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 .
Proof When , the proof is similar to the methods of [10–12]. So, we just give the proof of Theorem 3.1 for . The proof is divided into three parts.
-, -estimate and -estimate for v.
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 yields
Estimating the first term on the right-hand side of (3.4),
Substituting (3.5) into (3.4) yields
Select and denote . Notice the positive equilibrium point of (1.1) exists under condition , then
By Lemma 2.5, . Integrating the above inequality and using the Gronwall inequality, we get
Hence, there exists a positive constant such that . 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 satisfying . Substituting (3.10) into (3.9) and taking , we have
From (3.11), we know
When , it is easy to find that and . So,
Set . It follows from the -estimate for v that
By Lemma 2.4 and (3.12), we know
Obviously, and . It is easy to know that E is bounded by use of reduction to absurdity. Since , . So, is bounded, i.e., . Denote still as q. So,
Finally, when , with . For , taking in (3.9), it follows from (3.8) that there exists a positive constant such that
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  to (3.16), we need to prove the following conditions:
where ν, 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 for all , the condition (2) is verified. In view of the condition (3), we take appropriate q and r. Rewrite the first equation of system (1.1) as
When , , 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 [, 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 [, 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 [, p.181, Theorem 7.1], we can conclude that . From (2.1), there exists a positive constant such that
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 , u, v, ∇u and ∇v are all bounded. By the Schauder estimate , there exists such that
Furthermore, by the Schauder estimate, we obtain
Next, the regularity of v will be discussed. Set . So, satisfies
where . According to (3.22) to (3.24), we have , . Applying the Schauder estimate to (3.25), we know
From , 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. □