The cell density equation contains a diffusion term which writes as the Laplacian of a nonlinear term

GLOBAL EXISTENCE AND AGGREGATION

LAURENT DESVILLETTES, YONG-JUNG KIM, ARIANE TRESCASES, AND CHANGWOOK YOON

Abstract. We consider a chemotaxis model which belongs to the class of logarithmic models. It naturally appears when one drops the assumption that microscopic scale organisms measure a macroscopic scale chemical gradient.

We show that weak solutions exist globally in time in coefficient regimes that include an aggregation phenomenon in dimensionsn∈ {1,2,3}and for large initial data. We also show the instability of constant steady states and provide numerical simulations which illustrate the formation of aggregation patterns.

1. Introduction and conclusion

The purpose of this paper is to develop a chemotaxis theory which is based on a Fokker-Planck type diffusion equation,

(1.1) u_t= ∆(γ_c(v)u), x∈Ω, t >0,

whereuis the cell density,vis the chemical concentration, Ω is a smooth bounded domain ofRⁿ (n∈ {1,2,3}), andγc is the cell motility given by

(1.2) γ_c(v) = 1

c+v^k, k >0, c≥0.

The cell density equation contains a diffusion term which writes as the Laplacian of a nonlinear term. We do not assume in the modeling that microscopic scale organisms have to measure a macroscopic scale chemical gradient. However, technically, the model covers both bounded and unbounded chemosensitivity cases. By rewriting the diffusion as

u_t= ∆(γ_c(v)u) =∇ ·(γ_c(v)∇u+γ_c^′(v)u∇v)

=∇ ·( γ_c(v)

(∇u−k v^k c+v^k

u v∇v

) ) , (1.3)

we see that, ifc >0 andk≥1, the coefficient of the advection term is bounded. If c= 0, the Fokker-Planck type diffusion gives a logarithmic model type advection,

u_t=∇ ·( γ₀(v)

(∇u−ku v∇v

) ) ,

which has unbounded chemosensitivity asv→0.

One can consider this equation as a special case of the original model of Keller and Segel [14, (6) and (9)] when the effective body ratio isα= 0 (see Appendix B for an extra discussion). In this paper, we focus on the chemotactic self-aggregation phenomenon where the chemical concentrationvis assumed to satisfy

(1.4) v_t=ε∆v+u−v, x∈Ω, t >0.

1

This model equation indicates that the chemical is diffused with a constant dif- fusivity ε > 0, produced by cells with production rate one, and degraded with degradation rate one.

We study the problem with strictly positive smooth initial values, (1.5) u(x,0) =u0(x)>0, v(x,0) =v0(x)>0, x∈Ω.

The domain Ω⊂Rⁿ is smooth and bounded. We take the usual zero flux boundary condition,

∂_ν(γ_c(v)u) = 0, ∂_νv= 0 on∂Ω,

which is equivalent to the homogeneous Neumann boundary condition,

(1.6) ∂_νu= 0, ∂_νv= 0 on∂Ω.

The global existence of weak solutions to the system (1.1)–(1.6) is obtained in Section 2. Keller and Segel introduced their first chemotaxis equations to explain the initiation of the aggregation phenomenon of slime mold. They considered a logarithmic model [13, (2.3) and (2.4)] and viewed the aggregation as the insta- bility of constant steady states. They showed that the instability occurs when the chemosensitivity is greater than the diffusivity, which corresponds to the model equation (1.3) withk > 1 and c = 0. However, the global existence of strong or weak solutions for the logarithmic model has been obtained only for the cases when the constant steady state is stable, or in specific cases (low dimension, very weak (renormalized) solutions, etc. see discussion in Appendix C). Our main goal is to obtain the instability and the global existence at the same time. The global exis- tence obtained in this paper, Theorem 2.1, includes cases of multi-space dimensions and both bounded and unbounded chemosensitivity cases c ≥0. More precisely, the global existence is obtained for 0 < k <7/3 if n= 1, for 0< k <2 if n= 2, and for 0< k <4/3 ifn= 3 without any smallness assumption on the initial data or parameters.

The instability of homogeneous steady states is obtained in Section 3. For the case withc = 0, it is shown that constant steady states are unstable if ε < ^k_µ⁻¹

1 , whereµ₁>0 is the principal eigenvalue of the Laplace operator−∆ on Ω (see [35]).

