Global solutions and uniform boundedness of attractive/repulsive LV competition systems
- Yuanyuan Zhang^{1}Email author
https://doi.org/10.1186/s13662-018-1513-2
© The Author(s) 2018
Received: 12 September 2017
Accepted: 2 February 2018
Published: 8 February 2018
Abstract
Keywords
1 Introduction
It is proved in [1] that, when \(N=1\), (1.1) and its fully parabolic counter-part admit global classical solutions which are uniformly bounded, and when \(N\geq2\), (1.1) admit global and uniformly bounded classical solutions provided that \(\chi<0\) and \(b_{1}\) is sufficiently large. In this current work, we extend the results in [1] on the global existence and uniform boundedness of classical solutions to (1.1) and our first main result states as follows.
Theorem 1.1
Let \(\Omega\subset\mathbb{R}^{N}, N\geq2\) be a bounded domain and \(\chi>0\) be an arbitrary positive constant. Assumes that \(a_{i}\), \(b_{i}\), \(c_{i}\) and \(D_{i}\), \(i=1,2\), are positive constants. Then, for any nonnegative \(u_{0}\in C^{0}(\bar{\Omega})\), there exists one couple \((u,v)\) of nonnegative functions which solve (1.1) classically in \(\Omega\times(0,\infty)\). Moreover, the solutions are uniformly bounded in the following sense: \(0< v(x,t)<\frac{a_{2}}{c_{2}}, \forall (x,t)\in\Omega\times(0,\infty)\) and \(\Vert u(\cdot,t)\Vert _{L^{\infty}}\leq C, \forall t\in(0,\infty)\), for some positive constant C.
Equation (1.1) is very similar to the Keller–Segel type chemotaxis system which models the aggregated movement of cellular organisms towards the region high chemical concentration [2]. However, they have quite different kinetics in light that, in chemotaxis models, it is attraction that supports patterns, while here in (1.1) it is repulsion that supports patterns, as suggested by the analysis in [1]. It is well known that large advection rate usually supports blow-ups in chemotaxis system when there is no cellular growth [3]. On the other hand, the logistic growth tends to inhibit solutions from blowing up in finite or infinite time, however, this may not be sufficient when the diffusion is weak or chemotaxis is strong [4–6]. Theorem 1.1 shows that, for the attractive Lotka–Volterra competition system, the solutions are uniformly bounded and blow up in finite or infinite time cannot occur. It is worthwhile to mention that besides the competition model, the advection, which is referred to as the prey–taxis, has been studied in predator–prey models by various authors. See [7–14].
Theorem 1.2
Let all the conditions in Theorem 1.1 hold. Suppose further that \(b_{1}>\frac{a_{2}b_{2}\chi(N-2)}{c_{2}D_{2}N}\). Then (1.2) admits global and bounded classical solutions and the statements in Theorem 1.1 hold true for (1.2).
Remark 1
We would like to point out that when \(N=2\), Theorem 1.2 holds for any \(b_{1}>0\), and then this fact, together with Theorem 1.1, implies that both (1.1) and (1.2) admit global and uniformly bounded classical solutions regardless of the sign and size of the advection rate χ. We note that fully parabolic system of (1.2) was studied by [15] recently, and global existence and boundedness were established provided that the sensitivity function decays super linearly with respect to v. A bifurcation analysis is performed to establish nontrivial patterns.
2 Preliminaries
Our proof of the global existence of (1.1) and (1.2) starts with the local existence and its extensibility criterion due to the classical theory of Amann [16] (Theorem 3.5). Indeed, it is obvious that (1.1) and (1.2) are strictly parabolic systems, therefore both admit locally classical solutions in \(\Omega \times T_{\max}\), where \(T_{\max}\in(0,\infty]\) is the so called maximal existence time. Moreover, \(T_{\max}=\infty\) if \(\Vert u(\cdot ,t)\Vert_{L^{\infty}}< C\) for each \(t\in(0,T_{\max})\), or \(T_{\max }<\infty\) and \(\lim_{t\rightarrow T^{-}_{\max}} \Vert u(\cdot ,t)\Vert_{L^{\infty}}=\infty\). We first collect some important properties of local solutions \((u,v)\) to (1.1) or (1.2) in \(\Omega\times(0,T_{\max})\).
Lemma 2.1
Proof
3 Proof of Theorem 1.1
In order to prove Theorem 1.1, we shall see that it suffices to show the uniform boundedness of \(\Vert u(\cdot,t)\Vert_{L^{\infty}}\) for \(t\in(0,\infty)\). To this end, we first prove the boundedness of \(\Vert u(\cdot,t)\Vert_{L^{p}}\) for each finite p, then we can send p to ∞ by the Moser–Alikakos iteration [17].
Lemma 3.1
Proof
Proof of Theorem 1.1
Choosing \(p=N+1\) in (3.1), we conclude from (2.2) and the elliptic regularity argument that \(\Vert v\Vert_{W^{2,{N+1}}}\leq C\) for some positive constant C independent of t, then we conclude from the embedding (for ∇v) \(W^{1,N+1}(\Omega)\hookrightarrow L^{\infty}(\Omega)\) or Morrey’s inequality (e.g., p. 280 of [18])that \(\Vert\nabla v \Vert_{L^{\infty}}< C\).
4 Proof of Theorem 1.2
We proceed to prove Theorem 1.2. First we present the following result parallel to Lemma 3.1.
Lemma 4.1
Proof
Lemma 4.2
Proof
Proof of Theorem 1.2
By choosing \(p>N+1\) fixed in (4.5), we have \(\Vert\nabla v(\cdot ,t)\Vert_{L^{\infty}}< C\) for all \(t\in(0,\infty)\). Then by the same arguments as in (3.5) we can show the uniform boundedness of \(\Vert u(\cdot,t)\Vert_{L^{\infty}}\) for all \(t\in(0,\infty)\). Therefore \(T_{\max}=\infty\) and the local solution \((u,v)\) is global. Moreover, one can apply the standard regularity theory to show that both u and v are classical in \(\bar{\Omega}\times(0,\infty)\). □
Declarations
Acknowledgements
We thank the two anonymous reviewers for their time and patience in handling, carefully reading and meticulously scrutinizing our manuscript, which helped improve its presentation overall. We thank the first reviewer for the suggestions towards the original proof of Theorem 1.1, which make it easier for the reader.
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The author declares that they have no competing interests.
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Authors’ Affiliations
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