Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result

Concentration in Lotka-Volterra parabolic or integral equations:

a general convergence result

Guy Barles^∗, Sepideh Mirrahimi^†, Benoˆıt Perthame ^†‡

Abstract

We study two equations of Lotka-Volterra type that describe the Darwinian evolution of a population density. In the first model a Laplace term represents the mutations. In the second one we model the mutations by an integral kernel. In both cases, we use a nonlinear birth-death term that corresponds to the competition between the traits leading to selection.

In the limit of rare or small mutations, we prove that the solution converges to a sum of moving Dirac masses. This limit is described by a constrained Hamilton-Jacobi equation. This was already proved in [8] for the case with a Laplace term. Here we generalize the assumptions on the initial data and prove the same result for the integro-differential equation.

Key-Words: Adaptive evolution, Lotka-Volterra equation, Hamilton-Jacobi equation, viscosity solu- tions, Dirac concentrations.

AMS Class. No: 35B25, 35K57, 47G20, 49L25, 92D15

1 Introduction

We continue the study, initiated in [8], of the asymptotic behavior of Lotka-Volterra parabolic equa- tions. The model we use describes the dynamics of a population density. Individuals respond differ- ently to the environment, i.e. they have different abilities to use the available resources. To take this fact into account, population models can be structured by a parameter, representing a physiological (phenotypical) trait inherited from the parent, and that we denote by x ∈R^d. We denote by n(t, x) the density of trait x. The mathematical modeling in accordance with Darwin’s theory consists of two effects: natural selection and mutations between the traits (see [18, 24, 27, 25] for literature in adaptive evolution). We represent the birth and death rates of the phenotypical traits bya net growth rate R(x, I). The term I(t) is an ecological parameter that corresponds to a measure of the total population, whatever the trait, and that represents in the simpler possible way the resources (more precisely the inverse of it). We use two different models for mutations. A first possibility is to represent them by a Laplacian and, in an extreme and irrealistic simplification, we take them independent of birth, so as to write

∗Laboratoire de Math´ematiques et Physique Th´eorique, CNRS UMR 6083, F´ed´eration Denis Poisson, Universit´e Fran¸cois Rabelais, Parc de Grandmont, 37200 Tours, France. Email: barles@lmpt.univ-tours.fr

†Universit´e Pierre et Marie Curie-Paris 6, UMR 7598 LJLL, BC187, 4, place Jussieu, F-75252 Paris cedex 5. Email:

mirrahimi@ann.jussieu.fr

‡Institut Universitaire de France. Email: benoit.perthame@upmc.fr

(∂_tn−4n= ⁿR(x, I(t)), x∈R^d, t≥0,

n(t= 0) =n⁰ ∈L¹(R^d), n⁰ ≥0, (1)

I(t) = Z

R^d

ψ(x)n(t, x)dx. (2)

Here is a small term that we introduce to consider only rare mutations. It is also used to re-scale time to consider a much larger time than a generation scale.

A more natural way to model mutations is to use, instead of a Laplacian, an integral term that describes directly the mutation probability to generate a new-born of traitxfrom a mother with trait y. This yields

(∂tn= ⁿR(x, I(t)) + ¹ R ₁

^dK(^y−x )b(y, I)n(t, y)dy, x∈R^d, t≥0,

n(t= 0) =n⁰ ∈L¹(R^d), n⁰ ≥0, (3)

I(t) = Z

R^d

n(t, x)dx. (4)

Both types of models can be derived from individual based stochastic processes in the limit of large populations depending on the scales in mutations birth and death (see [13, 14]).

In this paper, we study the asymptotic behavior of equations (1)-(2) and (3)-(4) when vanishes.

Our purpose is to show that under some assumptions on R(x, I), n(t, x) concentrates as a sum of Dirac masses that are traveling. In biological terms, at every moment one or several dominant traits coexist while other traits disappear. The dominant traits change in time due to the presence of mu- tations.

We use the same assumptions as [8]. We assume that there exist two constantsψm,ψM such that 0< ψ_m< ψ < ψ_M <∞, ψ∈W^2,∞(R^d). (5) We also assume that there are two constants 0< I_m< I_M <∞ such that

min

x∈R^d

R(x, Im) = 0, max

x∈R^d

R(x, IM) = 0, (6)

and there exists constantsKi >0 such that, for any x∈R^d,I ∈R,

−K₁ ≤ ∂R

∂I(x, I)≤ −K₁⁻¹ <0, (7)

sup

Im

2 ≤I≤2I_M

kR(·, I)k_W2,∞(R^d)< K2. (8) We also make the following assumptions on the initial data

I_m ≤ Z

R^d

ψ(x)n⁰(x)≤I_M, and ∃A, B >0, n⁰ ≤e^−A|x|+B , (9)

and that there exist a pointx0∈R^d and positive constants L0 and M0 such that

e^−M0 ≤n⁰(x), for all |x−x0| ≤L0, (10) Note that assumption (10) means that initially we have some kind of biodiversity since it can be interpreted as different traits being sufficiently represented in the population.

Here we takeψ(x)≡1 for equations (3)-(4) because replacingnbyψnleaves the model unchanged.

