Large time behavior of solution to quasilinear chemotaxis system with logistic source

This paper deals with the quasilinear parabolic-elliptic chemotaxis system \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_{t} = \nabla\cdot(D(u)\nabla u)-\nabla\cdot(\chi u \nabla v)+\mu u- \mu u^{r}, \, \, \, x'>under homogeneous Neumann boundary conditions in a bounded domain \begin{document}$ \Omega\subset\mathbb{R}^{n} $\end{document} with smooth boundary, where \begin{document}$ \tau\in\{0, 1\} $\end{document} , \begin{document}$ \chi>0 $\end{document} , \begin{document}$ \mu>0 $\end{document} and \begin{document}$ r\geq2 $\end{document} . \begin{document}$ D(u) $\end{document} is supposed to satisfy \begin{document}$ \begin{equation*} \begin{split} D(u)\geq (u+1)^{\alpha} \, \, \, \text{with}\, \, \, \alpha>0. \end{split} \end{equation*} $\end{document} It is shown that when \begin{document}$ \mu>\frac{\chi^{2}}{16} $\end{document} and \begin{document}$ r\geq2 $\end{document} , then the solution to the system exponentially converges to the constant stationary solution \begin{document}$ (1, 1) $\end{document} .

In the last four decades, many biological phenomena such as angiogenesis, morphogenesis, immune system response have been described using mathematical model. As a subsystem, (1.1) contains the classical chemotaxis system obtained from (1.1) 1738 JIE ZHAO by setting µ = 0, which was established by Keller and Segel in 1970 [9] (see also [8,10]) to describe the collective behavior of cells type as follows    u t = ∇ · (D(u)∇u) − ∇ · (χu∇v), x ∈ Ω, t > 0, x ∈ Ω, t > 0, u(x, 0) = u 0 (x), v(x, 0) = v 0 (x), x ∈ Ω. (1.4) Since then, from a mathematical point of view, the system has been widely studied in the literature. For instance, when D(u) ≡ 1, it was shown in [17] that the system (1.4) admits a unique global solution under the condition n = 1; when τ = 1 and n = 2, it was known that the system (1.4) has a global bounded solution provided that Ω u 0 < 4π (see [16]); for the case τ = 1, n ≥ 3, there is no such threshold, when Ω is a disk, relying on a Lyapunov function, Winkler [29] showed that there exists radially symmetric solution blowing up in finite time with proper initial conditions. In general, the chemical substances are much smaller than the cells and therefore the chemicals diffuse much faster than cells. In this view, system (1.4) can be simplified to the case of a parabolic-elliptic system, i.e. (τ = 0), Nagai (see [14,15]) found a critical mass which determines the behavior of the solution. Namely, there is a critical mass m c > 0 such that the system (1.4) possesses a global and bounded solution if Ω u 0 < m c , whereas finite time blow up occurs when Ω u 0 > m c . In view of the underlying biological background, cell motility should be regarded as movement in a porous medium, for this reason, the cell movement can be described by a nonlinear function D(u) ∼ u α . The number 1 − 2 n was detected to be the critical exponent for system (1.4). More precisely, the influence of the diffusion term is greater than the attractive term and hence the solution to system (1.4) exists globally (see [2,6,20]) when α > 1 − 2 n . However, the attractive drift term prevails in the sense that α < 1 − 2 n and this leads to the solution to (1.4) blow-up in finite time (see [1,28]).
When µ > 0, a number of dynamical properties have been detected both numerically and also analytically. For example, the exceedance of corresponding carrying capacities seems possible (see [7,11,12,19,23,31,34,39]). For example, when µ > 0 is small, the chemotactic cross diffusion was shown to enforce the occurrence of solutions which attain possibly finite but arbitrarily large values (see [31]). In [19], the authors have proved that the population as a whole always persists in the sense that for any nonnegative global classical solution, there exists a lower bound for mass. In [23], it is showed that logistic dampening may prevent blow-up of solutions. Besides, when τ = 0, r = 2 and D(u) ≡ 1, the main results in [22] showed the prevention of blow up under the conditions n ≤ 2, µ > 0 or n ≥ 3, µ > n−2 n χ. For the case τ = 0 and D(u) ≥ u α , Wang et al. [24] established the boundedness of the solution to system (1.1). However, it is shown that in presence of merely certain subquadratic (generalized) logistic-type dampenings, even finite-time blow up will occur (see [32,35,39]). For instance, the author in [32] proved that when τ = 0 and r < 7 6 for n = 3, 4 or r < 1 + 1 2(n−1) for n ≥ 5, the solution to system (1.1) blows up at finite time. Under the assumptions τ = 1, D(u) ≡ 1, n ≥ 3 and Ω is a smooth bounded convex domain, Winkler [30] proved that sufficiently large µ ensures the global existence and boundedness of the solution to system (1.1). For more results on the classical Keller-Segel model and its variants, we refer the readers to [3-5, 21, 26, 27, 33, 35-37].