The population size is not involved in this unbounded chemosensitivity case. We ex- tend the instability analysis in Theorem 3.1 to include the bounded chemosensitivity case with c >0. If c >0, there is a minimum population size required for aggre- gation (that is,( _c

k−1

)¹_k

|Ω|). If the average of the population is u > u₁:=( _c

k−1

)¹_k andε < ^(k_µ^{−1)¯uk−c}

1(c+¯u^k) , then the constant state is unstable.

Numerical simulations for the aggregation phenomenon are given in Section 4.

We can observe from numerical simulations that the instability conditions in The- orem 3.1 provide sharp bounds on coefficients and population size for a pattern formation whenc̸= 0. If the domain size is small and the population size is bigger than a critical one for a given diffusivityε >0, we obtain a single hump solution.

If the domain size is large, multiple peaks appear in the first stage and then are combined to eventually form a larger single peak. We can observe a similar be- havior for the solutions for all space dimensions. Remember that the behavior of solutions for the minimal model (cf. Appendix B) is different when different space dimensions n ∈ {1,2,3} are considered. In Appendices B and C we discuss the results of this paper and compare them to the ones in the literature. In particular,

a brief comparison between the original Keller-Segel equations and the ones in this paper is given.

2. Global existence

In this section, we show the existence of global weak solutions to the initial- boundary problem (1.1)–(1.6) for arbitrary (nonnegative) initial data in a suitable functional space. This result of existence holds for arbitrary ε > 0 and c ≥ 0.

When the dimension isn= 1, we assume that 0< k <7/3. When the dimension is n = 2, we assume that 0 < k < 2. When the dimension is n = 3, we assume that 0< k <4/3. Note that for any dimensionn∈ {1,2,3}, the considered range of k includes real numbers which are strictly larger than 1, corresponding thus to the regime in which a steady state solution is not always stable. Therefore, an aggregation phenomenon may happen for small εor for large total population as shown in Theorem 3.1. The ratio of the chemosensitivity over the diffusivity of the model (1.3), which is k_c+v^vk−1k, is unbounded for small v and c = 0. This unboundedness had been up to now a key difficulty in obtaining the global existence of a logistic type equations when aggregation may occur.

We provide in Appendix C a discussion on the assumptions used in our work, and on the link with other works on the same subject, including the important recent paper [30] by Tao and Winkler.

We will use the following definition of (very) weak solution: for nonnegative initial data (u₀,v₀) in (L¹(Ω))², a pair of nonnegative functions (u, v) onR₊×Ω such thatu,vandγ_c(v)ulie inL¹([0, T]×Ω) for allT >0, is a (very) weak solution (global in time) of (1.1)–(1.6) onR₊×Ω if the equations

−

∫

R+

∫

Ω

u(t, x)∂tψ(t, x)dxdt−

∫

Ω

u0(x)ψ(0, x)dx

=

∫

R+

∫

Ω

γc(v(t, x))u(t, x) ∆ψ(t, x)dxdt,

−

∫

R₊

∫

Ω

v(t, x)∂_tψ(t, x)dxdt−

∫

Ω

v₀(x)ψ(0, x)dx

=

∫

R+

∫

Ω

ε v(t, x) ∆ψ(t, x)dxdt

+

∫

R+

∫

Ω

(u(t, x)−v(t, x))ψ(t, x)dxdt, (2.1)

are satisfied for all compactly supported test function ψ∈C_c²(R₊×Ω) such that

∂νψ= 0 onR+×∂Ω.

More precisely, we show the following theorem:

Theorem 2.1. Let Ω be a bounded smooth (C²) open subset of Rⁿ, for n ∈ {1,2,3}. We consider c ≥ 0, ε > 0 and 0 < k < 7/3 if n = 1, 0 < k < 2 if n= 2,0 < k <4/3 if n= 3. Let u0 :=u0(x)≥0 lying inL¹(Ω)∩H_m⁻¹(Ω) and v0:=v0(x)≥c0>0 lying in W^1,2(Ω) if n= 1, lying in W^1,2−δ(Ω)(for all δ >0) if n= 2, and lying in W^{1,10−5k5−2k−δ}(Ω) (for all δ >0) if n= 3. Then, there exists

a (very) weak (global in time) solution (u, v) such thatu≥0,v≥0 of the initial- boundary problem (1.1)–(1.6) onR+×Ω(and such thatu(0,·) =u0,v(0,·) =v0).

Moreover uandv satisfy the following bounds (for anyT >0 andη >0):

• When n = 1, v, v⁻¹ ∈ L^∞([0, T]×Ω), ∂xv ∈ L⁴([0, T]×Ω), ∂tv, ∂xxv ∈ L²([0, T]×Ω),u∈L²([0, T]×Ω),u∈L^∞([0, T];L¹(Ω)),u^r∈L²([0, T];H¹(Ω)) for all0< r <1/2.