For equation (3) we assume additionally that the probability kernelK(z) and the mutation birth rate b(z) verify

0≤K(z), Z

K(z)dz = 1, Z

K(z)e^|z|2dz <∞, (11) b_m ≤b(z, I)≤b_M, |∇_xb(z, I)|< L₁b(z, I), |b(x, I₁)−b(x, I₂)|< L₂|I₁−I₂|, (12) whereb_m,b_M,L₁ and L₂ are positive constants. Finally for equation (3) we replace (6) and (7) by

x∈minR^d

R(x, Im) +b(x, Im)

= 0, max

x∈R^d

R(x, IM) +b(x, IM)

= 0, (13)

|R(x, I₁)−R(x, I2)|< K3|I₁−I2| and −K4 ≤ ∂(R+b)

∂I (x, I)≤ −K₄⁻¹<0, (14) whereK₃ and K₄ are positive constants.

In both cases, in the limit we expect n(t, x) = 0 or R(x, I) = 0, where n(t, x) is the weak limit of n(t, x) as vanishes. If we suppose that the latter is possible at only isolated points, we expect n to concentrate as Dirac masses. Following earlier works on the similar issue [19, 7, 8, 28], in order to study n, we make a change of variable n(t, x) = e^u(t,x) . It is easier to study the asymptotic behavior ofuinstead ofn. In section 5 we study the asymptotic behavior ofu whilevanishes. We show that u, after extraction of a subsequence, converge to a functionu that satisfies a constrained Hamilton-Jacobi equation in the viscosity sense (see [3, 20, 16, 22] for general introduction to the theory of viscosity solutions). Our main results are as follows.

Theorem 1.1. Assume (5)-(10). Let n be the solution of (1)-(2), and u = ln(n). Then, after extraction of a subsequence,u converges locally uniformly to a functionu∈C((0,∞)×R^d), a viscosity solution to the following equation:







∂tu=|∇u|²+R(x, I(t)), max

x∈R^d

u(t, x) = 0, ∀t >0, (15)

I(t)−→

→0 I(t) a.e., Z

ψ(x)n(t, x)dx=I(t) a.e.. (16) In particular, a.e. in t, supp n(t,·) ⊂ {u(t,·) = 0}. Here the measure n is the weak limit of n

as vanishes. If additionally (u⁰) :=ln(n⁰) is a sequence of uniformly continuous functions which converges locally uniformly to u⁰ then u∈C([0,∞)×R^d) and u(0, x) =u⁰(x) in R^d.

Theorem 1.2. Assume (8)-(14), and(u⁰) is a sequence of uniformly Lipschitz-continuous functions which converges locally uniformly to u⁰. Let n be the solution of (3)-(4) with n⁰ = e^u

0

, and u = ln(n). Then, after extraction of a subsequence, u converges locally uniformly to a function u ∈ C([0,∞)×R^d), a viscosity solution to the following equation:











∂tu=R(x, I(t)) +b(x, I(t))R

K(z)e^∇u·zdz, max

x∈R^d

u(t, x) = 0, ∀t >0, u(0, x) =u⁰(x),

(17)

I(t)−→

→0 I(t) a.e., Z

n(t, x)dx=I(t) a.e.. (18)

In particular, a.e. in t, supp n(t,·)⊂ {u(t,·) = 0}. As above, the measure nis the weak limit of n

as vanishes.

These theorems improve previous results proved in [19, 8, 7, 29] in various directions. For the case where mutations are described by a Laplace equation, i.e. (1)-(2), Theorem 1.1 generalizes the assumptions on the initial data. This generalization derives from regularizing effects of Eikonal Hamil- tonian (see [26, 1, 2]). But our motivation is more in the case of equations (3)-(4) where mutations are described by an integral operator. Then we can treat cases where the mutation rateb(x, I) really depends onx, which was not available until now. The difficulty here is that Lipschitz bounds on the initial data are not propagated on u and may blow up in finite time (see [12, 5, 15] for regularity results for integral Hamiltonian). However, we achieve to control the Lipschitz norm by −u, that goes to infinity as|x|goes to +∞.

We do not discuss the uniqueness for equations (15) and (17) in this paper. The latter is studied, for some particular cases, in [8, 7].

A related, but different, situation arises in reaction-diffusion equations as in combustion (see [6, 9, 10, 21, 23, 30]). A typical example is the Fisher-KPP equation, where the solution is a progressive front. The dynamics of the front is described by a level set of a solution of a Hamilton-Jacobi equation.

The paper is organized as follows. In section 2 we state some existence results and bounds on n

and I. In section 3 we prove some regularity results foru corresponding to equations (1)-(2). We show that u are locally uniformly bounded and continuous. In section 4 we prove some analogous regularity results for u corresponding to equations (3)-(4). Finally, in section 5 we describe the asymptotic behavior of u and deduce the constrained Hamilton-Jacobi equation (15)-(16).

2 Preliminary results

We recall the following existence results for n and a priori bounds forI (see also [8, 17]).

Theorem 2.1. With the assumptions (5)-(8), and Im−C² ≤I(0)≤ I_M +C², there is a unique solution n∈C(R⁺;L¹(R^d)) to equations (1)-(2) and it satisfies

I_m⁰ =Im−C² ≤I(t)≤I_M +C²=I_M⁰ , (19) where C is a constant. This solution, n(t, x), is nonnegative for all t≥0.

We recall a proof of this theorem in Appendix A. We have an analogous result for equations (3)-(4):

Theorem 2.2. With the assumptions (8), (11)-(14), andIm ≤I(0)≤IM, there is a unique solution n∈C(R⁺;L¹∩L^∞(R^d)) to equations (3)-(4) and it satisfies

I_m ≤I(t)≤I_M. (20)

This solution, n(t, x), is nonnegative for allt≥0.

