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Boundedness in a quasilinear attraction–repulsion chemotaxis system with nonlinear sensitivity and logistic source
Boundary Value Problems volume 2019, Article number: 120 (2019)
Abstract
In this paper, we deal with the following quasilinear attraction–repulsion model:
with homogeneous Neumann boundary conditions in a smooth bounded domain \(\varOmega \subset R^{n}\) (\(n\geq 2\)). Let the chemotactic sensitivity \(\chi (v)\) be a positive constant, and let the chemotactic sensitivity \(\xi (w)\) be a nonlinear function. Under some assumptions, we prove that the system has a unique globally bounded classical solution.
1 Introduction
In this paper, we consider a quasilinear attraction–repulsion chemotaxis system with nonlinear sensitivity and logistic source
where \(\varOmega \subset R^{n}\) (\(n\geq 2\)) is a bounded domain with smooth boundary, and \(\frac{\partial }{\partial \nu }\) denotes the derivative with respect to the outer normal of ∂Ω, α, β, γ, and δ are positive parameters, and \(\chi (v)\) and \(\xi (w)\) represent chemosensitivity. We assume that the functions \(\chi (v)\) and \(\xi (w)\) satisfy the following hypotheses:
- \((H_{1})\) :
-
the function \(\chi (v)=\chi _{0}\), which is a positive constant;
- \((H_{2})\) :
-
the function \(\xi (w)=\frac{\xi _{0}}{w}\) for all \(w>0\), where \(\xi _{0}\) is a positive constant.
Here \(\chi _{0}\) is the strength of the attraction, and \(\xi _{0}\) is the strength of the repulsion, \(u(x,t)\), \(v(x,t)\), and \(w(x,t)\) denote the cell density, the concentration of the chemoattractant, and the concentration of the chemorepellent. We assume that
and there exist constants \(C_{D}>0\) and \(m\geq 1\) such that
The function \(f:[0,\infty )\rightarrow R\) is smooth and satisfies \(f(0)\geq 0\) and
with \(a\geq 0\), \(b>0\), and \(\eta >1\). The initial data comply with
Chemotaxis describes the oriented movement of cells along the concentration gradient of a chemical signal produced by cells. The prototype of the chemotaxis model, known as the Keller–Segel model, was first proposed by Keller and Segel [3] in 1970:
When \(\chi (v)\) is a positive constant, a global solution is studied by Osaki and Yagi [8] for \(n=1\); a global solution is investigated by Nagai et al. [7, 16] for \(n\geq 2\); the blowup solutions are proved by Herrero ea al. [2, 12]. For the case where \(\chi (v)\leq \frac{\chi _{0}}{(1+\alpha v)^{k}}\), \(\alpha >0\), and \(k>1\), the global classical solution is asserted by Winkler [17]. For the case \(\chi (v)=\frac{\chi }{v}\) with a positive constant \(\chi <\sqrt{\frac{2}{n}}\), a global classical solution is explored by Winkler [18].
Moreover, when \(D(u)=1\) and \(f(u)=0\), Tao and Wang [11] studied the following chemotaxis model:
The global boundedness of the solutions was obtained in high dimensions, and blowup solutions were identified in \(R^{2}\).
In the case where \(\chi (v)\) and \(\xi (w)\) are positive parameters in (1.7), \(D(u)\) satisfies (1.3), and \(f(u)\) satisfies (1.4), a unique global bounded classical solution was deduced by Wang [15]. When \(f(u)=0\) in (1.7), \(\chi (v)\) and \(\xi (w)\) are positive functions, \(D(u)\) satisfies (1.3), and \(f(u)\) satisfies (1.4), the global classical solutions are asserted by Wu and Wu [19], who obtained an important estimate of \(\int _{\varOmega } \vert \nabla v \vert ^{2}\,dx\). Note that this method is not applicable for the general \(f(u)\) in our paper. For more details about chemotaxis system, we refer the interested readers to [1, 5, 6, 9, 13, 14].
Motivated by [11, 15, 17,18,19], we consider a quasilinear attraction–repulsion chemotaxis system with nonlinear sensitivity and logistic source. Our main results are given as follows.
Theorem 1.1
Assume that (1.2)–(1.5), \((H_{1})\), and \((H_{2})\) are valid. Moreover, suppose that
and
-
(i)
If \(\sigma \in (1,\eta )\), then (1.1) admits a bounded global classical solution.
-
(ii)
If \(\sigma \in (\eta ,m)\), then (1.1) admits a bounded classical solution.
