By [34–36], we have the following result.

Theorem 4.1.

If

, then (1.4) has a unique nonnegative solution

, where

is the maximal existence time of the solution. If the solution

satisfies the estimate

then
. If, in addition,
, then

In this section, we consider the existence and the convergence of global solutions to the system (1.4).

Theorem 4.2.

Let
and the space dimension
. Suppose that
are nonnegative functions and satisfy zero Neumann boundary conditions. Then (1.4) has a unique nonnegative solution

In order to prove Theorem 4.2, some preparations are collected firstly.

Lemma 4.3.

Let

be a solution of (1.4). Then

where
.

Proof.

From the maximum principle for parabolic equations, it is not hard to verify that
and
is bounded.

Multiplying the second equation of (1.4) by

, adding up the first equation of (1.4), and integrating the result over

, we obtain

Using Young inequality and H

lder inequality, we have

where

It follows from (4.3) and (4.4) that

where

depends on

and coefficients of (1.4). In addition, there exists a positive constant

, such that

Integrating the first equation of (1.4) over

, we have

Integrating (4.8) from

to

, we have

According to (4.7), there exists a positive constant

, such that

Multiplying the second equation of (1.4) by

and integrating it over

, we obtain

Integrating the previous inequation from

to

, we have

Lemma 4.4.

Let

be a solution of (1.4),

, and

. Then there exists a positive constant

depending on

and

, such that

Furthermore
and

Proof.

satisfies the equation

where
are functions of
and so are bounded because of Lemma 4.3.

Multiply the second equation of (1.4) by

and integrate it over

to obtain

and
. From a disposal similar to the proof of Lemma 2.2 in [23], we have
. Using a standard embedding result, we obtain

Lemma 4.5 (see [23, Lemmas 2.3 and 2.4]).

Let

,

, and let

be any number which may depend on

. Then there is a constant

depending on

, and

such that

for any
with
for all
.

To obtain
-estimates of
, we establish
-estimates of
in the following lemma.

Lemma 4.6.

Let

,

, then there exist positive constants

and

, such that

Proof.

Multiply the first equation of (1.4) by

for

and integrate by parts over

to obtain

Integrating (4.19) from 0 to

, we have

Then substitution of

,

into (4.20) leads to

It follows from H

lder inequality and Lemma 4.3 that

Note that

, and

for

. From H

lder inequality, Young inequality, and Lemma 4.4, we have

Substitution of (4.22) and (4.23) into (4.21) leads to

where
is arbitrary and
.

Choose

such that

then it follows from (4.24) that

Then

for

According to Lemma 4.5 and the definition of

, we can see

Combining (4.26) and (4.29), we have

where
. Therefore
is bounded from (4.30).

From (4.29), we have
. Namely,
,
. Combining (4.28), we have
, where
.

Setting
in (4.20) (it is easily checked that
, i.e.,
), we have
.

Multiplying the second equation of (1.4) by

and integrating it over

, we have

The result of
can be obtained from an analogue of the previous proof of
's.

Lemma 4.7.

Let

, then there exists a positive constant

such that

Proof.

We will prove this lemma by [

37, Theorem 7.1, page 181]. At first, we rewrite the first two equations of (1.4) as

where
,
,
,
is
symbol. It follows from Lemma 4.6 that
,
.

By the third equation of (1.4), we have

It follows from Lemma 4.3 that

is bounded in

. Applying Theorem

[

37, Page 204] to (4.34), we have

Recall that

satisfy (4.14) in Lemma 4.4, that is,

where

is bounded. Since

by (4.35), applying Theorem

[

37, page 341-342] to (4.36), we have

It follows from [37, Lemma
, page 80] that
and so
. Recall from Lemma 4.6 that
, so that
by applying Theorem
[37, Page 181] to (4.33).

Proof of Theorem 4.2.

Firstly, Theorem 4.2 can be proved in a similar way as Theorem
in [21, 25] when the space dimension
.

Secondly, for

, applying Lemma

[

37, Page 80] to (4.36), we have

Since

, we obtain

The first two equations can be written in the divergence form as

where

. It follows from Lemmas 4.1, 4.5, and (4.39) that

are bounded. Thus applying Theorem

[

37, Page 204] to (4.40) leads to

We rewrite the third equation of (1.4) as

where

. Applying Schauder estimate [

29, Theorem

, page 114] to (4.42) gives

where

,

. From (4.41), we have

. It follows from (4.41) and (4.43) that

. Applying Schauder estimate to (4.45) gives

Solving equations (

4.44) for

, respectively, we have

In particular, to conclude
, we need to repeat the above bootstrap technique. Since
is arbitrary, so the classical solution
of (1.4) exists globally in time.

Now we discuss the global stability of the positive equilibrium
(see Section 2) for (1.4).

Theorem 4.8.

Assume that the all conditions in Theorem 4.2, (2.1), and

hold. Let

be the unique positive equilibrium point of (1.4), and let

be a positive solution for (1.4). Then

provided that
is large enough.

Proof.

Define the Lyapunov function

Let

be a positive solution of (1.4), Then

The first integrand in the right hand of the previous inequality is positive definite if

Therefore, when the all conditions in Theorem 4.8 hold, there exists a positive constant

such that

This implies that
. So the proof of Theorem 4.8 is completed.