This theorem can be proved with similar arguments as Theorem 2.1. A uniform BV bound onI(t) for equations (1)-(2) is also proved in [8]:

Theorem 2.3. With the assumptions (5)-(9), we have additionally to the uniform bounds (19), the locally uniform BV and sub-Lipschitz bounds

d

dtI(t)≥ − C+e^−Lt Z

ψ(x)n⁰(x)R(x, I⁰)

dx, (21)

d

dt%(t)≥ −Ct+ Z

(1 +ψ(x))n⁰(x)R(x, I⁰)

dx, (22)

where C and L are positive constants and %(t) =R

R^dn(t, x)dx. Consequently, after extraction of a subsequence,I(t) converges a.e. to a functionI(t), asgoes to 0. The limitI(t) is nondecreasing as soon as there exists a constant C independent of such that

Z

ψ(x)n⁰(x)R(x, I⁰)

≥ −Ce^o(1) . We also have a local BV bound onI(t) for equations (3)-(4):

Theorem 2.4. With the assumptions (8)-(14), we have additionally to the uniform bounds (20), the locally uniform BV bound

d

dtI(t)≥ −C⁰+e^−L

0t

Z

n⁰(x)R(x, I⁰) +b(x, I⁰)

dx, (23)

Z T 0

|d

dtI(t)|dt≤2C⁰T+C⁰⁰, (24)

where C⁰, C⁰⁰ and L⁰ are positive constants. Consequently, after extraction of a subsequence, I(t) converges a.e. to a function I(t), as goes to 0.

This theorem is proved in Appendix B.

3 Regularity results for equations (1)-(2)

In this section we study the regularity properties of u = lnn, where n is the unique solution of equations (1)-(2). We have

∂tn = 1

∂tue^u , ∇n = 1

∇ue^u , 4n = 1

4u+ 1

²|∇u|² e^u .

Consequentlyu is a smooth solution to the following equation

(∂tu−4u=|∇u|²+R(x, I(t)), x∈R, t≥0,

u(t= 0) =lnn⁰. (25)

We have the following regularity results for u.

Theorem 3.1. Assume (5)-(10) and let T >0 be given. Set D= B+ (A² +K₂)T. Then we have u ≤D². For all t₀>0, v =√

2D²−u are locally uniformly bounded and Lipschitz in [t₀, T]×R^d,

|∇v| ≤C(T)(1 + 1

√t₀), (26)

where C(T) is a constant depending on T, K1, K2, A and B. Moreover, if we assume that (u⁰) :=

ln(n⁰) is a sequence of uniformly continuous functions, then u are locally uniformly bounded and continuous in[0,∞[×R^d.

We prove Theorem 3.1 in several steps. We first prove an upper bound, then a regularizing effect inx, then local L^∞ bounds, and finally a regularizing effect in t.

3.1 An upper bound for u

From assumption (9) we have u⁰(x)≤ −A|x|+B. We claim that, with C=A²+K2,

u(t, x)≤ −A|x|+B+Ct, ∀t≥0. (27)

Define φ(t, x) =−A|x|+B+Ct. We have

∂_tφ−4φ− |∇φ|²−R(x, I(t))≥C+A(d−1)

|x| −A²−K₂ ≥0.

HereK2 is an upper bound forR(x, I) according to (8). We have also φ(0, x) =−A|x|+B≥u⁰(x).

So φ is a super-solution to (25) and (27) is proved.

3.2 Regularizing effect in space Letu=f(v), where f is chosen later. We have

∂_tu=f⁰(v)∂_tv, ∂_xu=f⁰(v)∂_xv, 4u=f⁰(v)4v+f⁰⁰(v)|∇v|². So equation (25) becomes

∂tv−4v−

f⁰⁰(v)

f⁰(v) +f⁰(v)

|∇v|²= R(x, I)

f⁰(v) . (28)

Define p=∇v. By differentiating (28) we have

∂_tp_i−4p_i−2

f⁰⁰(v)

f⁰(v) +f⁰(v)

∇v· ∇p_i−

f⁰⁰⁰(v)

f⁰(v) −f⁰⁰(v)²

f⁰(v)² +f⁰⁰(v)

|∇v|²p_i

=−f⁰⁰(v)

f⁰(v)²R(x, I)pi+ 1 f⁰(v)

∂R

∂xi

.

We multiply the equation bypi and sum over i:

∂_t|p|²

2 −X

(4p_i)p_i−2

f⁰⁰(v)

f⁰(v) +f⁰(v)

∇v· ∇|p|² 2 −

f⁰⁰⁰(v)

f⁰(v) −f⁰⁰(v)²

f⁰(v)² +f⁰⁰(v)

|p|⁴

=−f⁰⁰(v)

f⁰(v)²R(x, I)|p|²+ 1

f⁰(v)∇_xR·p.

First, we compute P

i(4p_i)p_i. X

i

(4p_i)pi =X

i

4p²_i 2 −X

|∇p_i|²

=4|p|²

2 −X

|∇p_i|²

=|p|4|p|+|∇|p||²−X

i

|∇p_i|².

We also have

|∇|p||² =X

i

|p·∂_xip|²

|p|² ≤X

i

|∂_xip|²=X

i,j

|∂_xipj|² =X

j

|∇p_j|².

It follows that

X

i

(4p_i)p_i ≤ |p|4|p|.

