1 Introduction
In this paper, we study the global existence for the following strongly coupled parabolic system of m equations (m≥2) for the unknown vector u=[ui]i=1m:
(1.1)
(
u
i
)
t
=
Δ
(
u
i
p
i
(
u
)
)
+
u
i
g
i
(
u
)
,
(
x
,
t
)
∈
Ω
×
(
0
,
∞
)
,
where pi,gi:ℝm→ℝ are sufficiently smooth functions (namely, pi∈C2(ℝm) and gi∈C(ℝm)), Ω is a bounded domain with smooth boundary in ℝN, N≥2. The system is equipped with Dirichlet boundary and sufficiently smooth initial conditions
{
u
i
=
0
on
∂
Ω
×
(
0
,
∞
)
,
u
i
(
x
,
0
)
=
u
i
,
0
(
x
)
,
x
∈
Ω
.
The consideration of (1.1) is motivated by the following extensively studied model in population biology introduced by Shigesada, Kawasaki and Teramoto in [11]:
(1.2)
{
u
t
=
Δ
(
d
1
u
+
α
11
u
2
+
α
12
u
v
)
+
k
1
u
+
β
11
u
2
+
β
12
u
v
,
v
t
=
Δ
(
d
2
v
+
α
21
u
v
+
α
22
v
2
)
+
k
2
v
+
β
21
u
v
+
β
22
v
2
.
Here, di,αij,βij and ki are constants with di>0. Dirichlet or Neumann boundary conditions were usually assumed for (1.2). This model was used to describe the population dynamics of two species densities u,v which move and interact with each other under the influence of their population pressures.
Of course, (1.2) is a special case of (1.1) with m=2 and
p
i
(
u
,
v
)
=
d
i
+
α
i
1
u
+
α
i
2
v
,
g
i
(
u
,
v
)
=
k
i
+
β
i
1
u
+
β
i
2
v
.
We will refer to the functions pi (respectively, gi) as the diffusion (respectively, reaction) rates (see [8] for further discussions).
Under suitable assumptions on the constant parameters αij, βij and that Ω is a planar domain (N=2), Yagi proved in [12] the global existence of (strong) positive solutions, with positive initial data. In this paper, we will extend this investigation to multi-species versions of (1.2) for more than two species on bounded domains of arbitrary dimension N.
The global existence problem of the systems like (1.1), a fundamental problem in the theory of pdes, is still in its infancy in comparison to the huge literature of scalar equations. We can write (1.1) in its general divergence form
(1.3)
u
t
=
div
(
A
(
u
)
D
u
)
+
f
(
u
)
.
This a strongly coupled parabolic system with the diffusion matrix A(u), the Jacobian of [uipi(u)]1m, being a full matrix. We say that the system is weakly coupled if A(u) is diagonal (i.e., pi depends only on ui).
The key point in the proof of global existence of strong solutions of (1.3) is the a priori estimate of their W1,p(Ω) norm for some p>N. In fact, it was established by Amann in [1] that (1.1) has a global strong solution u if there is some exponent p>N such that for any T∈(0,∞),
lim sup
t
→
T
-
∥
u
∥
W
1
,
p
(
Ω
)
<
∞
.
Since we consider only homogeneous Dirichlet boundary condition, a simple use of Sobolev inequality shows that we can use ∥Du∥Lp(Ω) for a norm of W1,p(Ω). Thus, we need only prove that
sup
t
∈
(
0
,
T
)
∥
D
u
∥
L
p
(
Ω
)
<
C
(
T
)
for all T∈(0,∞) and some p>N and C(T) being a continuous function on ℝ. With this a priori estimate, one can alternatively use the homotopy or fixed point approaches in [5, 6, 7], instead of semigroup theories in [1], to obtain the local/global existence of strong solutions.
The derivation of such estimates for (1.1) is a difficult issue when A(u) is full because the known techniques for scalar equations (m=1) are no longer applicable unless the matrix A(u) are of special form, e.g., diagonal or triangular, these techniques can be partly applied together with some ad hoc arguments (see [10]). In this paper, we will consider (1.1) with full diffusion matrix A(u) of special forms where some nontrivial modifications of the classic methods can apply and yield new affirmative answers to the problem.
