The main result about the global existence of non-negative classical solutions for the cross-diffusion system (1.1) is given as follows.
Theorem 3.1 Assume that and satisfy homogeneous Neumann boundary conditions and belong to for some . Then system (1.1) has a unique non-negative solution if the space dimension is .
, the proof is similar to the methods of [10
]. So, we just give the proof of Theorem 3.1 for
. The proof is divided into three parts.
Firstly, integrating the first equation of (1.1) over Ω, we have
Thus, for all
, we can obtain
Secondly, linear combination of the second and first equations of (1.1) and integrating over Ω yields
So, we get
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 right-hand side of (3.4),
Substituting (3.5) into (3.4) yields
. Notice the positive equilibrium point of (1.1) exists under condition
By Lemma 2.5,
. Integrating the above inequality and using the Gronwall inequality, we get
Hence, there exists a positive constant
. Furthermore, we have
Secondly, multiplying both sides of the second equation of system (1.1) by
) and integrating over Ω, we have
Integrating the above equation over
), it is clear that
By Lemma 2.2, it can be found that
. According to the Hölder inequality and Young inequality, we get
Choose an appropriate number ε
. Substituting (3.10) into (3.9) and taking
, we have
From (3.11), we know
, it is easy to find that
. It follows from the
-estimate for v
By Lemma 2.4 and (3.12), we know
. It is easy to know that E
is bounded by use of reduction to absurdity. Since
is bounded, i.e.
still as q
in (3.9), it follows from (3.8) that there exists a positive constant
By embedding theorem, we get
-estimate for v.
The second equation of system (1.1) can be written as the following divergence form:
where , and is the Kronecker sign.
In order to apply the maximum principle [13
] to (3.16), we need to prove the following conditions:
are positive constants and
Next, we will show that the above conditions (1) to (3) are satisfied for (3.16). When
, it is easy to find that the condition (1) is satisfied by use of (3.15). Since
, the condition (2) is verified. In view of the condition (3), we take appropriate q
. Rewrite the first equation of system (1.1) as
, it is clear that
has an upper bound over
by Lemma 2.1. Set
From (3.14), we have
. Therefore, all conditions of the Hölder continuity theorem [[5
], Theorem 10.1] hold for (3.18). Hence,
We will discuss (2.5) which is the corresponding form of (3.18). It follows from (2.1) and (3.14) that
. From (3.19), we obtain
. Thus, according to the parabolic regularity result of [[5
], pp.341-342, Theorem 9.1], we can conclude that
which implies that by Lemma 2.6.
Since , we have , i.e., . It means that . So, . From (2.1) and (3.14), .
Then the condition (3) and (3.17) are satisfied by choosing
. According to the maximum principle [[13
], p.181, Theorem 7.1], we can conclude that
. From (2.1), there exists a positive constant
Therefore, the global solution to the problem (1.1) exists.
The existence of classical solutions.
Under the conditions of Theorem 3.1, we consider above global solutions to be classical. By (3.20) and Lemma 2.6, we know , . It follows from Lemma 3.3 in  that . Since , we have .
Rewrite the second equation of system (1.1) as
Therefore, we can conclude that
are all bounded. By the Schauder estimate [13
], there exists
Furthermore, by the Schauder estimate, we obtain
Next, the regularity of v
will be discussed. Set
. According to (3.22) to (3.24), we have
. Applying the Schauder estimate to (3.25), we know
, we can see
Combining (3.24) and (3.26), we get
Therefore, the result of Theorem 3.1 can be obtained for , namely . When , namely , we have . (3.24) and (3.26) are obtained by repeating the above bootstrap argument and the Schauder estimate. This completes the proof of Theorem 3.1. □