• Whenn= 2,v∈L^1/η([0, T]×Ω)∩L^∞([0, T];L¹(Ω)),v⁻¹∈L^∞([0, T]×Ω),

∇xv∈L^4−η([0, T]×Ω),∂tv,∇x∇xv∈L^2−η([0, T]×Ω),u∈L^2−η([0, T]× Ω)∩L^∞([0, T];L¹(Ω)),u^r∈L²([0, T];H¹(Ω)) for all0< r <1/2.

• Whenn= 3,v∈L^10−5k−η([0, T]×Ω)∩L^∞([0, T];L¹(Ω)),v⁻¹∈L^∞([0, T]× Ω),∇xv ∈ L^{10−5k3−k −η}([0, T]×Ω), ∂_tv,∇x∇xv ∈L^{10−5k5−2k−η}([0, T]×Ω), u∈ L^{10−5k5−2k−η}([0, T] ×Ω)∩L^∞([0, T];L¹(Ω)), u^r ∈ L²([0, T];H¹(Ω)) for all 0< r <¹₂⁴₅⁻₋^3k_2k.

We list in the remarks below some direct extensions of this theorem.

Remark 2.2 (Weak solutions). When (n= 1 orn= 2) and when (n= 3 andk <1), the solutions obtained are weak solutions, in the sense that ∇v and ∇[γ_c(v)u]

lie in L¹([0, T]×Ω) for all T > 0, so that we can use an integration by parts (or Green’s identity) in the third terms of the two equations in the very weak formulation (2.1), and replace the set of test functions by C_c¹(R+×Ω). Indeed, writing∇u=¹_ru^1−r∇(u^r) and using H¨older’s inequality, we see that for allη >0,

∇uis bounded in L^4/3([0, T]×Ω) whenn= 1, inL^4/3−η([0, T]×Ω) when n= 2, and in L^{10−5k8−3k−η}([0, T]×Ω) whenn = 3 and k < 1. Then, writing ∇[γc(v)u] = u∇γc(v) +γc(v)∇u, we obtain thanks to H¨older’s inequality that for allη > 0,

∇[γ_c(v)u] is bounded inL^4/3([0, T]×Ω) when n= 1, inL^4/3−η([0, T]×Ω) when n= 2, and inL^{10−5k8−3k−η}([0, T]×Ω) whenn= 3 andk <1.

Remark 2.3 (Regularity of the initial data). The assumption that the initial datum v0 lies in W^1,pn(Ω) (with pn = 2 when n = 1, pn = 2−η when n = 2, and p_n = ¹⁰₅₋⁻_2k^5k −η when n = 3, fore some η > 0) is used in the proof to apply the maximal regularity of the heat equation in theL^pnspace to equation (1.4). However this assumption is not optimal whenp_n<2: indeed, it suffices to assume the initial datum to be in the fractional Sobolev spaceW^2−2/pn,pn(Ω) (see for example [15]).

Therefore, it is possible to replace in the theorem the assumption “v₀∈W^1,2−0(Ω)”

by the assumption “v0 ∈ W^{1−ν,2−ν}(Ω) for allν >0”, and the assumption “v0 ∈ W^{1,10−5k5−2k−η}(Ω)” (for allη >0) by the assumtion ”v0∈W^{1−10−5kk −ν,10−5k5−2k−ν}(Ω) for allν >0”.

Remark2.4 (LlogLestimate).Whenn= 1, if we furthermore assume thatu0logu0

lies inL¹(Ω), then we have that for all timeT >0,ulogu∈L^∞([0, T];L¹(Ω)) and

√u∈L²([0, T];H¹(Ω)). Indeed, it is a consequence of the following computation (at least at the formal level)

d dt

∫

Ω

ulogu dx =

∫

Ω

(1 + logu) ∆ ( u

c+v^k )

dx=−

∫

Ω

∇u u · ∇

( u c+v^k

) dx

= −

∫

Ω

|∇u|² u

1

c+v^k dx−

∫

Ω

∇u· ∇(

c+v^k)₋1

dx

≤ −1 2

∫

Ω

|∇u|² u

1

c+v^k dx+

∫

Ω

u(c+v^k)∇(

c+v^k)−1² dx,

where, after integration on some time interval (0, T), the last term is controlled thanks to the known estimates onuandv.