We deduce

∂t|p| −4|p| −2

f⁰⁰(v)

f⁰(v) +f⁰(v)

p· ∇|p| −

f⁰⁰⁰(v)

f⁰(v) −f⁰⁰(v)²

f⁰(v)² +f⁰⁰(v)

|p|³ (29)

≤ −f⁰⁰(v)

f⁰(v)²R(x, I)|p|+ 1

f⁰(v)∇_xR· p

|p|. From (27) we know that, for 0 ≤t ≤ T, u ≤D(T)², where D(T) = √

B+CT. Then we define f(v) =−v²+ 2D², for vpositive, and thus

D(T)≤v, f⁰(v) =−2v, and | 1

f⁰(v)|= 1 2v ≤ 1

2D,

f⁰⁰(v) =−2, and |f⁰⁰(v) f⁰(v)²|= 1

2v² ≤ 1 2D²,

f⁰⁰⁰(v) = 0, −

f⁰⁰⁰(v)

f⁰(v) −f⁰⁰(v)²

f⁰(v)² +f⁰⁰(v)

= 2 +1 v² >2.

From (29), Theorem 2.1, assumption (8) and these calculations we deduce

∂|p|

∂t −4|p| −2

f⁰⁰(v)

f⁰(v) +f⁰(v)

p· ∇|p|+ 2|p|³− K₂

2D²|p| − K₂ 2D ≤0.

Thus forθ(T) large enough we can write

∂|p|

∂t −4|p| −2

f⁰⁰(v)

f⁰(v) +f⁰(v)

p· ∇|p|+ 2(|p| −θ)³ ≤0. (30) Define the function

y(t, x) =y(t) = 1 2√

t +θ.

Since y is a solution to (30), and y(0) =∞ and |p|being a sub-solution we have

|p|(t, x)≤y(t, x) = 1 2√

t+θ.

Thus forv =√

2D²−u, we have

|∇v|(t, x)≤ 1 2√

t+θ(T), 0< t≤T. (31)

See Appendix C for more details on the comparison principle used above.

3.3 Regularity in space of u near t= 0

Assume that u⁰ are uniformly continuous. We show that u are uniformly continuous in space on [0, T]×R^d.

For δ > 0 we prove that for h small |u(t, x+h) −u(t, x)| < δ. To do so define w(t, x) = u(t, x+h)−u(t, x). Sinceu⁰ are uniformly continuous, for h small enough |w(0, x)|< ^δ₂. Besides w satisfies the following equation:

∂_tw(t, x)−4w(t, x)−(∇u(t, x+h) +∇u(t, x))· ∇w(t, x) =R(x+h, I(t))−R(x, I(t)).

From Theorem 2.1 and using assumption (8) we have

∂tw(t, x)−4w(t, x)−(∇u(t, x+h) +∇u(t, x))· ∇w(t, x)≤K2|h|.

Therefore by the maximum principle we arrive at max

R^d

|w(t, x)|<max

R^d

|w(0, x)|+K2|h|t.

So for h small enough|u(t, x+h)−u(t, x)|< δon [0, T]×R^d.

3.4 Local bounds for u

We show thatuare bounded on compact subsets of ]0,∞[×R^d. We already know from section 3.1 that u is locally bounded from above. We show that it is also bounded from below onC= [t0, T]×B(0, R), for all R >0 and 0< t₀< T.

From section 3.1 we have u(t, x) ≤ −A|x|+B+CT. So for R large enough there exists 0 such that for < ₀

Z

|x|>R

e^u dx <

Z

|x|>R

e^−A|x|+B+CT dx < I_m⁰ 2ψ_M. We have also from (19) that

Z

R^d

e^u dx > I_m⁰ ψ_M. We deduce that forR large enough and for all 0< < 0

Z

|x|<R

e^u dx > I_m⁰ 2ψ_M. Therefore there exists₁>0 such that, for all < ₁

∃x₀∈R^d; |x₀|< R, u(t, x0)>−1, thus v(t, x0)<p

2D²+ 1.

From Section 3.2 we know that v are locally uniformly Lipschitz

|v(t, x+h)−v(t, x)|< C(T) + 1 2√ t₀

|h|,

Thus for all (t, x)∈ C and < 1

v(t, x)< E(t₀, T, R) :=p

2D²(T) + 1 + 2 C(T) + 1 2√ t0

R.

It follows that

u(t, x)>2D²(T)−E²(t₀, T, R).

We conclude that u are uniformly bounded from below onC.

If we assume additionally that u⁰ are uniformly continuous, with similar arguments we can show thatu are bounded on compact subsets of [0,∞[×R^d. To prove the latter we use uniform continuity of u instead of the Lipschitz bounds of v.

3.5 Regularizing effect in time

From the above uniform bounds and continuity results we can also deduce uniform continuity in time i.e. for all η > 0, there exists θ > 0 such that for all (t, s, x) ∈ [0, T]×[0, T]×B(0,^R₂), such that 0< t−s < θ, and for all < 0 we have:

|u(t, x)−u(s, x)| ≤2η.

We prove this with the same method as that of Lemma 9.1 in [4] (see also [11] for another proof of this claim). We prove that for any η >0, we can find positive constantsA,B large enough such that, for any x∈B(0,^R₂),s∈[0, T] and for every < ₀,

u(t, y)−u(s, x)≤η+A|x−y|²+B(t−s), for every (t, y)∈[s, T]×B(0, R), (32) and

u(t, y)−u(s, x)≥ −η−A|x−y|²−B(t−s), for every (t, y)∈[s, T]×B(0, R). (33) We prove inequality (32), the proof of (33) is analogous. We fix (s, x) in [0, T[×B(0,^R₂). Define

ξ(t, y) =u(s, x) +η+A|y−x|²+B(t−s), (t, y)∈[s, T[×B(0, R),

where A and B are constants to be determined. We prove that, for A and B large enough, ξ is a super-solution to (25) on [s, T]×B(0, R) andξ(t, y)> u(t, y) for (t, y)∈ {s}×B(0, R)∪[s, T]×∂B(0, R).