Recently, there is an increasing interest in studying the Keller-Segel model and it is worthwhile and challenging to investigate the large time asymptotic behavior of the solution. Tello and Winkler [22] showed that the solution to system (1.1) fulfilling provided that τ = 0, D(u) ≡ 1, r = 2 and µ > 2χ. Furthermore, Wang et al. [24] extended the results given in [22] and they proved that the quasilinear parabolic-elliptic system (1.1) admits the positive constant equilibrium (1, 1) as a global attractor when µ > 2χ. For the case τ = 1, Lin [13] established the large time asymptotic behavior of the solution to the parabolic-parabolic system (1.1). When D(u) ≡ 1, by constructing a Lyapunov function, the main results in [4] showed that the solution to the system (1.1) exponentially converges to the constant stationary solution (1,1).
Due to the lack of an effective way, the large time behavior, especially the convergence rate of the solution to quasilinear system (1.1) is still open, our first aim in the present paper is to obtain the condition about the convergence rate for solution to parabolic-elliptic quasilinear system (1.1) i.e. (τ = 0). In order to prove our main result in this direction, we construct two new Lyapunov functions and for α ∈ (0, 1) and α > 1, respectively. Relying on an estimate of the corresponding energy inequality, we can first obtain the convergence of (u, v) in L 2 (Ω) as well as L ∞ (Ω). Finally, we we establish the convergence rate of (u, v) by means of some inequalities.
Our first result reads as follows: Moreover, we can find two positive constants c(p) and δ(p) such that for all p ∈ [1, ∞), the classical solution of (1.1) satisfies Remark 1.1. Theorem 1.1 removes the assumption that r = 2, which was essential in [22,24].
Remark 1.2. Our result in this paper, together with the previous results in [24], show that the solution to system (1.1) enjoys the property (1.5) when r = 2 and µ > min{2χ, χ 2 16 }. As an interesting by-product of Theorem 1.1, let us write down To the best of our knowledge, few rigorous result seems to be known about the large time behavior and convergence rate of the solution to quasilinear parabolicparabolic model (1.1). With regard to this, the goal of the present work is to make a substantial step forward towards the large time behavior and convergence rate in quasilinear parabolic-parabolic setting. Motivated by the arguments in [21], fortunately, we construct new functions which act as Lyapunov functions α ∈ (0, 1) and α > 1, respectively. By means of the energy function above, we can establish following convergence and hence obtain In light of these premises, it seems natural and inevitable that our second result addressing asymptotic homogenization of the solution can be proven. Now we state the second result as follows. Assume the parameter µ fulfills µ > χ 2 16 and then for any initial data u 0 ∈ C 0 (Ω), the solution (u, v) of model (1.1) is global bounded and satisfies In addition, for p ∈ [1, ∞), the classical solution of (1.1) enjoys the following property where c := c(p) and δ := δ(p) are two positive constants. Similarly, from Theorem 1.2, we can obtain the following corollary.
In the proof of the main result, we will frequently use the following two inequalities.
Lemma 2.4. Suppose the assumptions as that in Theorem 1.1 hold, then the solution to system (1.1) fulfills (1.5).

Proof. Let
and it is easy to verify that s − 1 − ln s ≥ 0 for s > 0. We collect (2.1) and the first equation of (1.1) to see that for all t ∈ (0, ∞). Employing the Young's inequality, we see on another hand, we can find hold for all u ≥ 0 and r ≥ 2. A combination of (2.3)-(2.5) yields for all t ∈ (0, ∞). To estimate Ω |∇v| 2 , we test the second equation of (1.1) by (v − 1) and integrate by parts to compute (2.7) Applying the Young's inequality again, we discover (2.8) Applying (2.6) and (2.8), we deduce for all t ∈ (0, ∞), where := µ − χ 2 16 > 0. Integrating (2.9) from t 0 > 0 to t, we infer that Thanks to A(t) ≥ 0 and the boundedness of u, we conclude for all t > t 0 > 0, from [18] and the condition (1.3), we have u and u − 1 are uniform continue and hence According to (2.7) and using the Young's inequality, we arrive at and hence If (1.5) was false, we could find a positive constant l > 0, {t k } k∈N ⊂ (1, ∞), and {x k } k∈N ⊂ Ω such that t k → ∞ as k → ∞ and |u(x k , t k ) − 1| ≥ l for all k ∈ N.