Proof of Thm. 2.1. We present here thea prioriestimates related to the problem (1.1)–(1.6). Existence will then be obtained thanks to a suitable approximation process. In all the sequel,T >0 will denote any strictly positive time, andCT will denote a strictly positive constant (changing from line to line) depending onT and on the parameters of the problem (Ω,ε, c,k, nand the initial data).

First step (first estimates): Thanks to an integration w.r.t. x∈Ω, we immedi- ately see that

(2.2) sup

t∈[0,T]

||u(t,·)||L¹(Ω)+ sup

t∈[0,T]

||v(t,·)||L¹(Ω)≤CT.

Then, observing that ∂tv−ε∆v ≥ −v and using the minimum principle, we get the estimate

(2.3) inf

t∈[0,T],x∈Ωv(t, x)≥C_T⁻¹, whereC_T⁻¹:=c0exp(−T).

Second step (use of the duality lemma): Observing that

∂tu−∆(A u) = 0, ∂ν(A u)|∂Ω= 0,

whereA:= (c+v^k)⁻¹≥0 lies inL¹([0, T]×Ω) thanks to estimate (2.3), a standard duality lemma (cf. [27]) implies that

(2.4)

∫ T 0

∫

Ω

u²

c+v^k dxdt≤CT, since the initial datumu₀≥0 lies inH_m⁻¹(Ω).

Third step (estimate of the r.h.s of the equation for v): We consider m∈]1,2[

(the constants C_T will depend upon m in the sequel). We observe that, using H¨older’s inequality and estimate (2.4):

∫ T 0

∫

Ω

|u−v|^mdxdt≤C_T ( ∫ T

0

∫

Ω

( u

√c+v^k )m

(c+v^k)^m/2dxdt+

∫ T 0

∫

Ω

v^mdxdt )

(2.5) ≤CT

( 1 +

[ ∫ T 0

∫

Ω

v^2−mkm dxdt ]1−m/2

+

∫ T 0

∫

Ω

v^mdxdt )

.

Fourth step (use of the properties of the heat kernel relative to theL^p spaces):

For any ˜m∈[1,^m_n/2+1^(n/2+1)_−m[ when m≤1 +n/2 and ˜m=∞whenm >1 +n/2, we know (cf. for example [1]) that

||v||L^m˜([0,T]×Ω)≤CT

(

||∂tv−ε∆v||L^m([0,T]×Ω)+||v(0,·)||W^1,m(Ω)

) .

Observing (when ˜m > m) that for anyδ >0, there existsCδ >0 such that

∫ T 0

∫

Ω

v^mdxdt≤Cδ+δ

∫ T 0

∫

Ω

v^m˜dxdt,

and using estimate (2.5), we end up with the estimate (2.6) ||v||L^m˜([0,T]×Ω)≤CT



1 +||v(0,·)||W^2,m(Ω)+ (∫ T

0

∫

Ω

v^2−mkm dxdt

)^2−m_2m 

, which holds ifm∈]1,2[ and ˜m∈]m,^m(n/2+1)_n/2+1−m[ whenm≤1 +n/2, and if m∈]1,2[

and ˜m=∞whenm >1 +n/2.

Whenn= 1, using the bounds (2.2) and (2.6), we see that form∈]3/2,2[,

||v||L^∞([0,T]×Ω)≤CT

(

1 +||v||^(k/2_L∞([0,T^−2−m2m]×Ω)⁾⁺

) .

Remembering that in that case, we assumed thatk < 7/3, we see that by taking m >3/2 close enough to 3/2, we obtain the final estimate

(2.7) ||v||L^∞([0,T]×Ω)≤CT.

Whenn = 2, we assumed that k∈]0,2[. Then, we can takem ∈]2−k,2[ and

˜

m= ₂^km_−m, and use the bound (2.6) in order to get

||v||

L2−m^km ([0,T]×Ω)≤CT

(

1 +||v||^k/2

L2−m^km ([0,T]×Ω)

) ,

so that

||v||

L2−m^km ([0,T]×Ω)≤CT.

Then, selectingm <2 close to 2, we see that for allq∈[1,∞[, (2.8) ||v||L^q([0,T]×Ω)≤C_T.

