Asymptotic profile of solutions to a certain chemotaxis system

We consider a Cauchy problem for a two-dimensional model of chemotaxis and we show that large time behavior of solution is given by a multiple of the heat kernel.


1.
Introduction. Chemotaxis i.e. biological process of directed movement (e.g. of cells) towards a chemically more favorable environment appears to be in a great interest of mathematical study in recent years (see i.e. survey [3] and references therein). The most classical model describing this phenomenon is the system of parabolic equations derived by Keller and Segel ([4]). Usually, the chemical substrate is produced by cells. In this paper, we consider the opposite case i.e. when the chemical substrate is consumed by them. A typical example of such a substrate is an oxygen (see [9]). More precisely, we consider the following system of partial differential equations, describing a population of bacteria supplemented with nonnegative initial conditions n(x, 0) = n 0 (x), c(x, 0) = c 0 (x).
This system of equations was extensively studied in different contexts. First of all, it can be treated as a fluid-free version of the following, so called chemotaxis-Navier-Stokes model, initially proposed by Tuval et al. [9] n t + u · ∇n = ∆n − ∇ · (n∇c), c t + u · ∇c = ∆c − nc, u t + u · ∇u = ∆u − ∇P − n∇ϕ, ∇ · u = 0. (4) In this model the additional equation describing the velocity field (denoted by u) of the fluid determined by the incompressible Navier-Stokes model with some given gravitational force ∇ϕ and scalar function P (representing the pressure of the fluid, recovered from n and u via Calderón-Zygmund operators) is added in order to cover the situation in which both bacteria and oxygen are also transported with fluid. It is easy to see that the system (1)-(2) can be obtained by putting u(x, 0) = ∇ϕ(x) = 0 in (4) and hence results to the full chemotaxis-Navier-Stokes model applies also to our problem (1)-(2).
The models above were mostly studied in bounded domain with the standard Neuman boundary conditions. The existence and boundedness results for (1)- (2) for n ≥ 2 can be found in [7]. Considering the system (4) and its generalisation we refer the reader to [6,10,11] and references therein for results devoted to the problem of local and global-in-time existence and also uniqueness of solutions together with some asymptotic behavior properties.
The issue in the whole plane is considered in a much smaller number of publications. The global existence of the solution to the problem (4) was proved by Zhang and Zheng in [12]. The global existence of classical bounded solutions for the same system was proved in [6]. Under different initial conditions the authors obtained solutions (n, c, u) in C 2+θ,1+ θ 2 (R 2 × (0, ∞)) for some θ ∈ (0, 1). The model (4) enriched with the additional equation describing the evolution of the concentration of chemical attractant was examined by Kozono, Miura and Sugiyama in the paper [5]. Except the existence of the mild solutions they proved the estimates for decay rate for n(t) and ∇c(t). For n ≥ 2 and sufficiently small initial data it occurs For some generalisation of the system (4) the existence of classical solution and temporal decay was proved under some smallnes assumptions on L ∞ -norm of c 0 ([2]).
Notations. Throughout the paper R + = (0, ∞) and C stands for a constant which may vary from line to line. The function G denotes the heat kernel ( G(x, t) = (4πt) −1 e − |x| 2 4t ) and the symbol e t∆ denotes the heat semigroup given as a convolution with the heat kernel G(x, t).

2.
Results and comments. Since the considered model is simplification of that one investigated in [12], we recall that existence result and concentrate on large time asymptotics of the solutions, what is the main subject of the paper.
Restricting the space for the initial conditions let us define (following [12]) the space Thus, we rewrite [12, Theorem 1.1] as follows Then the initial value problem (1)-(3) has the unique, nonnegative, global-in-time solution (n, c) such that Note that with more assumptions on the initial conditions (see [6]) we get the regular, classical solutions to the problem.
The goal of this work is to study large time behaviour of such obtained solution.
In the proof of the main result, we use the mild solutions, i.e. solutions which fulfill the following Duhamel formula Thus, let us formulate the main result of this work.
Theorem 2.2 (Large-time asymptotics). Assume that (n 0 , c 0 ) ∈ X 0 and denote by (n(t), c(t)) the corresponding solution of problem (1)-(3) provided by Theorem 2.1. Then, for large times, the solutions n = n(x, t) and c = c(x, t) behave as a multiple of a heat kernel i.e. for every p ∈ [1, ∞] and every q ∈ [1, ∞) there exists constant C p > 0 such that where the number is nonnegative and finite.
With the additional condition on the second moment of the initial condition n 0 , i.e. assuming n 0 ∈ L 1 (R 2 ; (1 + |x| 2 )dx) we get with