According to section 3.4,u are locally uniformly bounded, so we can take A a constant such that for all < ₀,

A≥ 8kuk_L^∞_{([0,T]×B(0,R))}

R² .

With this choice, ξ(t, y)> u(t, y) on [0, T]×∂B(0, R), for all η,B andx∈B(0,^R₂). Next we prove that, for A large enough, ξ(s, y) > u(s, y) for all y ∈ B(0, R). We argue by contradiction. Assume that there exists η >0 such that for all constantsA there existsy_A, ∈B(0, R) such that

u(s, y_A,)−u(s, x)> η+A|y_A,−x|². (34) It follows that

|y_A,−x| ≤ r2M

A ,

where M is a uniform upper bound for k u k_L∞([0,T]×B(0,R)). Now let A → ∞. Then for all ,

|y_A,−x| → 0. According to Section 3.3, u are uniformly continuous on space. Thus there exists h >0 such that if|y_A,−x| ≤hthen|u(s, y_A,)−u(s, x)|< ^η₂, for all. This is in contradiction with (34). Therefore ξ(s, y) > u(s, y) for all y ∈ B(0, R). Finally, noting that R is bounded we deduce that forB large enough,ξ is a super-solution to (25) in [s, T]×B(0, R). Sinceu is a solution of (25) we have

u(t, y)≤ξ(t, y) =u(s, x) +η+A|y−x|²+B(t−s) for all (t, y)∈[s, T]×B(0, R).

Thus (32) is satisfied for t≥s. We can prove (33) for t≥s analogously. Then we putx =y and we conclude taking θ < _B^η.

4 Regularity results for equations (3)-(4)

In this section we study the regularity properties of u = lnn, where n is the unique solution of equations (3)-(4) as given in Theorem 2.2. From equation (3) we deduce that u is a solution to the following equation

(

∂tu =R(x, I(t)) +R

K(z)b(x+z, I)e^{u(t,x+z)− u(t,x)}dz, x∈R, t≥0,

u(t= 0) =lnn⁰. (35)

We have the following regularity results for u.

Theorem 4.1. Let n be the solution of (3)-(4) with n⁰ =e^u

0

, and u=ln(n). With the assump- tions (8)-(14), and if we assume that (u⁰) is a sequence of uniformly bounded functions in W^1,∞, thenu are locally uniformly bounded and Lipschitz in [0,∞[×R^d.

As in section 3 we prove Theorem 4.1 in several steps. We first prove an upper and a lower bound on u, then local Lipschitz bounds in space and finally a regularity result in time.

4.1 Upper and lower bounds on u

From assumption (9) we have u⁰(x)≤ −A|x|+B. As in section 3.1 we claim that

u(t, x)≤ −A|x|+B+Ct, ∀t≥0. (36)

Define v(t, x) =−A|x|+B+Ct, whereC=b_MR

K(z)e^A|z|dz+K2. Using (8) and (12) we have

∂tv−R(x, I(t))− Z

K(z)b(x+z, I)e

v(t,x+z)−v(t,x)

dz ≥C−K2−bM

Z

K(z)e^A|z|dz ≥0.

We also have v(0, x) = −A|x|+B ≥u⁰(x). So v is a supersolution to (35). Since (3) verifies the comparison property, equation (35) verifies also the comparison property, i.e. if v and u are respec- tively super and subsolutions of (35) then u≤v. Thus (36) is proved.

To prove a lower bound onu we assume thatu⁰ are locally uniformly bounded. Then from equation (35) and assumption (8) we deduce

∂_tu(t, x)≥ −K₂, and thus

u(t, x)≥ −ku⁰k_L∞(B(0,R))−K₂t, ∀x∈B(0, R).

Moreover,|∇u⁰|being bounded, we can give a lower bound inR^d

u(t, x)≥inf

u⁰(0)− k∇u⁰k_L^∞|x| −K₂t, ∀x∈R^d. (37) 4.2 Lipschitz bounds

Here we assume thatu is differentiable inx (See [15]). See also Appendix D for a proof without any regularity assumptions onu.

Letp =∇u·χ, whereχis a fixed unit vector. By differentiating (35) with respect toχ we obtain

∂tp(t, x) =∇R(x, I(t))·χ+ Z

K(z)∇b(x+z, I)·χ e^{u(t,x+z)− u(t,x)}dz +

Z

K(z)b(x+z, I)p(t, x+z)−p(t, x)

eu(t,x+z)−u(t,x)

dz.

Thus, using assumptions (8) and (12), we have

∂tp(t, x)≤K2+L1

Z

K(z)b(x+z, I)e^u(t,x+z)−u(t,x)

dz (38)

+ Z

K(z)b(x+z, I)p(t, x+z)−p(t, x)

e^{u(t,x+z)− u(t,x)}dz.

