1. Introduction

Chemotaxis systems have received considerable attention because they describe several biological phenomena such as leukocyte movement, self-organization during embryonic development, wound healing and cancer growth ^[8,9]. These are phenomena where a population of cells moves towards a chemical signal emitted by a substance, or another population, called chemoattractant. Various forms of the system and boundary condition have been studied (cf. ^[5,3,6,12]).

Of special interest is the following Chemotaxis system:

where, (N = 1, 2, 3) is a bounded domain with smooth boundary , denotes the derivative with respect to the outer normal of and T > 0 is a fixed time.

The above problem arises from the study of pattern formation on animal coats, where pigment cells both respond to and produce their own chemoat-tractant [¹¹,¹⁰,⁷]. In the biological interpretation p = p(x, t) and c = c(x, t) represent the pigment cell density and the chemoattractant concentration respectively at position x and time t. The constants Dp and Dc are the cells and chemoattractant diffusion coefficient respectively, and α is the chemotaxis coefficient. It is assumed that cell population grows logistically where is the linear mitotic growth rate with r and both nonnegative constants. The chemoattractant production by the cells is given by a simple Michaelis-Menten kinetics and its consumption is linear. The constants s, β and are nonnegative.

Concerning to the well-posedness of the system (1)-(4) many advances have been done in the recent years ^[13,2] and ^[1]. Specially, in ^[1] is proven the existence and uniqueness of classical solution for all positive values of α, and r. The proof uses semigroup techniques, parabolic Schauder estimates and contraction arguments.

The aim of this paper is to get the local-in-time existence and uniqueness of a weak solution to (1)-(4) in one, two and three dimensions with proper assumptions on the initial data. Before stating our main results, we give the definition of a weak solution.

Definition 1.1. A weak solution of (1) - (4) is a pair (c, p) of functions satisfying the following conditions, c(x, t) ≥ 0 and p(x, t) ≥ 0, for a.e (x, t) Є

and for all

a.e. in [0,T].

The main result is the following existence and uniqueness theorem for weak solutions.

Theorem 1.2. with 0 ≤ c₀ and 0 ≤ p ₀ ≤ in Ω, then there exists T > 0 such that the system (1) - (4) has a unique weak solution in the sense of Definition 1.1. Furthermore, c and p belong to the space

Our proof is based on generate a convergent sequence of approximate solutions of the nonlinear system (1)-(4). To this aim, we perform a successive substitution strategy, such that the nonlinear system (1)-(4) is replaced by a sequence of linear partial differential equations.

We start taking as initial value of the iteration the weak solutions c0,p0 Є of the homogeneous system

In addition, for be the weak solutions to the nonhomogeneous system

To prove theorem 1.2 we first prove existence and uniqueness of weak solutions to the homogeneous problems (7) and (8) by applying the standard theory for linear PDE. These solutions c0 and p0 are sufficient regular, that the standard theory for linear PDE guarantee the existence and uniqueness of the successive iterates (ck,pk) k = 1, 2,... . Next, we show that the generated solutions sequence is a bounded Cauchy sequence, and its limit is the solution of (1)-(4).

2. Detail of Proof

Lemma 2.1. (Properties of iterative Sequence). Under the assumptions of theorem 1.2, there exists T > 0 such that:

(i) There exists a unique weak solution to the system (7)-(8) and (9)-(10) with conditions (3) and (4) and for every k Є it holds that

For adequate constants (Ω,T) the following estimates are satisfied

(ii) The functions p ^k , c ^k satisfy for all k Є, the following inequalities

Proof. The proof is by induction on k.

Verification for k=0: We prove, that the lemma holds for the system (7)-(8). If we write then

which simplifies to

Hence, c ⁰ (x,t) equals some solution u(x, t) of the diffusion solution, multiplied by an exponentially decay term. Since c ₀ Є H ³ (Ω) and the compatibility conditions are fulfilled trivially, the regularity theory of linear parabolic equations ^[4] implies that

and

That c ⁰ (x, t ) ≥ 0 a.e in Ω x (0, T) follows from the maximum principle for the diffusion equation.

To prove that p ⁰ satisfies the lemma, we start writing the equation (8) as follows

Since c ⁰ is known, equation (19) is linear. To show existence and uniqueness of it is sufficient to see that

(a) The coefficients and belong to

and

(b) There exist some μ > 0 and K ≥ 0 such that for all 0 ≤ t ≤ T

Where B[p, v,t] denotes the bilinear form

for p,v Є H ¹ (Ω), a.e. 0 ≤ t ≤ T.

Item (a) follows from the fact that and the Sobolev embedding of H ² () in C(Ω) for Ω open subset of , N =1, 2, 3.

In order to prove (b), first observe that by the uniformly elliptic property, there exists a constant θ > 0 such that

Furthermore, for all ε > 0,

and

Thus, for all the inequality (20) holds, with

Applying, the theory of linear parabolic equations in ^[4], we get the existence and uniqueness of the weak solution p ⁰ . In addition, since the intial data p ₀ is in H³(Ω), theorem 7.16 in ^[4] implies that p ⁰ satisfies (11), (12) and (14).

The task is now to show that p ⁰ ≥ 0. We test with (p ⁰ )^-: = min(p ⁰ , 0) the variational formulation of (7), then

After adding to both sides of (24), and applying property (20), we get

By Gronwall's lemma, we can now deduce that

since (p ⁰ )^-(0) = p ₀ ≥ 0 by assumption. Then (p ⁰ )^-(t) = 0 almost everywhere in Ω x (0, T), and therefore p ⁰ ≥ 0 almost everywhere in Ω x (0, T).