Whenn = 3, using the bound (2.6), an interpolation and the bound (2.2), we obtain

||v||L^m˜([0,T]×Ω) ≤ C_T (

1 +||v||

2−mkm−(_2−m^km−1)₊

1−1

˜ m

L¹([0,T]×Ω) ||v||

(_2−m^km−1)₊

1−1

˜ m

L^m˜([0,T]×Ω)

)^2−m_2m

≤ C_T (

1 +||v||^m−1˜m˜ (^k2−^2−m2m )₊

L^m˜([0,T]×Ω)

) ,

for all ˜m∈] max(1,₂^km_−m),₅₋^5m_2m[, which is non empty when we takem∈]1,¹⁰₅₋⁻_2k^5k[.

Note that indeed ¹⁰₅₋⁻_2k^5k >1 since by assumptionk <5/3. Now selecting ˜m <₅₋^5m_2m close to ₅₋^5m_2m, we have for some smallν >0,

||v||L^m˜([0,T]×Ω)≤CT

(

1 +||v||(_7m−5^{5m +ν})(^k₂−²_2m^−m)₊

L^m˜([0,T]×Ω)

) .

Recalling that by assumptionk <9/5, we see that whenν >0 is small enough, the exponent in the right-hand-side satisfies

( 5m

7m−5+ν) (_k

2−²_2m^−m)

+ <1, so that for

˜

m < ₅₋^5m_2m close enough to ₅₋^5m_2m, we have

||v||L^m˜([0,T]×Ω)≤C_T.

Finally, selectingm < ¹⁰₅₋⁻_2k^5k close to ¹⁰₅₋⁻_2k^5k (therefore ˜m < ₅₋^5m_2m <10−5kis close to 10−5k), we end up with, for allq∈[1,10−5k[,

(2.9) ||v||L^q([0,T]×Ω)≤C_T.

Note that until now we only used the fact that k < 5/3. The assumption that k <4/3 will be used in the next step.

Fifth step (use of a multiplicator foru): We first consider the case whenn= 1.

Thanks to estimates (2.4) and (2.7), we see that (2.10) ||u||L²([0,T]×Ω)≤CT. Then

||∂tv−ε∆v||L²([0,T]×Ω)≤CT(||u||L²([0,T]×Ω)+||v||L²([0,T]×Ω)).

Using estimates (2.7), (2.10), and the maximal regularity of the heat equation, we obtain the bound

(2.11) ||∂tv||L²([0,T]×Ω)+||∂xxv||L²([0,T]×Ω)≤CT.

We interpolate then between the bounds (2.7) and (2.11), and see that (2.12) ||∂xv||L⁴([0,T]×Ω)≤CT.

We now use the equation foru, and compute, for anyq∈]0,1[, d

dt

∫

Ω

u^q

q dx= (1−q)

∫

Ω

u^q−2∂_xu²(c+v^k)⁻¹dx−k(1−q)

∫

Ω

u^q−1∂_xu ∂_xv v^k−1 (c+v^k)²dx.

Using Young’s inequality and estimate (2.3), we see that k(1−q)

∫

Ω

u^q−1∂xu ∂xv v^k−1 (c+v^k)²dx

≤1 2(1−q)

∫

Ω

u^q−2∂xu²(c+v^k)⁻¹dx +CT

( ∫

Ω

|∂xv|⁴dx+

∫

Ω

u^2qdx )

.

Then, integrating w.r.t. time on [0, T], thanks to the bounds (2.2), (2.10) and (2.12),

∫ T 0

∫

Ω

u^q−2∂xu²(c+v^k)⁻¹dx≤CT, so that, using again estimate (2.3), we end up with the bound, (2.13)

∫ T 0

∫

Ω

∂x(u^r)²dxdt≤CT, for allr∈]0,1/2[.

We then turn to the case whenn= 2. Thanks to estimates (2.4) and (2.8), we see that

(2.14) ||u||L^2−ν([0,T]×Ω)≤CT, for allν >0 small enough. Then

||∂_tv−ε∆v||L^2−ν([0,T]×Ω)≤C_T (

||u||L^2−ν([0,T]×Ω)+||v||L^2−ν([0,T]×Ω)

) .

Using estimates (2.8) and (2.14), and the maximal regularity of the heat equation, we obtain the bound (fori, j= 1,2)

(2.15) ||∂tv||L^2−ν([0,T]×Ω)+||∂x_ix_jv||L^2−ν([0,T]×Ω)≤CT,

for all ν > 0 small enough. We interpolate then between the bounds (2.8) and (2.15), and see that

(2.16) ||∇v||L⁴−ν([0,T]×Ω)≤C_T.

