of f by means of its entropy dissipation. This is a functional estimate, that is, the function f in this lemma does not depend on t and has not necessarily something to do with the solution of our equation. Lemma 2. Assume (1.3). Let f := f (x , y) ? 0 be a measurable function with moments satisfying 0 < M? ? M (x ) ? M L?x and 0 < N? = N . Then, the following entropy-entropy dissipation estimate holds: with a constant C (M?, N?, a?, P( )) depending only on M?, N?, a? and the Poincaré constant P( ).

Proof. Step 1. We start with the right-hand side of (2.4) by using the additivity (1.21) and calculating and further, we obtain (2.6) by expanding ?N ? M 2L2 and Young?s inequality x ?

N ? M ? M + M 2L2 .

x M H ( f | f?N ,N ) d x + 2 M ? ?N 2L2 x 1 ? 2 N ? M 2L2x ? Thus, we obtain (using 0 < M? < M )

M ? M 2L2x ? by the inequality (1.19) f ? f f ln

d y d y d x f f f (2.4) (2.5) (2.6) (2.7) Step 3. Next, the variance of M , i.e. the last term on the right-hand side of (2.7) is controlled by the first, ?Fisher?-type term of (2.3). Denoting with P( ) the constant of

0 provides a bound which does not seem sufficient to conclude as in Step 2.

Now, let us directly apply this entropy-entropy dissipation estimate to prove the constant diffusion part of Theorem 1. In the constant diffusion case, the equations for the first two moments M (t, x ) and N (t, x ) become the closed system (1.13)-(1.14). The existence and uniqueness of global, classical solutions with global L? bounds from below and above are standard thanks to the maximum principle applied to the equations for N and further for M . We refer, for instance, to [Ro,Ki] for details, to conclude with: Lemma 3. Let be a smooth bounded connected open set of Rd , d ? 1 and let us assume that the initial data M0(x ) and N0(x ) = 0 are nonnegative L?( )-functions. Then, there exist increasing functions t ? M?(t ), N?(t ) and decreasing functions t ? M ?(t ), N ?(t ) such that the unique global bounded solutions of the system (1.13)-(1.14) satisfy 0 < M?(t ) ? M (t, x ) ? M ?(t ) < ?, 0 < N?(t ) ? N (t, x ) ? N ?(t ) < ?, (2.9) (2.10) for all t > 0.

Proof of Theorem 1. Case a(y) = a constant, d ? 1. Let us fix t? > 0. From (2.9)(2.10), we have 0 < M? ? M (t, x ) ? M? < ? and 0 < N? ? N (t, x ) ? N ? < ? for all t ? t?, and thus for all t ? t? due to (2.4) with the constant C (M?, N?, a?, P( )) given in Lemma 2.

for all t ? t?. Gronwall?s lemma implies estimate (1.22).

Next, convergence in L1 as stated in Theorem 1 follows from the functional inequality of Csiszar-Kullback type [Cs,Ku]: f (t, ·, ·) ? f? 2L1x,y ? 2

M (t, x ) d x +

N?

H ( f (t )| f?) d x .

(2.11) The proof is standard, see [CCD] for related inequalities, and it is shown via a Taylor expansion of the function ?( f ) = f ln( f ) ? f up to second order around f?. Indeed, for a function ? (x , y) ? (inf{ f (x , y), f?(y)}, sup{ f (x , y), f?(y)}), we get 0 0 ? ? and the first term vanishes due to the conservation law (1.9). For the second term, we apply Hölder?s inequality f ? f? 2L1x,y ? ? L1x,y ( f ? f?)2 d y d x with ? L1x,y ?

M d x +

N .

? Noticing that t ? [0, t?] ? f (t, ·, ·) ? L 1x,y is bounded, we finally get (1.23), which concludes the proof of Theorem 1.

Remark 1. As a consequence of the previous result, we also showed that the unique global bounded solutions of the system (1.13)-(1.14) satisfy M (t, x ) ? M? = ?N? and N (t, x ) ? N? as t ? ? in L1( ) exponentially fast with explicit constants. In fact, we first remark that H ( f | f?) = H ( f | f M,N ) + H ( f M,N | f?) with E S :=

H ( f M,N | f?)d x =

N (? ln ? ? ? + 1)+

N?

N ?

N? ? 2 d x , where ? = ?MN and f M,N (t, x , y) =

M 2 N

e? MN y .