Define w(t, x) = p(t, x) +L₁u(t, x) and ∆(t, x, z) = ^{u(t,x+z)−u (t,x)}. From (38) and (35) we deduce

∂_tw−K₂(1 +L₁)− Z

K(z)b(x+z, I)w(t, x+z)−w(t, x)

e^∆(t,x,z)dz

≤2L1

Z

K(z)b(x+z, I)e^∆(t,x,z)dz

−L₁ Z

K(z)b(x+z, I)∆(t, x, z)e^∆(t,x,z)dz

=L₁ Z

K(z)b(x+z, I)e^∆(t,x,z) 2−∆(t, x, z) dz

≤L1bMe,

noticing that e is the maximum of the function g(t) = e^t(2−t) in R. Therefore by the maximum principle, withC₁=K₂(1 +L₁) +L₁b_Me, we have

w(t, x)≤C₁t+ max

R^d

w(0, x).

It follows that

p(t, x)≤C₁t+k ∇u⁰ k_L^∞ +L₁(B+Ct) +L₁ k∇u⁰k_L^∞|x|+K₂t−u⁰(x= 0)

(39)

=C2t+C3|x|+C4,

whereC₂,C₃ andC₄ are constants. Since this bound is true for any|χ|= 1, we obtain a local bound on |∇u|.

4.3 Regularity in time

In section 4.2 we proved thatu is locally uniformly Lipschitz in space. From this we can deduce that

∂_tu is also locally uniformly bounded.

LetC= [0, T]×B(x0, R) andS1 be a constant such thatkuk_L^∞_(C)< S1 for all >0. Assume that R⁰ is a constant large enough such that we haveu(t, x)<−S₁ in [0, T]×R^d\B(x₀, R⁰). According to (36) there exists such constant R⁰. We choose a constant S₂ such that k ∇u k_L∞([0,T]×B(x0,R⁰))< S₂ for all >0. We deduce

|∂_tu| ≤ |R(x, I(t))|+ Z

K(z)b(x+z, I)eu(t,x+z)−u(t,x)

1|x+z|<R⁰+1|x+z|≥R⁰

dz

≤K₂+b_M Z

K(z)e^S2|z|1|x+z|<R⁰dz+b_M Z

K(z)1|x+z|≥R⁰dz

≤K2+bM 1 + Z

K(z)e^S2|z|dz . This completes the proof of Theorem 4.1.

5 Asymptotic behavior of u

Using the regularity results in sections 3 and 4, we can now describe the asymptotic behavior of u

and prove Theorems 1.1 and 1.2. Here we prove Theorem 1.1. The proof of Theorem 1.2 is analogous, except the limit of the integral term in equation (17). The latter has been studied in [19, 12, 7, 29].

Proof of theorem 1.1. step 1 (Limit) According to section 3, u are locally uniformly bounded and continuous. So by Arzela-Ascoli Theorem after extraction of a subsequence, u converges locally uni- formly to a continuous functionu.

step 2 (Initial condition)We proved that if u⁰ are uniformly continuous then u will be locally uniformly bounded and continuous in [0, T]×R^d. Thus we can apply Arzela-Ascoli neart= 0 as well.

Therefore we haveu(0, x) = lim

→0u(0, x) =u⁰(x).

step 3 max

x∈R^d

u = 0

Assume that for some t, x we have 0 < a ≤ u(t, x). Since u is continuous u(t, y)≥ ^a₂ on B(x, r), for somer >0. Thus we haven(t, y)→ ∞, while→0. ThereforeI(t)→ ∞ while →0. This is a contradiction with (19).

To prove that max

x∈R^d

u(t, x) = 0, it suffices to show that lim

→0n(t, x)6= 0, for somex∈R^d. From (27) we have

u(t, x)≤ −A|x|+B+Ct.

It follows that for M large enough

lim→0

Z

|x|>M

n(t, x)dx≤lim

→0

Z

|x|>M

e^−A|x|+B+Ct = 0. (40)

From this and (19) we deduce

→0lim Z

|x|≤M

n(t, x)dx≥ I_m⁰ ψ_M. If u(t, x) < 0 for all |x| < M then lim

→0 e^u(t,x) = 0 and thus lim

→0

R

|x|≤Mn(t, x)dx = 0. This is a contradiction with (40). It follows that max

x∈R^d

u(t, x) = 0, ∀t >0.

step 4 (supp n(t,·)⊂ {u(t,·) = 0})Assume thatu(t0, x0) =−a <0. Sinceuare uniformly contin- uous in a small neighborhood of (t0, x0), (t, x)∈[t0−δ, t0+δ]×B(x₀, δ), we haveu(t, x)≤ −^a₂ <0 for small. We deduce that R

[t0−δ,t₀+δ]×B(x₀,δ)n dtdx=R

[t0−δ,t₀+δ]×B(x₀,δ)lim

→0e^u(t,x) dtdx= 0. Therefore we have supp n(t,·)⊂ {u(t,·) = 0} for almost everyt.

step 5 (Limit equation)Finally we recall, following [8], how to pass to the limit in the equation.

Since u is a solution to (25), it follows that φ(t, x) = u(t, x)−Rt

0R(x, I(s))ds is a solution to the following equation

∂_tφ(t, x)−4φ(t, x)− |∇φ(t, x)|²−2∇φ(t, x)· Z t

0

∇R(x, I(s))ds

= Z t

0

4R(x, I(s))ds+| Z t

0

∇R(x, I(s))ds|².

