Abstract
A strongly coupled self and crossdiffusion predatorprey system with Holling type II functional response is considered. Using the energy estimate, Sobolev embedding theorem and bootstrap arguments, the global existence of nonnegative classical solutions to this system in which the space dimension is not more than five is obtained.
MSC: 35K50, 35K55, 35K57, 92D40.
Keywords:
predatorprey system; crossdiffusion; global solution1 Introduction
In this paper, we consider the global existence of nonnegative classical solutions to the following diffusion predatorprey system with Holling type II functional response:
where Ω is a bounded region in
Obviously, the nonnegative equilibrium solutions of system (1.1) are
(1) If
(2) If
(3) If
In view of the study of dynamic behavior of a predatorprey reactiondiffusion system
with Holling type II functional response, a natural problem is what the global behavior
for a predatorprey crossdiffusion system (1.1) is. To the best of our knowledge,
the existing results are very few. In this paper, we consider the space dimension
to be less than six, and initial function
We denote
2 Auxiliary results
Lemma 2.1Let
Proof Firstly, the existence of local solutions for system (1.1) is given in [46]. Roughly speaking, if
If
then
Choose
Lemma 2.2Let
where
Furthermore,
Proof From
where
Using the Hölder inequality and Young inequality to estimate the righthand side of (2.6), we have
with some
where
From (2.5), we have
Lemma 2.3Assume that
with the boundary condition
The proof of the above lemma can be found in [[8], Proposition 2.1].
The following result can be derived from Lemma 2.3 and Lemma 2.4 of [9].
Lemma 2.4Let
and there exist positive constants
Finally, one proposes some standard embedding results which are important to obtain
the
Lemma 2.5There exists a constant
Proof Let
By Lemma 2.3 with
Lemma 2.6Let
(1)
(2)
(3)
whereCis a positive constant dependent onq, n, Ω, T.
3 The existence of classical solutions
The main result about the global existence of nonnegative classical solutions for the crossdiffusion system (1.1) is given as follows.
Theorem 3.1Assume that
Proof When
(i)
Firstly, integrating the first equation of (1.1) over Ω, we have
Thus, for all
where
Furthermore,
Secondly, linear combination of the second and first equations of (1.1) and integrating over Ω yields
So, we get
Further,
Then multiplying both sides of the second equation of system (1.1) by v and integrating over Ω, we obtain
Integrating the above expression in
Estimating the first term on the righthand side of (3.4),
Substituting (3.5) into (3.4) yields
Select
By Lemma 2.5,
Hence, there exists a positive constant
Secondly, multiplying both sides of the second equation of system (1.1) by
Integrating the above equation over
By Lemma 2.2, it can be found that
Choose an appropriate number ε satisfying
Let
From (3.11), we know
When
Set
By Lemma 2.4 and (3.12), we know
Obviously,
Finally, when
By embedding theorem, we get
(ii)
The second equation of system (1.1) can be written as the following divergence form:
where
In order to apply the maximum principle [13] to (3.16), we need to prove the following conditions:
(1)
(2)
(3)
where ν,
Next, we will show that the above conditions (1) to (3) are satisfied for (3.16).
When
When
From (3.14), we have
We will discuss (2.5) which is the corresponding form of (3.18). It follows from (2.1)
and (3.14) that
which implies that
Since
Then the condition (3) and (3.17) are satisfied by choosing
Therefore, the global solution to the problem (1.1) exists.
(iii) The existence of classical solutions.
Under the conditions of Theorem 3.1, we consider above global solutions
So,
Rewrite the second equation of system (1.1) as
Therefore, we can conclude that
Furthermore, by the Schauder estimate, we obtain
Next, the regularity of v will be discussed. Set
where
From
Combining (3.24) and (3.26), we get
Therefore, the result of Theorem 3.1 can be obtained for
Competing interests
The author declares that he has no competing interests.
Authors’ contributions
The work was realized by the author.
Acknowledgements
The author thanks the anonymous referee for a careful review and constructive comments. This work was supported by the Qinghai Provincial Natural Science Foundation (2011Z915).
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