Precisely, we study the case when either the diffusion or reaction rates are identical. Considering the first case and being inspired by the standard (SKT) system (1.2), where pi are a linear function in u, we consider a function Ψ on ℝ, a linear combination L(u) of ui, L(u)=∑iaiui, and assume that for i=1,…,m,
(1.4)
p
i
(
u
)
=
λ
0
+
Ψ
(
L
(
u
)
)
.
We also assume that the reaction rates gi satisfy the control growth |gi(u)|≤C+c0Ψ(|L(u)|) for some positive constants C,c0. We will establish the global existence of nonnegative strong solutions to (1.1) with nonnegative initial data.
Clearly, (1.2) is the case when d1=d2, αi1=αj1, αi2=αj2 and Ψ(s)=s.
Assume that there are constants Cij,cij≥0 such that
(1.5)
|
g
i
(
u
)
|
≤
∑
j
(
C
i
j
+
c
i
j
Ψ
(
|
u
i
|
)
)
.
Suppose also that the function
(1.6)
F
(
v
,
p
)
:=
∫
0
v
Ψ
1
2
(
s
)
s
p
-
1
𝑑
s
∼
p
-
1
Ψ
1
2
(
v
)
|
v
|
2
p
for all
p
and
v
≥
0
.
For examples, this condition is clearly satisfied if Ψ has a polynomial growth.
We have our first main result as follows.
Theorem 1.1.
Assume (1.5) and (1.6). If c0=maxcij is sufficiently small, then (3.1) has a unique nonnegative strong solution.
The key idea of the proof is that one can show that if u is bounded, then L(u) satisfies a scalar equation from which we can derive Harnack inequalities to show that L(u) is Hölder continuous and then it is nonnegative and its derivative can be bounded (if c0 is sufficiently small). The nonnegativity of each components can be deduced from this structure and plays a important role in the argument that follows. The remain of the proof is to show that this boundedness can be transferred to the components of u. Thanks to the special structures of each equation of the system, they can be written as scalar equations of similar structures with coefficients depending on L(u). We can repeat the same argument to obtain bounds for the derivatives of u.
One should remark that this argument crucially bases on the assumption that the reaction terms gi have growth rates that are comparable to the diffusion ones and the boundary conditions are homogeneous Dirichlet. The argument in this paper fails if we drop the term Ψ in the diffusitivities or replace the homogeneous Dirichlet boundary condition by Neumann, even in the case of scalar equations. In this case, one cannot even obtain (or we do not know of) an estimate of ∥u∥L1(Ω).
Thanks to the referee, the following example should be discussed to point out in comparison with the conditions here. It concerns the scalar system
{
u
t
=
Δ
(
u
+
|
a
1
u
|
p
-
1
u
)
+
λ
u
+
ε
|
u
|
p
-
1
u
in
Ω
×
(
0
,
T
)
,
u
=
0
on
∂
Ω
×
(
0
,
T
)
,
u
(
x
,
0
)
=
u
0
(
x
)
in
Ω
.
When a1>0 and ε>0, the reaction has the same growth rate like the diffusion and one can establish that this system possesses a global solution, Tmax=∞. This is also corresponding to what we consider here. However, if a1=0 and ε>0, then the diffusion is weaker than the reaction. The solution blows up in finite time for sufficiently large enough u0. This is also reflected in the following system considered in [2]:
{
u
t
=
Δ
u
+
λ
u
-
u
2
+
b
u
v
in
Ω
×
(
0
,
T
)
,
v
t
=
Δ
v
+
μ
v
-
v
2
+
c
u
v
in
Ω
×
(
0
,
T
)
,
u
=
0
,
v
=
0
on
∂
Ω
×
(
0
,
T
)
,
u
(
x
,
0
)
=
u
0
(
x
)
,
v
(
x
,
0
)
=
v
0
(
x
)
in
Ω
.