M 2

N It is obvious that E S is nonnegative since the minimum of ? ln ? ? ? + 1 is zero, and it can be written as

H ( f M,N | f?) d x =

M ln ? 2(M ? ?

N ) + 2

N? ? ?

N d x , by using the conservation of mass (1.9). Since H ( f | f M,N ) ? 0, then (1.22) implies the exponential convergence to zero of E S by the above additivity property. Finally, a simple Taylor expansion shows that, for all t ? t?,

N (t ) ? M (t ) 2L2 + x

N (t ) ?

N? 2L2x ? L

H ( f M,N (t )| f?) d x , with

L = max 1, ?

M?

N? , ?

N ? , that implies by trivial arguments the exponential convergence in L1( ) towards equilibrium for M and N . In fact, the system (1.13)-(1.14) might have been studied by a direct application of the techniques in [DF05,DF06].

In the sequel, we shall discuss the general diffusion coefficient, i.e., size dependent verifying (1.3) but we restrict to the one dimensional case, d = 1 (we shall not recall this fact in the various lemmas). We begin the proof of Theorem 1.

Lemma 4. Assume that f0 = 0 is a non-negative initial datum such that (1 + y) f0 ? L1((0, 1)×(0, ?)). Then, there exists M0? > 0 such that solutions of (1.1)?(1.8) satisfy f (t, x , y) d y d x ?

M (t, x ) d x ? M0?.

(3.1) Proof. We estimate the L1( )-norm of M (t, x ) by integrating equality (1.14), obtaining sup t?0 d dt 0 ? by Hölder?s inequality and the conservation of mass (1.9). Therefore, for all t ? 0, M (t, x ) d x ? max Lemma 5. Assuming that the nonnegative initial datum f0 = 0 satisfies (1 + y + ln f0) f0 ? L1((0, 1) × (0, ?)). Then, the number density of polymers M ? L1 + L?(0, ?; L?(0, 1)). More precisely, there exist m? > 0 and an L1+(0, ?)-function m1(t ) such that the solution of (1.1)?(1.8) satisfies (3.2) (3.3) and as a consequence, a.e. t ? 0.

?

sup f (t, x , y) d y ? m? + m1(t ), 0 0<x<1

M (t, ·) L?x ? m? + m1(t ) Proof. In order to estimate the L ?x-norm of M (t, x ), we first use the entropy dissipation (2.3) of H ( f | f?) to deduce that 2 0 0 0

2 ?x f d y d x dt ? 2 a? 2 a? Now, we integrate f (t, x , y) ? f (t, x?, y) =

f (t, ?, y) d? x ? x 2 ?x ? 0 1 := m1(t ) In particular, we have, due to (3.1) and (3.3), that with respect to x? ? (0, 1) and estimate sup 0<x<1 f (t, x , y) ?

f (t, x?, y) d x? 0 1 Hence, after further integration with respect to y ? (0, ?), we apply Young?s and Hölder?s inequalities to show ?

sup f (t, x , y) d y ? 2 0 0<x<1 0 0 ?x

2 f (t, ?, y) d? d y + 2 ?x f (t, ?, y)

d?.

2 0 0 f (t, x?, y) d x? d y . ? m?

0 ? m1(t ) dt ? µ1 =

H ( f0| f?) , 2 a?

m? = 2M0?.

M (t, ·) L?x ? ?

sup f (t, x , y) d y, 0 0<x<1 which completes the proof of Lemma 5.

Note that the estimates (3.1) and (3.2) are somehow in duality, a fact that will become essential below. We now prove a lemma showing that the total number of clusters 01 M (t, x ) d x is bounded below by a strictly positive constant: Lemma 6. Assume that f0 = 0 is a nonnegative initial datum such that (1 + y + ln f0) f0 ? L1((0, 1) × (0, ?)). Then, there exists a constant M0? > 0 such that for all times t ? 0, one has

M (t, x ) d x ? M0?, (3.4) where f is a solution of (1.1)?(1.8).

Proof. We recall that d dt 0

1 so that d dt 0 1 0

0 1 and, recalling µ1 ? 0+? m1(s) ds), we deduce Distinguishing here between t < 1 and t ? 1, for instance, we obtain 0 1 0 t e?(t?s) m??µ1 ds 1 ? e?m? t m? 0 1 0 1 which concludes the proof of Lemma 6.

Next, we show the uniform control in time of all moments with respect to size y of the solutions. Let us define the moment of order p > 1 by

M p( f )(t ) := 1 0 0

?y p f (t, x , y) d y d x for all t ? 0.