Note that we have I(s) → I(s) for all s≥ 0 as goes to 0, and on the other hand, the function R(x, I) is smooth. It follows that we have the locally uniform limits

→0lim Z t

0

R(x, I(s))ds= Z t

0

R(x, I(s))ds,

→0lim Z t

0

∇R(x, I(s))ds= Z t

0

∇R(x, I(s))ds,

lim→0

Z t 0

4R(x, I(s))ds= Z t

0

4R(x, I(s))ds,

for all t ≥ 0. Moreover the functions Rt

0R(x, I(s))ds, Rt

0∇R(x, I(s))ds and Rt

0 4R(x, I(s))ds are continuous. According to step 1,u(t, x) converge locally uniformly to the continuous functionu(t, x) as vanishes. Therefore φ(t, x) converge locally uniformly to the continuous function φ(t, x) = u(t, x)−Rt

0 R(x, I(s))ds asvanishes. It follows that φ(t, x) is a viscosity solution to the equation

∂tφ(t, x)− |∇φ(t, x)|²−2∇φ(t, x)· Z t

0

∇R(x, I(s))ds

=| Z t

0

∇R(x, I)ds|². In other wordsu(t, x) is a viscosity solution to the following equation

∂_tu(t, x) =|∇u(t, x)|²+R(x, I(t)).

A Proof of theorem 2.1

A.1 Existence

LetT >0 be given and A be the following closed subset:

A ={u∈C [0, T], L¹(R^d)

, u≥0, ku(t,·)k_L1≤a},

wherea= R n⁰dx

e^K2T. Let Φ be the following application:

Φ : A→A u7→v, wherev is the solution to the following equation

(∂tv−4v= ^vR(x, I¯ u(t)), x∈R, t≥0,

v(t= 0) =n⁰. (41)

Iu(t) = Z

R^d

ψ(x)u(t, x)dx, (42)

and ¯R is defined as below

R(x, I) =¯







R(x, I) if ^I₂^m < I <2I_M, R(x,2I_M) if 2I_M ≤I, R(x,^Im₂ ) if I ≤ ^I₂^m. We prove that

(a) Φ defines a mapping ofA into itself, (b) Φ is a contraction forT small.

With these properties, we can apply the Banach-Picard fixed point theorem and iterate the con- struction with T fixed.

Assume that u ∈A. In order to prove (a) we show that v, the solution to (41), belongs to A. By the maximum principle we know thatv≥0. To prove the L¹ bound we integrate (41)

d dt

Z

vdx= Z v

R(x, I¯ _u(t))dx≤ 1 max

x∈R^d

R(x, I¯ _u(t)) Z

vdx≤ K₂

Z vdx,

and we conclude from the Gronwall Lemma that kvk_L1≤

Z n⁰dx

e^K2T =a.

Thus (a) is proved. It remains to prove (b). Let u1, u2 ∈A, v1= Φ(u1) and v2= Φ(u2). We have

∂t(v1−v2)−4(v₁−v2) = 1

(v1−v2) ¯R(x, Iu1) +v2 R(x, I¯ u1)−R(x, I¯ u2) .

Noting that k v₂ k_L1≤ a, and |R(x, I¯ _u1)−R(x, I¯ _u2)| ≤ K₁|I_u1 −I_u2| ≤ K₁ψ_M k u₁−u₂ k_L1 we obtain

d

dt kv₁−v₂k_L1≤ K₂

kv₁−v₂ k_L1 +aK₁ψ_M

ku₁−u₂ k_L1 . Usingv1(0,·) =v2(0,·) we deduce

kv1−v2k_L^∞

t L¹_x≤ aK₁ψ_M K2

(e

K2T

−1)ku1−u2 k_L^∞

t L¹_x . Thus, for T small enough such that e^K2T(e^K2T −1)< _2K ^K2

1ψ_MR

n⁰, Φ is a contraction. Therefore Φ has a fixed point and there existsn∈A a solution to the following equation

(∂_tn−4n = ⁿR(x, I(t)),¯ x∈R,0≤t≤T, n(t= 0) =n⁰.

I(t) = Z

R^d

ψ(x)n(t, x)dx,

With the same arguments as A.2 we prove that ^I₂^m < I(t) < 2I_M and thus n is a solution to equations (1)-(2) for t∈ [0, T]. We fix T small enough such that e^K2T(e^K2T −1)< _4K^K2ψm

1ψMIM. Then we can iterate in time and find a global solution to equations (1)-(2).

A.2 Uniform bounds on I(t) We have

dI dt = d

dt Z

R^d

ψ(x)n(t, x)dx= Z

R^d

ψ(x)4n(t, x)dx+ 1

Z

R^d

ψ(x)n(t, x)R(x, I(t))dx.

We define ψ_L = χ_L·ψ ∈ W^∞_2,c(R^d), where χ_L is a smooth function with a compact support such thatχ_L|_B(0,L)≡1,χ_L|_R\B(0,2L)≡0. Then by integration by parts we find

Z

R^d

ψL(x)4n(t, x)dx= Z

R^d

4ψ_L(x)n(t, x)dx.

AsL→ ∞,ψ_L converges toψ inW_loc^2,∞(R^d). Therefore we obtain

L→∞lim Z

R^d

4ψ_L(x)ndx= Z

R^d

4ψ(x)ndx,

L→∞lim Z

R^d

ψ_L(x)4n(t, x)dx= Z

R^d

ψ(x)4n(t, x)dx.

From these calculations we conclude dI

dt = Z

R^d

4ψ(x)n(t, x)dx+1

Z

R^d

ψ(x)n(t, x)R(x, I(t))dx.

It follows that

−C1

ψ_mI+1 Imin

x∈R^d

R(x, I)≤ dI

dt ≤C1

ψ_mI+ 1 Imax

x∈R^d

R(x, I).