On the other hand, we can relax the assumption that the diffusion rates are identical as in (1.4). We then discuss an example which relax the assumption of equal diffusion rates pi. The trade off is that the reaction rates gi are identical and satisfying the control growth (1.5) and we have to consider equal reaction rates gi and assume that the system consists of two equations. We will have to assume further that Ψ is a C1 function on ℝ such that
(1.7)
Ψ
(
s
)
,
Ψ
′
(
s
)
≥
0
and
Ψ
(
s
)
≥
s
for
s
≥
0
.
Our next main result concerns the following system:
(1.8)
{
u
t
=
Δ
(
λ
0
u
+
u
Ψ
(
L
(
u
,
v
)
)
+
ε
0
a
Δ
(
u
v
)
+
u
g
(
u
,
v
)
,
v
t
=
Δ
(
λ
0
v
+
v
Ψ
(
L
(
u
,
v
)
)
-
ε
0
b
Δ
(
u
v
)
+
v
g
(
u
,
v
)
.
Here, L(u,v)=bu+av and λ0,ε0,a,b are positive constants. Regarding the reaction term, we also assume that there are positive constants C,c0 such that (compare with (1.5))
(1.9)
|
g
(
u
,
v
)
|
≤
C
+
c
0
Ψ
(
|
L
(
u
,
v
)
|
)
for all
u
,
v
∈
ℝ
.
Note that the diffusions in (1.8) are different but the reaction rates are the same. Then by appropriate modifications of the previous argument we can prove that
Theorem 1.2.
If ε0,c0 are sufficiently small and the initial data u0,v0≥0, then system (1.8) has a unique global strong solution (u,v) with u,v≥0.
The paper is organized as follows. In Section 2, we discuss some regularity positivity results for strong solutions to scalar parabolic equations. Our main results on system (1.1) will be proved in Section 3.
2 Some Facts on Scalar Equations
In this section we consider the following scalar equation:
(2.1)
v
t
=
Δ
(
P
(
v
)
)
+
div
(
v
b
(
v
)
)
+
v
g
(
v
)
in
Q
=
Ω
×
(
0
,
T
)
and study the smoothness, uniform boundedness and positivity of its strong solutionv under some special conditions on P,g which will serve our purpose in discussing cross diffusion systems later.
To proceed, we first need the following parabolic Sobolev imbedding inequality.
Lemma 2.1.
Let r*=pN if N>p and let r* be any number in (0,1) if N≤p. For any sufficiently nonnegative smooth functions g,G and any time interval I there is a constant C such that
(2.2)
∬
Ω
×
I
g
r
*
G
p
𝑑
z
≤
C
sup
I
(
∫
Ω
×
{
t
}
g
𝑑
x
)
r
*
∬
Ω
×
I
(
|
D
G
|
p
+
G
p
)
𝑑
z
.
If G=0 on the parabolic boundary ∂Ω×I, then the integral of Gp over Ω×I on the right-hand side can be dropped.
Furthermore, if r<r*, then for any ε>0 we can find a constant C(ε) such that
(2.3)
∬
Ω
×
I
g
r
G
p
𝑑
z
≤
C
sup
I
(
∫
Ω
×
{
t
}
g
𝑑
x
)
r
∬
Ω
×
I
(
ε
|
D
G
|
p
+
C
(
ε
)
G
p
)
𝑑
z
.
Proof.
For any r∈(0,1) and t∈I we have via Hölder’s inequality
(2.4)
∫
Ω
g
r
G
p
𝑑
x
≤
(
∫
Ω
g
𝑑
x
)
r
(
∫
Ω
G
p
1
-
r
𝑑
x
)
1
-
r
.
If r=r*, then p1-r=N*=pNN-p, the Sobolev conjugate of p if N>p (the case N≤p is obvious), so that the Sobolev inequality gives
(
∫
Ω
G
p
1
-
r
𝑑
x
)
1
-
r
≤
C
N
∫
Ω
(
|
D
G
|
p
+
G
p
)
𝑑
x
.