Lemma 7. We assume that f0 = 0 is a nonnegative initial datum such that (1 + y + ln f0) f0 ? L1((0, 1) × (0, ?)). Then, the solution f of (1.1)?(1.8) has moments M p( f )(t ) uniformly bounded in time t > t? > 0 and for any p > 1, i.e., there exist explicit constants M?p( f0, m?, m1, p) such that

M p( f )(t ) ? M?p, for a.e. t > t? > 0.

(3.5)

Step 1. We first assume that M p( f )(t?) < ? for certain p > 1 and t? > 0. Using the weak formulation (1.16), it is easy to check that < Q( f, f ), y p >= ? 2 Taking into account Lemma 5 and (y + z) p ? C p (y p + z p), we deduce < Q( f, f ), y p > ? 2(C p ? 1) ?y p f (y) d y [m? + m1(t )] ? pp ?+ 11 0?f (y) y p+1 d y for all p > 1. Integrating in space, we find that the evolution of the moment of order p > 1 is given by d dt (3.6) Trivial interpolation of the p + 1-order moment with the moment of order one implies 1 for a.e. t > t?. According to Duhamel?s formula,

M p( f )(t ) ? M p( f )(t?) exp 2(C p ? 1) + D exp 2(C p ? 1) m1(? ) d? ? ds,

(3.7) s t t t? m1(s) ds ? t ? t?

2 t ? s 2 that which shows that the moment M p( f )(t ) is bounded by a constant M?p for a.e. t > t? since m1(t ) ? L1((0, ?)) by Lemma 5.

Moreover, it follows from (3.6) that the boundedness of M p(t?) immediately implies for all T > 0, and thus the finiteness of M p+1( f )(t ) for a.e. t > t? and a simple induction argument enables then to conclude the bounds on all higher moments. Step 2. It remains to show that for given nontrivial initial data y f0 ? L 1x,y and for a p > 1 and a time t? > 0 we have that M p(t?) < ?. We start with the following observation [MW, Appendix A]: For a nonnegative integrable function g(y) = 0 on (0, ?), there exists a concave function (y), depending on g, smoothly increasing from (0) > 0 to (?) = ? such that

M p+1( f )(t ) dt < ?

(y) g(y) d y < ?. t t?

T t? for 0 < y < y with C not depending on g. We refer to [MW, Appendix A] for all the details of this ?by-now standard? construction.

To show now that M p(t?) < ? for a p > 1 and a time t? > 0, we take functions (x , y) constructed for nontrivial y f0(x , y) ? L1y (0, ?) a.e. x ? (0, 1) and calculate - similar to Step 1 - the moment

M1, ( f )(t ) =

y (x , y) f (x , y) d y d x .

For the fragmentation part, we use (3.8) for 0 < y < y and estimate y (y) Q f ( f ) = 2 y ( (y ) ?

(y)) d y f (y) 1 ? Moreover, the function can be constructed to satisfy (y) ? (y ) ? C for all ? > 0 and a positive constant C?, where the (t, x )-dependence has been dropped for notational convenience. Hence, by estimating the coagulation part similar to Step 1, making use of the concavity of , we obtain that d dt

M1, ( f )(t ) ? 3(m? + m1(t ))M1, ( f )(t ) ? C? M2??( f )(t ), and boundedness of the moment M1, follows by interpolation as well as the finiteness of M2??( f )(t?) analogously to Step 1.

Next, we show that M and N are bounded below uniformly (with respect to t and x ) for all t ? t? > 0.

Proposition 2. Under the assumptions of Theorem 1, let t? > 0 be given. Then, there are strictly positive constants M? and N? such that for all t ? t? > 0, M (t, x ) ? M? and

N (t, x ) ? N?.

where g1 is nonnegative. Then ?t f ? a(y) ?x x f = g1 ? y f ?

M (t, ·) L?x f, (?t + a(y) ?x x ) f et y+ 0t M(s,·) L?x ds = g2, where g2 is nonnegative.

Now, we recall that the solution h := h(t, x ) of the heat equation

?t h ? a ?x x h = G, with homogeneous Neumann boundary condition on the interval (0, 1), where a > 0 is a constant and G := G(t, x) ? L1, is given by the formula

1 h(t, x) = 2?? 1 ?1

h?(0, z) with h? and G? denoting the ?evenly mirrored around 0 in the x variable? functions h and G.