LetC = ^C_ψ^1K1

m . As soon asI overpasses I_M +C², we haveR(x, I)<−^C_K²

1 =−²_ψ^C1

m and thus ^dI_dt becomes negative. Similarly, as soon as I becomes less than I_m −C², ^dI_dt becomes positive. Thus (19) is proved.

B A locally uniform BV bound on I

for equations (3)-(4)

In this appendix we prove Theorem 2.4. We first integrate (3) overR^d to obtain d

dtI(t) = 1

Z

n(t, x) R(x, I(t)) +b(x, I(t)) dx.

Define J(t) = _dt^dI(t). We differentiateJ and we obtain d

dtJ(t) = 1 J(t)

Z

n(t, x)∂(R+b)

∂I (x, I(t))dx + 1

² Z

R(x, I) +b(x, I)

n(t, x)R(x, I) + Z

K(y−x)b(y, I)n(t, y)dy dx.

We rewrite this equality in the following form d

dtJ(t) = 1 J(t)

Z

n(t, x)∂(R+b)

∂I x, I(t)

dx+ 1 ²

Z

n(t, x) R x, I(t)

+b x, I(t)2

dx + 1

² Z Z

K(y−x) R x, I(t)

−R y, I(t)

b y, I(t)

n(t, y)dydx + 1

² Z Z

K(y−x) b x, I(t)

−b y, I(t)

b y, I(t)

n(t, y)dydx.

It follows that d

dtJ(t)≥ 1 J(t)

Z

n(t, x)∂(R+b)

∂I x, I(t) dx+ 1

² Z

n(t, x) R x, I(t)

+b x, I(t)2

dx

−K₂+b_ML₁

Z Z

K(z)|z|b x+z, I(t)

n(t, x+z)dzdx

≥ 1 J(t)

Z

n(t, x)∂(R+b)

∂I x, I(t) dx+ 1

² Z

n(t, x) R x, I(t)

+b x, I(t)2

dx−C₁ , whereC1 is a positive constant. Consequently, using (14) we obtain

d

dt(J(t))−≤ C1

−C2

(J(t))−, with (J(t))−= max(0,−J(t)). From this inequality we deduce

(J(t))−≤ C1

C2

+ (J(0))−e⁻

C2t . With similar arguments we obtain

(J(t))₊≥ −C₁⁰

C₂⁰ + (J(0))₊e⁻

C0 2t ,

with (J(t))₊ = max(0, J(t)). Thus (23) is proved. Finally, we deduce the locally uniform BV bound (24)

Z T 0

|d

dtI(t)|dt= Z T

0

d

dtI(t)dt+ 2 Z T

0

(d

dtI(t))−dt

≤IM −Im+ 2C⁰T +O(1).

C Complement to the proof of the regularizing effect (26)

In this section, we provide some details for the comparison principle used in the proof of (26). In Subsection 3.2 we proved thatp=∇v satisfies the following (see the inequality (30))

∂|p|

∂t −4|p| −2h v −2vi

p· ∇|p|+ 2(|p| −θ)³≤0.

To apply the comparison principle we first claim the following lemma that we will prove at the end of this section.

Lemma C.1. Assume (8) and (10). Then, there exist positive constants A1, B1 and D1 such that, for all t₁ >0 and ≤1,

−A₁|x|²+B₁+D₁t

t₁ ≤u(t, x), for (t, x)∈(t₁,+∞)×R^d. (43) The above lemma implies that

D(T)≤v≤ r

2D²+ 1

t₁(B₁+D₁T +A₁|x|²), for (t, x)∈(t₁,+∞)×R^d. We deduce that, for some positive constants A2 andD2(T),

∂|p|

∂t −4|p| − 1

√t1

[A₂|x|+D₂(T)]|p| · ∇|p|+ 2(|p| −θ)³ ≤0, for (t, x)∈(t₁,+∞)×R^d. (44) Define, for (t, x)∈(t1, T]×BR(0) and for A3 to be chosen later,

z(t, x) = 1 2√

t−t₁ + A₃R²

√t₁(R²− |x|²) +θ.

We prove that, forA₃ =A₃(T) chosen large enough,zis a strict supersolution of (44) in (t₁, T]×B_R(0).

To this end, we compute

∂_tz(t, x) =− 1 4(t−t1)√

t−t1

,

∇z(t, x) = 2A3R²x

√t₁(R²− |x|²)²,

∆z(t, x) = 2A3R²

√t₁(R²− |x|²)² + 8A3R²|x|²

√t₁(R²− |x|²)³. We then replace this in (44) to obtain

∂z

∂t −∆z−^√1

t1(A2|x|+D2(T))|z∇z|+ 2(z−θ)³ =

−_4(t−t¹

1)√

t−t₁ − ^√_t ^2A3R2

1(R²−|x|²)² +^√_t^8A3R2|x|2

1(R²−|x|²)³

−^√1

t1(A₂|x|+D₂)(₂^√_t−t¹

1 +^√_t ^A3R2

1(R²−|x|²)+θ)^√_t^2A3R2|x|

1(R²−|x|²)²

+2(₂^√_t−t¹

1 +^√_t^A3R2

1(R²−|x|²))³ ≥

− ^{√ 2A3R2}

t1(R²−|x|²)² +^√_t ^8A3R4

1(R²−|x|²)³

−^√1

t1(A₂R+D₂)(₂^√_t−t¹

1 + ^√_t ^A3R2

1(R²−|x|²)+θ)^√_t ^2A3R3

1(R²−|x|²)²

+(^√_t−t³

1)_t ^A23R4

1(R²−|x|²)² + 2_t ^A33R6

1

√t1(R²−|x|²)³,