Using the above in (2.4) and integrating over I, we easily obtain (2.2). On the other hand, if r<r*, then p1-r<N*. A simple contradiction argument and the compactness of the imbedding of W1,p(Ω) into Lp1-r(Ω) imply that for any ε>0 there is C(ε) such that
(
∫
Ω
G
p
1
-
r
𝑑
x
)
1
-
r
≤
ε
∫
Ω
|
D
G
|
p
𝑑
x
+
C
(
ε
)
∫
Ω
G
p
𝑑
x
.
We then obtain (2.3). ∎
We now have the following a priori boundedness of solution of (2.1).
Theorem 2.2.
Consider a (weak or strong) solution v to (2.1) in Q=Ω×(0,T). Assume that there are a function λ(v) and a number λ0 such that λ(v)≥λ0>0 and
P
v
(
v
)
≥
λ
(
v
)
,
|
b
(
v
)
|
≤
g
1
λ
(
v
)
,
|
g
(
v
)
|
≤
g
2
λ
(
v
)
,
where g1,g2 are functions such that g12+g2∈Lq(Q) for some q>N2+1. For v∈R and p≥1 consider the function
(2.5)
F
(
v
,
p
)
=
∫
0
v
λ
1
2
(
s
)
s
p
-
1
𝑑
s
,
and assume that there are universal constants c1,c2 such that for all p and v∈R,
(2.6)
c
1
p
-
1
λ
1
2
(
v
)
|
v
|
p
≤
|
F
(
v
,
p
)
|
≤
c
2
p
-
1
λ
1
2
(
v
)
|
v
|
p
.
If ∥vλ(v)∥L1(Q) is finite, then v,Dv are bounded and Hölder continuous in Ω×(τ,T) for any τ∈(0,T). Their L∞ norms depend on ∥v2λ(v)∥L1(Q).
Condition (2.6) is clearly verified if λ(v) has a polynomial growth in |v|.
Proof.
We test the equation with |v|2p-2v and use integration by parts
∫
Ω
Δ
(
P
(
v
)
)
|
v
|
2
p
-
2
v
𝑑
x
=
-
∫
Ω
P
v
(
v
)
D
v
D
(
|
v
|
2
p
-
2
v
)
𝑑
x
,
∫
Ω
div
(
v
b
(
v
)
)
|
v
|
2
p
-
2
v
𝑑
x
=
-
∫
Ω
v
b
(
v
)
D
(
|
v
|
2
p
-
2
v
)
𝑑
x
.
Because D(|v|2p-2v)=(2p-1)|v|2p-2Dv and by the assumptions on Qv(v) and b(v),g(v), we easily get for allp≥1,
sup
(
0
,
T
)
1
2
p
∫
Ω
|
v
|
2
p
𝑑
x
+
(
2
p
-
1
)
∬
Q
λ
(
v
)
|
v
|
2
p
-
2
|
D
v
|
2
𝑑
z
≤
C
∬
Q
g
1
λ
(
v
)
|
v
|
2
p
-
1
|
D
v
|
𝑑
z
+
C
∬
Q
g
2
λ
(
v
)
|
v
|
2
p
𝑑
z
.
Applying Young’s inequality g1λ(v)|v|2p-1|Dv|≤ε|v|2p-2|Dv|2+C(ε)g12|v|2p for ε small,
sup
(
0
,
T
)
1
2
p
∫
Ω
|
v
|
2
p
𝑑
x
+
(
2
p
-
1
)
∬
Q
λ
(
v
)
|
v
|
2
p
-
2
|
D
v
|
2
𝑑
z
≤
C
∬
Q
(
g
1
2
+
g
2
)
λ
(
v
)
|
v
|
2
p
𝑑
z
.
As c3λ(v)|v|2p-2≤Fv2(v,p)≤c4λ(v)|v|2p-2 for some universal constants c3,c4 by the definition in (2.5), for g3=g12+g2 the above is
sup
(
0
,
T
)
1
2
p
∫
Ω
|
v
|
2
p
d
x
+
(
2
p
-
1
)
∬
Q
|
D
(
F
(
v
,
p
)
|
2
d
z
≤
C
∬
Q
g
3
λ
(
v
)
|
v
|
2
p
d
z
.