Therefore, for all t1, t ? 0, and x ? (0, 1), y ? R+, f (t1 + t, x, y) e(t1+t) y+ 0t1+t M(s,·) L?x ds so that when t ? [t?, 2t?] (and since |x ? z| < 2): and thus, for any A > 0, we deduce due to the conservation law (1.9) and

N (t1 + t, x) ? C e?2t? A Choosing now A, we get that N (t1 + t, x ) ? N? for some N? > 0 which does not depend on t1. Using Lemma 6,

M (t1 + t, x ) ? C 1 ? Once again choosing A, we get that M (t1 + t, x ) ? M?. Since M? does not depend on t1, we get Proposition 2. 4. Proofs of Theorem 1 and Proposition 1 if = (0, 1).

With Proposition 2 and Lemma 5 providing the moment bounds required by the entropyentropy dissipation Lemma 2 in the one dimensional case = (0, 1), we turn now to the Proof of Theorem 1. Case

= (0, 1). According to Lemma 2, H ( f | f?) d x ? ?D( f ) ? ?

Knowing that m1(t ) ? Lt1 with 0? m1(t ) dt ? µ1, we consider the sets A := {s > 0 : m1(s) ? 1} and Bt := {s ? [0, t ] : m1(s) < 1}. We readily find that m1(t ) dt ? µ1 and |Bt | = t ?

ds ? t ? µ1.

A?[0,t] 0 1 d dt 0

1 | A| =

A

ds ?

t t? M L?x ds ?

Bt ?

C C

M L?x ds ? ? (1 + m?) (t ? µ1), finishing the proof of (1.22). The proof of the L1-decay estimate (1.23) follows the same arguments as in the case of constant diffusion done in Sect. 2 using Csiszar-Kullback type inequalities.

Finally, we show Proposition 1. Let us denote by CT any constant of the form C (t ) (1+ T )s , where s ? R and C (t ) is bounded on any interval [t?, +?) with t? > 0. 0 0 0 0

According to the properties of the heat kernel (cf. [DF06] for example), we know that for any ? > 0 and t? > 0, f (·, ·, y) L3??([t?,T ]× ) ? CT f (0, ·, y) L1 + Q+( f, f )(·, ·, y) L1([0,T ]× ) .

x Then, for all r ? [2, 3[, 0 ? 0 ? (1 + y)q

Q+( f, f )(·, ·, y) Lr/2([t?,T ]× ) d y

(1 + y)q f (·, ·, y) L3??([t?,T ]× ) d y ? CT . Proof of Proposition 1. We observe using the bounds (3.5) and (3.1) that for all q ? 0, ? (1 + y)q+1 ? +

0 ? CT + 0 ?

q + 1 (1 + y)q ?

? 0 0 0 ? 0 ? 0 ? 0 ? f (·, ·, y) Lr ([t?,T ]× ) d y

f (·, ·, y ) f (·, ·, y ? y ) d y Lr/2([t?,T ]× ) d y (1 + y + z)q f (·, ·, y) f (·, ·, z) Lr/2([t?,T ]× ) d yd z ? CT + 2q?1 (1 + y)q f (·, ·, y) Lr ([t?,T ]× )d y 2 ? CT .

Using again the properties of the heat kernel (still described in [DF06]), we see that for any s ? [1, ?) and t? > 0, The above argument can now be used with r = 4 and shows that

(1 + y)q f (·, ·, y) Ls ([t?,T ]× )d y ? CT . (1 + y)q

Q+( f, f )(·, ·, y) L2([t?,T ]× ) d y ? CT .

As a consequence, the standard energy estimate on the heat kernel implies that

(1 + y)q f (T , ·, y) Hx1 d y ? CT . Then, using a Gagliardo-Niremberg type interpolation and Theorem 1, we obtain ? 0 ?

0 × ? × ? 0 0 ? (1 + y)q f (T , ·, y) ? f?(y) L?x d y (1 + y)q f (T , ·, y) ? f?(y) 3H/x14 which concludes the proof of Proposition 1.

Acknowledgements. JAC acknowledges the support from DGI-MEC (Spain) project MTM2005-08024. KF is partially supported by the WWTF (Vienna) project ?How do cells move?? and the Wittgenstein Award 2000 of Peter A. Markowich. JAC and KF appreciate the kind hospitality of the ENS de Cachan. The authors want to express their gratitude to the reviewer who helped us to improve this work. [AB]

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