Since u and ∇v are bounded in Ω × (1, ∞) according to Lemma 2.2, from two straightforward applications of well-known Hölder estimates for scalar parabolic problems (see [18]), we can claim that u and u − 1 are uniformly continuous in Ω × (1, ∞). Therefore, there exist two positive constants m and ρ fulfill for arbitrary k ∈ N . Due to the smoothness of the boundary ∂Ω, we can pick a positive constant c := inf k∈N |B ρ (x k ) ∩ Ω|, from this we infer that However, on the other hand, from Lemma 3.1, we must have as k → ∞, which is absurd and hence establishes (1.5). Lemma 2.5. Suppose the assumptions as that in Theorem 1.2 hold, then the solution to system (1.1) fulfills (1.8).

Same arguments give
Similar to the proof of Lemma 2.4, we arrive at (1.8) immediately.
3. Proof of Theorem 1.1. In this section, we treat the asymptotic behavior of the solution to system (1.1) under the condition τ = 0. Motivated by the methods in [4,21], we can derive the following estimates of the solution to (1.1) by making full use of the new Lyapunov function F (t) and G(t) as follows.
Proof. Let and it is easy to see that s α − αs + α − 1 ≥ 0 for s > 0, thus we have G(t) ≥ 0 for all t > 0. By (1.1)-(1.3) and (3.17), we observe that for all t > 0. In light of Young's inequality, we thereby conclude for all t > 0. A combination of (3.18) and (3.19) yields for all t > 0. Next, we obtain from (3.5) and (3.20) that for all t > 0. By L'Hospital's rule, we can see combine with the condition µ > χ 2 16(r−1) , it immediately implies the existence of t 3 > 0 and 4 > 0 such that for all t > t 3 and µ(r − 1)(1 − 4 ) > χ 2 16 . Thus, from (3.21)-(3.23), we arrive at d dt where 5 is a positive constant. Using L'Hospital's rule again, we can find the fact that (1.5) and (3.25) ensures the existence of t 4 > 0 such that and this entails (α − 1)α 4 Recalling (3.24) and (3.26), we derive d dt (3.27) and consequently, Collecting (3.26) and (3.28), we get and it follows that for all t > t 4 . Using the interpolation inequality and the uniformly boundedness of u to obtain where p ∈ (2, ∞) and a = 2 p ∈ (0, 1). By means of Holder's inequality, we can conclude This section deals with the parabolic-parabolic case τ = 1 and we prove the following inequality, which plays an important role to obtain the convergence rate of the solution to (1.1).
Proof. We construct a new function and it is easy to observe that H(t) ≥ 0 for all t > 0. It follows from (1.2), (1.3), (4.1) and the first equation of (1.1) that for all t > 0. We invoke Young's inequality to see that for all t > 0. Inserting (4.3) back into (4.2), we conclude for all t > 0. (3.7) and µ > χ 2 16(r−1) ensure the existence of positive constant t 5 , 6 and such that for all t > t 6 . In view of the interpolation inequality and the boundedness of u, for any p ∈ (2, ∞) there is a positive constant c such that for all t > t 6 , where a = 2 p ∈ (0, 1) and c = c( u L ∞ (Ω) ). Noting the Holder's inequality, for the case p ∈ (1, 2), we can conclude for p ∈ (1, 2) and t > t 6 . Collecting (4.16)-(4.19), we arrive at (1.9). Next, we investigate the case α > 1.
Proof. We construct a function  for all t > t 8 . Similar arguments give the desired estimates of v, combine with (4.32) and (4.33), we obtain (1.9) immediately. Finally, we are in a position to complete the proof of Theorem 1.2.
Proof of Theorem 1.2. With the aid of Lemma 2.5, we can find that the solution to system (1.1) fulfills (1.8). In addition, by Lemma 4.1 and Lemma 4.2, we obtain the desired result (1.9).