In this section, we discuss the existence of nonnegative classical global solutions and the global asymptotic stability of unique positive equilibrium point of system (1.6).
Some notations throughout this section are as follows:
,
means that
for any
with
,
,
means that
and
are in
,
and
with
.
To obtain
normal estimates of the solution for (1.6), we present a series of lemmas in the following.
Lemma 4.1.
Let
be the solution of (1.6). Then there exists a positive constant
(
1) such that
Proof.
By applying the comparison principle [
20] to system (1.6), we have
and
in
. To prove that
in the following, we consider the auxiliary problem
Notice that the functions
and
are sufficiently smooth in
, and are quasimonotone in
. Let
and
be a pair of upperlower solutions for (4.2), where
and
are positive constants. Direct calculation with inequalities
yields
and
. It follows that there exists
for any
, where
is a big enough positive constant such that (4.1) holds.
Lemma 4.2.
Let
, and
for the solution to following equation:
where
,
are positive constants and
. Then there exists a positive constant
, depending on
and
, such that
Proof.
It is easy to check, from
, that
where
and
.
and
are bounded in
from (4.1). Multiplying (4.7) by
, and integrating by parts over
, yields
Using Hölder inequality and Young inequality to estimate the right side of (4.8), we have
with some
. Substituting (4.9) into (4.8) yields
where
depends on
and
. Since
, the elliptic regularity estimate [
10, Lemma
] yields
From (4.7), we have
. Hence,
. Moreover, the Sobolev embedding theorem shows that (4.6) holds.
Lemma 4.3 (Lemma
can be presented by combining Lemmas
and
in [11]).
Let
, and let
satisfy
and there exist positive constants
and
such that
. Then there exists a positive constant
independent of
but possibly depending on
,
,
,
and
such that
Finally, one proposes some standard embedding results which are important to obtain the
normal estimates of the solution for (1.6).
Lemma 4.4.
Let
be a fixed bounded domain and
. Then for all
with
, one has
(1)
(2)
(3)
where
is a positive constant dependent on
and
.
The main result about the global existence of nonnegative classical solution for the crossdiffusion system (1.6) is given as follows.
Theorem 4.5.
Assume that
and
satisfy homogeneous Neumann boundary conditions and belong to
for some
. Then system (1.6) has a unique nonnegative solution
when the space dimension is
.
Proof.
Step 1.


Estimates and

Estimates of
. Firstly, integrating the third equation of (1.6) over
, we have
Integrating (4.14) in
and moving terms yield
Secondly, multiplying the third equation of (1.6) by
and integrating over
, we have
Integrating the above expression in
yields
Since
from Lemma 4.2, and using Hölder inequality and Young inequality, we have
From (4.1) and
, it holds that
Taking
and selecting a proper
such that
, then applying (4.20) and (4.21) to (4.19) yields
Denote that
. Then it follows from (4.22) that
It is easy to see that
and
for any
; hence
Take
. Then it follows from
estimates of
namely (4.15), that
It follows from Lemma 4.3 and (4.24) that
Since
,
is bounded by contrary proof. It follows that
is bounded, that is,
. It is easy to check that
for all
still denote
by
, then
Finally, we observe that
satisfies
with
. So take
for (4.17) and (4.19). Then there exists a positive constant
such that
Step 2.

Estimates of
. We rewrite the third equation of (1.6) as a linear parabolic equation
where
,
,
are Kronecker symbols.
To apply the maximum principle [
15, Theorem
, page 181] to (4.15) to obtain
, we need to verify that the following conditions hold: (1)
is bounded; (2)
(3)
, where
and
are positive constants, and
and
satisfy
Next we verify conditions (1)–(3) in turn. From (4.28), condition (1) is true for
. One can choose
such that condition (2) holds. To verify condition (3), the first equation of (1.6) is written in the divergence form
where
is bounded in
by Lemma 4.1, and
for
from (4.27). Application of the Hölder continuity result [
15, Theorem
, page 204] to (4.19) yields
Returning to (4.7), since
for any
by (4.1) and (4.27), and
by (4.32), then by applying the parabolic regularity theorem [
15, Theorem
, pages 341342] to (4.7) we have
Hence
from Lemma 4.4, which shows that
. Similarly,
by the second equation of (1.6). Now we can show that
, which imply that
. In addition,
obviously belongs to
. It follows that one can select
. Now the above three conditions are satisfied, and
from [
15, Theorem
, page 181]. Recalling Lemma 4.1, thus there exists a positive constant
for any
such that
Step 3.
The Proof of the Classical Solution
of (1.6)
in
for Any
. Because
, we have from (4.34) that
for any
. So
for all
. It follows from [
15, Lemma
, page 80] that
. And direct calculation
yields
. So we have
The third equation of (1.6) can be written as
Summarizing the above conclusions that are proved, we know that
and
are all bounded in
. It follows from [
15, Theorem
page 204] that there exists
such that
The proof of Lemma 4.2 is similar. Then we have
, that is,
. Applying the [
13, Theorem
page 204] to the second equation (
1.6), there exists
such that
Furthermore, applying Schauder estimate [
15, page 320321] yields
for
. Selecting
and using Sobolev embedding theorem, we have
. Still applying Schauder estimate, we have
Let
. Then
satisfies
where
. By (4.35)–(4.38), we have
. So applying Schauder estimate to (4.40) yields
. Since
, we have
The first equation of (1.6) can be written as
where
. By (4.35), (4.39), and (4.41), we have
. So applying Schauder estimate to (4.42) yields
In particular, if
, then
; in other words, Theorem 4.5 is proved. For the case
, from Sobolev embedding theorem, we have
. Repeating the above bootstrap and Shauder estimate arguments, this completes the proof of Theorem 4.5. About space dimension
, see [21].
Theorem 4.6.
System (1.6) has the unique positive equilibrium point
when one of the conditions (i), (ii), (iii), (iv), (v), and (vi) in Section 2 holds. Let the space dimension be
, and let the initial values
be nonnegative smooth functions and satisfy the homogenous Neumann boundary conditions. If the following condition (4.44) holds, then the solution
of (1.6) converges to
in
:
where
and
.
Proof.
Define the Lyapunov function
where
and
have been given in Theorem 4.6. Obviously,
is nonnegative, and
if and only if
and
. When
is a positive solution of system (1.6),
is well posed for all
from Theorem 4.5. According to system (1.6), the time derivative of
satisfies
It is easy to check that the final three integrands on the right side of the above expression are positive definite because of the electing of
, and the sufficient and necessary conditions of the first integrand being positive definite are the following:
Noticing that (4.44) is the sufficient conditions of (4.47), so there exists a positive constant
such that
Similar to the tedious calculations of
, using integration by parts, Hölder inequality, and (4.34), one can verify that
is bounded from above. Thus we have from (4.48) and Lemma 3.2 in Section 3 that
In addition,
is decreasing for
, so we can conclude that the solution
is globally asymptotically stable. The proof of Theorem 4.6 is completed.