We now use the equation foru, and compute, for anyq∈]0,1[, d

dt

∫

Ω

u^q

q dx= (1−q)

∫

Ω

u^q−2|∇u|²(c+v^k)⁻¹dx−k(1−q)

∫

Ω

u^q−1∇u·∇v v^k−1 (c+v^k)²dx.

Using Young’s inequality and estimate (2.3), we see that k(1−q)

∫

Ω

u^q−1∇u· ∇v v^k−1 (c+v^k)²dx

≤1 2(1−q)

∫

Ω

u^q−2|∇u|²(c+v^k)⁻¹dx

+CT

( ∫

Ω

|∇v|^4−ν1dx+

∫

Ω

u^2−ν2dx )

,

for some ν₁, ν₂ >0 small enough (and depending onq). Then, integrating w.r.t.

time on [0, T], and thanks to the bounds (2.14) and (2.16),

∫

Ω

u^q

q (0)dx−

∫

Ω

u^q

q (T)dx+1 2

∫ T 0

∫

Ω

u^q−2|∇u|²(c+v^k)⁻¹dx≤CT, so that, remembering that q <1 and using estimates (2.2), (2.3), we end up with the bound

(2.17)

∫ T 0

∫

Ω

|∇(u^r)|²dxdt≤C_T, for allr∈]0,1/2[.

We finally treat the case whenn= 3. By estimates (2.4) and (2.9), we have that

(2.18) ||u||

L^10−5k5−2k−ν([0,T]×Ω)≤CT, for allν >0 small enough. Then

||∂_tv−ε∆v||

L^10−5k5−2k−ν([0,T]×Ω)≤C_T (

||u||

L^10−5k5−2k−ν([0,T]×Ω)+||v||

L^10−5k5−2k−ν([0,T]×Ω)

) ,

so that by the maximal regularity of the heat equation, using estimates (2.9) and (2.18), we have (fori, j∈ {1,2,3})

(2.19) ||∂tv||

L^10−5k5−2k−ν([0,T]×Ω)+||∂x_ix_jv||

L^10−5k5−2k−ν([0,T]×Ω)≤CT,

for allν >0 small enough. Interpolating between this bound and (2.9), we obtain

(2.20) || |∇v|²||

L^10−5k6−2k−ν([0,T]×Ω)≤CT.

Note that ¹⁰₆₋⁻_2k^5k >1 thanks to the assumption thatk < 4/3. We compute using the equation foruand the lower bound (2.3), for anyq∈]0,1[,

−d dt

∫

Ω

u^q

q dx+ (1−q)

∫

Ω

u^q−2|∇u|²(c+v^k)⁻¹dx≤ CT

∫

Ω

|∇v|²u^qdx,

so that by Young’s inequality, assuming furthermore thatq < ⁴₅⁻₋^3k_2k,

−d dt

∫

Ω

u^q

q dx + (1−q)

∫

Ω

u^q−2|∇u|²(c+v^k)⁻¹dx

≤ C_T ( ∫

Ω

|∇v|^{210−5k6−2k−ν1}dx+

∫

Ω

u^{10−5k5−2k−ν2}dx )

,

for some small ν1, ν2 >0 depending on q. We now integrate on [0, T], using the bounds (2.18), (2.20), (2.2), to obtain

∫ T 0

∫

Ω

u^q−2|∇u|²(c+v^k)⁻¹dxdt≤ CT. Sinceq < ⁴₅⁻₋^3k_2k, using furthermore the bound (2.3), we have (2.21)

∫ T 0

∫

Ω

|∇(u^r)|²dxdt≤CT, for allr∈]0,¹₂⁴₅⁻₋^3k_2k[.

The above a priori estimates are applied uniformy w.r.t. δ ∈]0,1[, to the ap- proximated system

∂tuδ = ∆

( uδ

c+ (δ+vδ)^k )

, (2.22)

∂tvδ = ε∆vδ+ u_δ

1 +δ uδ −vδ, (2.23)

together with Neumann boundary conditions and smoothed initial data uin,δ≥δ, vin,δ≥δcompatible with the Neumann boundary conditions.