-
(iii)
If \(m>\max \{1,\frac{n\sigma +2-2\sigma }{n+2}, \frac{n\sigma -2}{n}\}\), then (1.1) admits a bounded global classical solution.
The local existence and uniqueness of system (1.1) can be derived from Lemma 2.1 in [4], and hence we only state the result and omit its proof.
Lemma 1.1
([4])
Suppose that (1.2)–(1.5) are valid. Then there exist a maximal existence time \(T_{\max }\in (0,+\infty )\) and a unique triplet (u,v,w) of functions that satisfy
with \(l>n\) and
In addition, if \(T_{\max }<+\infty \), then
Lemma 1.2
Let \((u,v,w)\) be the solution of system (1.1). Then there exist a constant \(m^{*}\) such that
Proof
Integrating the first equation of system (1.1) over Ω, we have
Due to \(\eta >1\) and Young’s inequality, we derive
Combining with (1.12), we have
which yields (1.11). □
Lemma 1.3
(Gagliardo–Nirenberg inequality)
Let \(r\in (0,\alpha )\) and \(\psi \in W^{1,2}(\varOmega )\cap L^{r}(\varOmega )\). Then there exists a constant \(C_{\mathrm{GN}}>0\) such that
with
Lemma 1.4
Let Ω be a bounded domain in \(R^{n}\) with smooth boundary, and let \(v_{0}\in W^{1,\infty }(\varOmega )\). Suppose that there exists a constant \(C_{1}\) such that
For the problem
-
(i)
if \(1\leq k< n\), then
$$\begin{aligned} \bigl\Vert v(t) \bigr\Vert _{W^{1,j}(\varOmega )}\leq C \quad \textit{for all } j \in \biggl(0,\frac{nk}{n-k} \biggr); \end{aligned}$$(1.16) -
(ii)
if \(k=n\), then (1.16) holds for all \(j\in (0,\infty )\);
-
(iii)
if \(k>n\), then (1.16) holds for \(j=\infty \).
Lemma 1.5
([20])
For any \(h\in [1,\frac{n}{n-1})\), there exists a constant \(C_{2}>0\) such that
Lemma 1.6
([21])
For any \(h\in [1,\frac{n\eta }{(n+2-\eta )^{+}})\), there exists a constant \(C_{3}>0\) such that
2 A priori estimates
Lemma 2.1
Suppose
where
If \(\sigma \in (1,\eta )\), then there exist constants \(E_{1}>0\) and \(E_{2}>0\) such that
for sufficiently large k.
Proof
Since \(\sigma \in (1,\eta )\), by Young’s inequality we have
and
Combining (2.1), (2.3), and (2.4), we get that there are positive constants \(E_{1}\) and \(E_{2}\) such that
□
Lemma 2.2
Suppose
where
If \(\sigma \in (\eta ,m)\), then there exist constants \(E_{3}>0\) and \(E_{4}>0\) such that
Proof
By Lemma 1.2 and the Gagliardo–Nirenberg inequality there exists a constant \(C_{8}>0\) such that
where
By Young’s inequality we obtain
Since \(\sigma \in (\eta ,m)\), by Young’s inequality there exist \(C_{11}>0\) and \(C_{12}>0\) such that
Hence, combining (2.5), (2.8), and (2.9), we obtain (2.6). □
Lemma 2.3
Suppose
where
If \(m> \max \{1,\frac{n\sigma +2-2\sigma }{n+2},\frac{n\sigma -2}{n} \}\), then there exist constants \(E_{5}>0\) and \(E_{6}>0\) such that
Proof
By the Gagliardo–Nirenberg inequality there exists \(C_{13}>0\) such that
where
The condition \(m>\max \{1,\frac{n\sigma +2-2\sigma }{n+2}\}\) and sufficiently large k guarantee that
Hence \(\lambda _{1}\in (0,1)\).
Since \(m>\max \{1,\frac{n\sigma -2}{n}\}\), we obtain
By Young’s inequality we derive
Therefore (2.10) and (2.13) yield (2.11). □
Lemma 2.4
Let \(n\geq 2\). Defining
and
we have
-
(a)
if \(\eta \in (1,\frac{n+2}{n}]\), \(s<\frac{m+\eta }{2}- \frac{n-1}{n}\), then for sufficiently large k, there exist \(\beta >2\) and \(h\in [1,\frac{n}{n-1})\) such that
$$\begin{aligned} \delta _{i}(k,\beta ;h)\in (0,1) \quad \textit{and} \quad w_{i}(k,\beta ;h)< 2, \quad i=2,3. \end{aligned}$$(2.16) -
(b)
if \(\eta \in (\frac{n+2}{n},n+2)\), \(s<\frac{m}{2}+ \frac{\eta (n+4)}{2(n+2)}-1\), then for sufficiently large k, there exist \(\beta >2\) and \(h\in (\frac{n}{n-1},\frac{n\eta }{n+2-\eta })\) such that (2.16) holds.