To show the upper bound of p ⁰ , we use the same trick but test now with ()⁺: = max(). As is a constant we have 0 and therefore

is equivalent to

Property (20) of B implies

Now, we use Gronwall's lemma and the fact that p₀ < p_o to deduce

Therefore ()⁺ = 0 almost everywhere in x (0, T), which yields almost everywhere in Ω x (0,T).

Induction hypothesis: Assume the lemma holds for k.

Induction step (k → k + 1): By induction hypothesis 0 ≤ p ^k (x, t) < p _∞ for a.e x Є Ω, t Є [0,T], then it is easy to see that the right hand sides

of equations (9) and (10) belong to the space L ² (0,T; L ² (Ω)). Indeed

and

Now the linear theory yields the existence of a unique weak solution of (9) and (10) with initial data (4) and boundary conditions (3). The solution (c ^k+1 ,p ^k+1) ) satisfies

In order to see that c ^k+1 and p ^k+1 satisfy the regularity properties (13) and estimate (14), we apply theorem 7.1.6 in ^[4]. Then, it is sufficient to prove that f (p ^k ) and g(p ^k ) belongs to the space L²(0, T; H²(Ω)) and (p ^k ), (p ^k ) Є L ² (0,T; L ² (Ω)). To this end, we observe that:

• The functions f (x) and g(x) in (27) are continuous differentiable functions for all x Є .

• By induction hypothesis ck and pk belong to H4(Ω) and the Sobolev embedding H4(Ω) ⊂ C2(), we have that ck and pk are C2() functions. Further, ≥ p ^k ≥ 0 almost everywhere in Ω x (0, T).

Hence f (p ^k (x,t)) and g(p ^k (x,t)) belong to H ² (Ω) a.e. t Є [0, T] and

i.e., .

In addition, since

We next show that belongs to L ² (0,T; L ² (Ω)):

We now turn to show that ck+1(x, t) ≥ 0. Consider the weak formulation of (9) and test with (ck+1)- :=min(ck+1, 0), then

Hence

which gives by integration in time

As (ck+1)-(0) = (c0)- = 0 and pk ≥ 0 by induction hypothesis, we deduce

that is to say that (ck+1)- = 0 a.e in (0,T) x Ω and therefore ck+1 ≥ 0 a.e in Ω x (0, T).

Remark 2.2. If ≥ 1 then c ^k+1 (x,t) ≤ S a.e in Ω x (0, T).

It remains to show that 0 ≤ p ^k+1 ≤ p_∞. For the positivity of p ^k+1 , we use the variational formulation of (10) and test with (p ^k+1 )^-: = min(p ^k+1 ,0), this yields

By induction hypothesis 0 ≤ p ^k ≤ p _∞ , then from (49) we get that

Adding to both sides with K as in (23), we obtain

and applying Gronwall's lemma, we can deduce that

since p ^k+1 (0) = p ₀ ≥ 0 by assumption. This yields (p ^k+1 (t))^- = 0 a.e in O Ω x (0,T) and therefore p ^k+1 ≥ 0 a.e in Ω x (0,T).

Finally, we have to show that p^k+1 is bounded from above by p_∞ a.e on Ω x (0,T). Testing the variational formulation of (10) with (p ^k+1 - p ^∞ )⁺, we find by the rules of calculus Sobolev spaces that

After adding to both sides of (50), taking in account the inequality (20) and that p ^k is bounded, we have

Gronwall's inequality and the fact that p ^k+1 (0) ≤ p _∞ imply

Thus p ^k+1 ≤ p _∞ a.e in Ω x (0,T).

This completes the induction proof.

Proof. of Theorem 1.2

Existence: We show that the iterative sequence constructed above is a Cauchy sequence, which will lead to the existence of the solution (c,p) as its limit.

Let k Є be arbitrary. Since c ^k and c ^k+1 solve (9) with the same initial data and c ^k , c ^k+1 Є L ² (0, T, H ² (Ω)) ∩ L^∞(0, T, H ¹ (Ω)), (by Lemma 2.1), then theorem 7.1.5 in ^[4] implies

for 0 < T ≤ T ₁ with T ₁ = min

Similarly, due to (10) and theorem 7.1.5 in ^[4], we estimate

Now we estimate each of the three terms separately

As n ≤ 3, by the Sobolev embedding there exist C₁ > 0 such that

for all w Є H ¹ (Ω). Then

for

Further, for

and

for

Altogether, (53), (58), (59) and (61) yield

whenever 0 < T ≤ T ₄ . That is, for T: = T ₄ the sequences {c ^k } and {p ^k } are Cauchy sequences in L^∞(0,T; H ¹ (Ω)) and there are functions c and p in L ^∞ (0,T; H ¹ (Ω)) such that

Since L ² (0,T; H ⁴ (Ω)) and L ² (0,T; H ² (Ω)) are Hilbert spaces, the uniform bounds (13) and (14) imply that for subsequences c ^kl and p ^kl

Using all these convergences in the weak formulation of (9), (10) and letting l → ∞, we conclude that (c, p) is a weak solution to (1)-(4) and also satisfies (11)-(15).

Uniqueness if (c ₁ ,p ₁ ) and (c ₂ ,p ₂ ) are two weak solutions of (1)-(4), they satisfy (62). Then

for T: = T ₄ . Therefore, both solutions coincide.

Hence, the proof of theorem 1.2 is complete.