R. CELIŃSKI AND A. RACZYŃSKI
Let us notice that the decay estimates received in the paper [5] were obtained under smallness assumption on initial data while the decay rate for n(t) and c(t) (compare the estimates (5), (6)) are worse than these proved in this paper.
The reminder of this paper is devoted to the proof of Theorem 2.2. In Section 3 we provide decay estimates of solutions using energy methods. In Section 4 we use an integral formulation of the initial value problem (1)-(3) to prove the main result.
3. Global-in-time estimates. We begin with the proof of the certain global-intime estimates of solutions to problem (1)-(3) which will be necessary in our study of their large time asymptotics.
Lemma 3.1. Let (n(t), c(t)) be the solution obtained by Theorem 2.1 for given n 0 , c 0 ∈ X 0 . Then for all t > 0 we have Proof. This is an immediate consequence of a standard reasoning for parabolic equation, see [12, Proposition 3.1] for more details.
Lemma 3.2. Let n(t) and c(t) be the solutions obtained by Theorem 2.1 for given n 0 (x) and c 0 (x). The solution n(t) as a function of t conserves the "mass", i.e.
while the solution c(t) is bounded from above, i.e.
Proof. The proof of the equality (13) is standard for chemotaxis models and obtained by integration over the whole plane of the first equation of the system, while the estimates (14) is based on the positivity of n(x, t), n 0 (x) and c(x, t) applied to Duhamel formula for the solution c(t).
The next step is estimating the decay of the norms n(t) p and c(t) p for all p ∈ [1, ∞).
First, we recall classical estimates for solutions to the heat equation.
for each f ∈ L q (R 2 ).
We have Proof. Since c(x, t) and n(x, t) are positive function, using the comparison principle for the heat equation we obtain 0 ≤ c(x, t) ≤ e t∆ c 0 for x ∈ R 2 , t > 0.
The decay of the heat semigroup (15) implies the required statement and finishes the proof of this Lemma.
Moreover for p ∈ [1, 2] we have To prove Lemma 3.5 we need boundedness of some norm of the solution c(t).
Lemma 3.6. Assume that c(t) is the solution of problem (1)-(3). Then, there exist constant C > 0 such that Proof. This estimate is direct consequence of an energy inequality (see inequality (21) below) which has been often used in the study of problem (1) In the proof we will rely on (and recall) the reasoning from [12], (where authors considered the same system) which leads to similar estimate. Note that in [12] the authors obtained only the exponentially growth of the considered term while we need its boundedness. Note that the calculations and estimates below should be done on the level of the regularized problem (as in [12]), which gives more regularity than claimed in the statement of Theorem 2.1. For simplicity we write it without introducing standard mollifier ρ ε .

R. CELIŃSKI AND A. RACZYŃSKI
Similarly, using the equality we rewrite the equation (2) as Multiplying the above equation by −∆ √ c and integrating over R 2 we get Next, by calculations analogous to those in [12, Proposition 4.1] we have Hence, plugging inequalities (25) into equation (24) we obtain By adding equation (22) to inequality (26) we get d dt R 2 (n + 1) log(n + 1) dx + 2 To estimate the integral of right-hand side of the inequality (27) we integrate by parts and use the Cauchy inequality in the following way since n is positive. Now, we multiple equation (2) by c and integrate over R 2 to obtain d dt Adding inequality (29) to (27) and using estimate (28) we obtain (21).
Integrating (21) over the interval (0, T ) we get Since all the above components are nonnegative the proof of estimate (20) is an immediate consequence of inequality below (due to (14)) Next, we prove a technical lemma which will be subsequently used later.
Then there exist C = C(p, C 0 , C 1 , C 2 ) > 0 such that Proof. For G(t) = C 2 t 0 g(s) ds, we rewrite inequality (31) in the form Since h is nonnegative, solving differential inequality (33) we obtain Applying the estimate 1 ≤ e G(t) ≤ e C2C0 we finally get Now, we are ready to prove decay estimates of solutions to problem (1)-(3) in L p -norms. First, we deal with the densities of bacteria n(x, t).

R. CELIŃSKI AND A. RACZYŃSKI
Proof of Lemma 3.5. Decay for p = 2. We multiply equation (1) by n and integrate over R 2 . After integration by parts and using the Cauchy inequality, we obtain Next, using the Hölder and Sobolev inequalities we get ≤ C ∇c 2 4 ∇n 2 n 2 . Again using the Cauchy inequality we obtain Now, by the following Nash inequality where M = n(t) 1 is independent of t. Note, that the regularity assumed in Lemma 3.7 can be obtain on the level of approximation and next passing to the limit.
Notice that, since n 0 ∈ L 1 ∩ L 2 , Theorem 2.1 together with the estimates above implies that n(t) p ≤ C(1 + t) −(1− 1 p ) for all 1 ≤ p ≤ 2, which gives us the second statement of Lemma 3.5. This implies also that there is no blow-up at t = 0 in (18).
Decay for p = 2 k for each k ∈ N. We proceed by induction assuming that decay estimate (18) holds true for p = 2 k−1 . Let p = 2 k . Similarly as above, we multiply equation (1) by n p−1 and integrate over R 2 . After integration by parts, using the Cauchy inequality, we obtain 1 p d dt R 2 n p dx + (p − 1)