Proof
By computation we verify that (2.16) is equivalent to
Thus it is sufficient to ensure that
(a) For \(h\in [1,\frac{n}{n-1}]\), by the continuity of h it suffices to prove the case \(h=\frac{n}{n-1}\). To prove (2.17), we need to prove
Since \(s<\frac{m+\eta }{2}-\frac{n-1}{n}\), there exists
such that
so (2.18) and (2.19) are satisfied. Hence (2.17) holds.
(b) We note that \(\eta \in (\frac{n+2}{n},n+2)\) ensures the interval \(h\in (\frac{n}{n-1},\frac{n\eta }{n+2-\eta })\). By the continuity of h, let \(h=\frac{n\eta }{n+2-\eta }\). To prove (2.17), we need to show that
and
Since \(s<\frac{m}{2}+\frac{\eta (n+4)}{2(n+2)}-1\), there exists
such that
Then (2.20) and (2.21) are satisfied, and hence (2.17) holds. □
Lemma 2.5
For the second equation in (1.1), \(E>0\), and \(\beta >2\) we have
for all \(t\in [0,T_{\max })\).
Proof
The proof can be found in [18]. □
Lemma 2.6
Under assumptions (1.2)–(1.5), \((H_{1})\), and \((H_{2})\), let \(n\geq 2\) satisfy
and let \(S(u)\) and \(F(u)\) satisfy (1.8). If \(\sigma \in (1, \eta )\), there exist sufficiently large k and \(t\in [0,T_{\max })\) such that
Proof
Multiplying by \((u+1)^{k-1}\) the both sides of the first equation in (1.1), we have
for all \(t\in (0,T_{\max })\). Since \((u+1)^{\eta }\leq 2^{\eta -1}(u ^{\eta }+1)\) for \(\eta >1\), this implies that
Then (2.25) can be rewritten as
where
Similarly, we have
and then
For all \(t\in (0,T_{\max })\) with \(C_{16}>0\), we obtain
Combining (2.23), (2.27), (2.28), (2.29), and Young’s inequality, we deduce
with \(C_{17}, C_{18}, C_{19}>0\) and \(\lambda _{2}\), \(\lambda _{3}\) as shown in Lemma 2.4 for all \(t\in (0,T_{\max })\). By Lemma 1.5, Lemma 1.6, and the Gagliardo–Nirenberg inequality we have
with \(\lambda _{i}\), \(\delta _{i}\) as in Lemma 2.4, where \(i=2,3\). Since \(w_{i}=\frac{\delta _{i}\lambda _{i}}{\beta }<2\), by Young’s inequality we have
From (2.30) and (2.31) we have that there exist constants \(D_{1},D_{2},D_{3}>0\) such that
where \(D_{1}=\frac{2C_{D}k(k-1)}{(k+m-1)^{2}}\) and \(D_{2}=\frac{C_{F} \xi _{0}k(k-1)}{k+\sigma -1}\). By Lemma 2.1 we have
for all \(t\in (0,T_{\max })\). By an ODE comparison argument we obtain (2.24).
For \(\eta \in [n+2,\infty )\), from the Lemma 1.6 we have
In addition, \(s<\frac{m+\eta }{2}\) is equivalent to \(k+2s-m-1< k+ \eta -1\), so by (2.30), (2.34), and Lemma 2.1, using an ODE comparison argument, we derive (2.24). □
Remark 2.1
If \(\sigma \in (\eta ,m)\) in Theorem 1.1, then by (2.32), Lemma 2.2, and Lemma 2.4 we obtain (2.24).
Remark 2.2
If \(m>\max \{1,\frac{n\sigma +2-2\sigma }{n+2},\frac{n\sigma -2}{n}\}\) in Theorem 1.1, then by (2.32), Lemma 2.3, and Lemma 2.4 we obtain (2.24).