The existence of a strong (classical) solution uδ ≥ 0, vδ ≥ 0 to the system above can be obtained thanks to Schauder fixed-point procedure. More precisely, we denote by L²₊([0, T]×Ω) the set of nonnegative functions uin L²([0, T]×Ω), and we consider the applicationT₁ defined onL²₊([0, T]×Ω) by

T1:u7→v=T1(u) = (∂t−ε∆ +Id)⁻¹ ( u

1 +δu )

,

(to be understood with initial conditionv(0,·) =vin,δand homogeneous Neumann boundary conditions), and we denote byC₊^0t,1x,1−0([0, T]×Ω) the set of nonnegative¯ functions on [0, T]×Ω which are, and whose first space derivatives are,¯ β-H¨older continuous for any 0 < β < 1, and we consider the application T2 defined on C₊^0t,1x,1−0([0, T]×Ω) by¯

T2:v7→u˜=T2(v) = (

∂t− ∇ ·

[ 1

c+ (δ+v)^k∇ − k(δ+v)^k−1∇v (c+ (δ+v)^k)²Id

])−1

(0), (to be understood with initial condition ˜u(0,·) = uin,δ and homogeneous Neu- mann boundary conditions). For any u∈ L²₊([0, T]×Ω), by the minimum prin- ciple v ≥ 0, and by the maximal regularity of the heat equation, v = T1(u) is in C₊^0t,1x,1−0([0, T]×Ω) with bounds which do not depend upon¯ u (and also in L^1/η([0, T];W^2,1/η(Ω))∩W^1,1/η([0, T]×Ω) for all η >0). In particular the com- position T2◦T1 is defined, and, using the minimum principle, ˜u is nonnegative.

Furthermore, by the maximal regularity of linear parabolic equations (see [15]), we see that ˜u=T2◦T1(u) is inL^1/η([0, T];W^2,1/η(Ω))∩W^1,1/η([0, T]×Ω) for all η >0, with bounds which do not depend uponu(and also inC₊^0t,1x,1−0([0, T]×Ω)).¯ Thanks to a Sobolev embedding, we conclude that the image ofL²₊([0, T]×Ω) by T₂◦T₁ is relatively compact inL²₊([0, T]×Ω). To check the continuity of T₂◦T₁,

we compute for any u1, u2∈L²₊([0, T]×Ω), notingvi :=T1(ui) and ˜ui:=T2(vi), i= 1,2,

d dt

∫

Ω

(˜u1−˜u2)²

2 dx

= −

∫

Ω

∇(˜u₁−u˜₂)·

( 1

c+ (δ+v1)^k∇u˜₁− 1

c+ (δ+v2)^k∇u˜₂ )

dx

−

∫

Ω

∇(˜u₁−u˜₂)·

(k(δ+v₂)^k−1∇v₂

(c+ (δ+v2)^k)² u˜₂−k(δ+v₁)^k−1∇v₁ (c+ (δ+v1)^k)² u˜₁

) dx,

so that integrating w.r.t. time on (0, T) and using the initial condition,

∫

Ω

(˜u1−u˜2)²

2 (T)dx+

∫ T 0

∫

Ω

|∇(˜u1−u˜2)|² 1

c+ (δ+v₁)^kdxdt

= −

∫ T 0

∫

Ω

∇(˜u1−u˜2)· ∇u˜2

( 1

c+ (δ+v1)^k − 1 c+ (δ+v2)^k

) dxdt

−

∫ T 0

∫

Ω

∇(˜u1−u˜2)· ∇(v2−v1) k(δ+v1)^k−1

(c+ (δ+v1)^k)²u˜1dxdt

−

∫ T 0

∫

Ω

∇(˜u₁−u˜₂)· ∇v₂

( k(δ+v₂)^k−1

(c+ (δ+v2)^k)² − k(δ+v₁)^k−1 (c+ (δ+v1)^k)²

)

˜ u₁dx

−

∫ T 0

∫

Ω

∇(˜u₁−u˜₂)· ∇v₂ k(δ+v₂)^k−1

(c+ (δ+v2)^k)²(˜u₂−u˜₁)dxdt.

Using Young’s inequality, the smoothness of ˜u₁, ˜u₂, v₁ and v₂, and the Lipschitz continuity of the functionsw7→ _c+(δ+w)¹ k andw7→ _(c+(δ+w)^k(δ+w)k−1k)², we end up with

∫

Ω

(˜u₁−u˜₂)²

2 (T)dx+1 2

∫ T 0

∫

Ω

|∇(˜u1−u˜2)|²dx

≤4C_T {∫ T

0

∫

Ω

(v₁−v₂)² dx+

∫ T 0

∫

Ω

|∇(v₂−v₁)|²dx+

∫ T 0

∫

Ω

(˜u₂−u˜₁)² dxdt }

.