Proof of Theorem 1.1
For \(k>\frac{n}{2}\), by Lemmas 1.4 and 2.6 there exists a positive constant \(C_{27}\) such that
Using the elliptic regularity theory, we have
Then, for a sufficiently large k, by the Sobolev embedding theorem there exists a positive constant \(C_{29}\) such that
By using Lemma A.1 in [10] we conclude that u is uniformly bounded in \(\varOmega \times (0,T_{\max })\). Thus there exists a positive constant \(C_{30}\) such that
that is, \((u,v,w)\) is a global bounded classical solution to (1.1). □
References
Cieslak, T., Stinner, C.: New critical exponents in a fully parabolic quasilinear Keller–Segel system and applications to volume filling models. J. Differ. Equ. 258, 2080–2113 (2015)
Herrero, M., Velázquez, J.: Singularity patterns in a chemotaxis model. Math. Ann. 306, 583–623 (1996)
Keller, E., Segel, L.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)
Liu, J., Zheng, J., Wang, Y.: Boundedness in a quasilinear chemotaxis–haptotaxis system with logistic source. Z. Angew. Math. Phys. 67, 1–33 (2016)
Magdalena, L., Alexandra, C.R., Leah, E.K., Mogilner, A.: Chemotactic signaling, microglia, and Alzheimer disease senile plaques: is there a connection? Bull. Math. Biol. 65, 693–730 (2003)
Masaaki, M., Tomomi, Y.: A unified method for boundedness in fully parabolic chemotaxis systems with signal-dependent sensitivity. Math. Nachr. 290, 2648–2660 (2017)
Nagai, T., Senba, T., Yoshida, K.: Application of the Trudinger–Moser inequality to a parabolic system of chemotaxis. Funkc. Ekvacioj 40, 411–433 (1997)
Osaki, K., Yagi, A.: Structure of the stationary solution to Keller–Segel equation in one dimension. Surikaisekikenkyusho Kokyuroku 1105, 1–9 (1999)
Tao, Y., Wang, Z.: Competing effects of attraction vs. repulsion in chemotaxis. Math. Models Methods Appl. Sci. 23, 1–6 (2013)
Tao, Y., Winkler, M.: Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity. J. Differ. Equ. 252, 692–715 (2012)
Tao, Y., Winkler, M.: Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 30, 157–178 (2013)
Wang, G.: Blow-up in a chemotaxis model without symmetry assumptions. Eur. J. Appl. Math. 12, 159–177 (2001)
Wang, L., Mu, C., Hu, X., Tian, Y.: Boundedness in a quasilinear chemotaxis–haptotaxis system with logistic source. Math. Methods Appl. Sci. 40, 3000–3016 (2017)
Wang, Q.: Global solutions of a Keller–Segel system with saturated logarithmic sensitivity function. Commun. Pure Appl. Anal. 14, 383–396 (2015)
Wang, Y., Liu, J.: Boundedness in a quasilinear fully parabolic Keller–Segel system with logistic source. Nonlinear Anal., Real World Appl. 38, 113–130 (2017)
Winkler, M.: Global diffusive behavior in the higher-dimensional Keller–Segel model. J. Differ. Equ. 248, 2889–2905 (2010)
Winkler, M.: Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity. Math. Nachr. 283, 1664–1673 (2010)
Winkler, M.: Global solutions in a fully parabolic chemotaxis system with singular sensitivity. Math. Methods Appl. Sci. 34, 176–190 (2011)
Wu, S., Wu, B.: Global boundedness in a quasilinear attraction repulsion chemotaxis model with nonlinear sensitivity. J. Math. Anal. Appl. 442, 554–582 (2016)
Zhang, Q., Li, Y.: Boundedness in a quasilinear fully parabolic Keller–Segel system with logistic source. Nonlinear Anal., Real World Appl. 38, 113–130 (2017)
Zheng, J., Wang, Y.: Boundedness and decay behavior in a higher-dimensional quasilinear chemotaxis system with nonlinear logistic source. Comput. Math. Appl. 72, 2604–2619 (2016)
Acknowledgements
We would like to thank the referees for their valuable comments and suggestions to improve our paper.
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Project Supported by the National Natural Science Foundation of China (Grant No. 11571093).
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This work was carried out in collaboration between both authors. ZY designed the study and guided the research. LY performed the analysis and wrote the first draft of the manuscript. ZY and LY managed the analysis of the study. Both authors read and approved the final manuscript.
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Yan, L., Yang, Z. Boundedness in a quasilinear attraction–repulsion chemotaxis system with nonlinear sensitivity and logistic source. Bound Value Probl 2019, 120 (2019). https://doi.org/10.1186/s13661-019-1232-y
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DOI: https://doi.org/10.1186/s13661-019-1232-y