Applying the maximal regularity for linear parabolic equations to the equation

∂t(v1−v2)−ε∆(v1−v2) + (v1−v2) =_1+δu^u1

1 −_1+δu^u2 ₂, we see that

||v1−v2||L²([0,T]×Ω)+||∇(v1−v2)||L²([0,T]×Ω)≤CT||u1−u2||L²([0,T]×Ω), so that we can conclude thanks to a Gronwall lemma that

∫

Ω

(˜u1−u˜2)²(T)dx≤CT

∫ T 0

∫

Ω

(u1−u2)² dxdt,

which implies the continuity ofT2◦T1onL²₊([0, T]×Ω). ConsideringT2◦T1on the closure of the convex hull ofT2◦T1(L²₊([0, T]×Ω)), we can now apply Schauder’s theorem, which gives the existence of a strong nonnegative solution (u_δ, v_δ) in (L^1/η([0, T];W^2,1/η(Ω))∩W^1,1/η([0, T]×Ω))² for all η >0, to the approximated system (2.22).

We now show how to pass to the limit in the system (2.22) when δ→ 0. Let pn, qn >1 be defined by

(p_n, q_n) = (2,∞) ifn∈ {1,2}, (p_n, q_n) = ((10−5k)/(5−2k),10−5k) ifn= 3.

According to the a priori estimates shown above, we know that we can extract from (uδ, vδ)δ>0a subsequence still denoted by (uδ, vδ)δ>0converging towards (u, v) weakly in L^pn−η([0, T]×Ω)×L^qn−η([0, T]×Ω) for all η > 0, with u, v ≥ 0. In particular, the second and third terms appearing in the second equation of the weak formulation (2.1) pass to the limit weakly.

Since we know moreover that (v_δ)_δ>0 is bounded in W^1,pn−η([0, T]×Ω) for all η > 0, we see that (v_δ)_δ>0 also converges a.e. towards v. As a consequence, (δ+c+v^k_δ)⁻¹converges a.e. towards (c+v^k)⁻¹(and is bounded inL^∞([0, T]×Ω)).

Then we can pass to the weak limit in the product (δ+c+v_δ^k)⁻¹uδ and get its convergence towards (c+v^k)⁻¹uinL^pn−η([0, T]×Ω), for allη >0.

The two remaining terms appearing in the weak formulation (2.1) also clearly pass to the limit weakly thanks to the regularity of the initial data. Therefore, (u, v) is a (very) weak solution of (1.1)–(1.6) on [0, T]×Ω. A standard Cantor’s diagonal argument allows to extend it into a global in time (very) weak solution on R+×Ω.

Finally, the estimates on the solution (u, v) are obtained by passing to the limit in the uniform estimates on (uδ, vδ) using Fatou’s lemma.

3. Instability of constant steady states

In this section we find the instability condition of a constant steady state solution to the Neumann boundary value problem, (1.1)–(1.6). The condition depends on the size of cell population, the size of chemical diffusivity, and the sensitivity of motility. However, the condition does not depend upon the size of cell diffusion.

To see it more clearly, we introduce an extra parameterεu>0 and set the motility function as

γc(v) = εu

c+v^k.

The new parameter εu controls the diffusion size of cell population. Notice that this parameter is not involved in the instability condition of the following theorem.

Theorem 3.1(Instability of a constant steady state). Letc≥0,k >1,u¯= ¯v >0, Ω be smooth and bounded, and µ1 >0 be the principal eigenvalue of the Laplace operator −∆ on Ω. Suppose thatu > u¯ 1:= (_k−^c₁)^1k. Then,ε1(¯u) := ^(k_µ^{−1)¯vk−c}

1(c+¯v^k) >0 and, if0< ε < ε1(¯u), the constant state(u, v) = (¯u,v)¯ is a linearly unstable steady state solution of (1.1)–(1.6). If ε > ε1(¯u), then (¯u,v)¯ is linearly asymptotically stable.

Proof. The case whenc= 0 has been shown in [34] and we consider the other case (c >0). The positivity of ε₁ is clear. Denoteu= ¯u+u₁,v = ¯v+v₁and consider a linearized problem,

(3.1) ∂

∂t (u₁

v₁ )

=A(¯u,¯v) (u₁

v₁ )

,

where

A(¯u,¯v) =

(γc(¯v)∆ γ_c^′(¯v)¯u∆

1 ε∆−1

) .

Let{µ, ϕ}be an eigenpair of −∆ in Ω with the homogeneous Neumann boundary condition. If{λ,c}is an eigenpair of a related matrix

B(¯u,¯v) =

(−γc(¯v)µ −γ^′_c(¯v)¯uµ

1 −εµ−1

) ,