Thus, for p≥1,
sup
(
0
,
T
)
∫
Ω
|
v
|
2
p
𝑑
x
,
∬
Q
|
D
(
F
(
v
,
p
)
)
|
2
𝑑
z
≤
C
p
∬
Q
g
3
λ
(
v
)
|
v
|
2
p
𝑑
z
.
By applying the parabolic Sobolev inequality in Lemma 2.1 with g=|v|p and G=F(v,p), the above estimate yields for r=2N,
(
∬
Q
|
v
|
2
p
r
|
F
(
v
,
p
)
|
2
𝑑
z
)
1
1
+
r
≤
C
p
1
+
2
1
+
r
∬
Q
g
3
λ
(
v
)
|
v
|
2
p
𝑑
z
.
By (2.6), we then obtain for γ=1+r=1+2N,
(
∬
Q
|
v
|
2
p
γ
λ
(
v
)
𝑑
z
)
1
γ
≤
C
p
1
+
3
γ
∬
Q
g
3
λ
(
v
)
|
v
|
2
p
𝑑
z
.
Hölder’s inequality yields
∬
Q
g
3
λ
(
v
)
|
v
|
2
p
𝑑
z
≤
C
(
∬
Q
g
3
q
λ
(
v
)
𝑑
z
)
1
q
(
∬
Q
|
v
|
2
p
q
′
λ
(
v
)
𝑑
z
)
1
q
′
.
Let dμ=λ(v)dz. As we assume that g12,g2∈Lq(Q,dμ), g3∈Lq(Q,dμ) and the first factor on the right-hand side is finite. The above inequality is
∥
v
∥
L
2
p
γ
(
Q
,
d
μ
)
≤
(
2
C
p
)
(
1
+
3
γ
)
1
2
p
∥
v
∥
L
2
p
q
′
(
Q
,
d
μ
)
.
Because q>N2+1, we have q′<γ=1+2N. By replacing p by pq′ and defining γ0=γq′>1, it follows that
∥
v
∥
L
2
p
γ
0
(
Q
,
d
μ
)
≤
(
2
C
p
)
(
q
′
+
3
γ
0
)
1
2
p
∥
v
2
∥
L
2
p
(
Q
,
d
μ
)
.
Because γ0>1, we can apply the Moser iteration argument to show that v is bounded. Indeed, by taking 2p=γ0i with i=1,2,… to the above estimate implies
∥
v
∥
L
γ
i
(
Q
,
d
μ
)
≤
(
2
C
)
γ
1
γ
γ
2
∥
v
2
∥
L
1
(
Q
,
d
μ
)
with
γ
1
=
(
q
′
+
3
γ
0
)
∑
i
=
0
∞
γ
0
-
i
,
γ
2
=
(
q
′
+
3
γ
0
)
∑
i
=
0
∞
i
γ
0
-
i
.
Letting i→∞ and using the fact that limp→∞∥v∥Lp(Q,dμ)=∥v∥L∞(Q,dμ) (we will show that dμ is finite below) we obtain for some constant C0 that ∥v∥L∞(Q,dμ)≤C0∥v2λ(v)∥L1(Q,dμ).
As λ(v) is bounded below by a positive constant, this implies that v is bounded if v∈L1(Q,dμ) is bounded. Furthermore, we now show that dμ is finite. Because
∬
|
v
|
≥
1
λ
(
v
)
𝑑
z
≤
∥
v
λ
(
v
)
∥
L
1
(
Q
)
and λ(u) is bounded on the set |v|<1, we see that dμ is finite if ∥vλ(v)∥L1(Ω) is.
Once we show that v is bounded, we obtain the local Harnack inequality (using both positive and negative powers p and cutoff functions) to show that v is Hölder continuous. The argument is now classical and we refer the readers to the classical books [4, 9] for details. It also follows that Dv is bounded and Hölder continuous in Ω×(τ,T) for any τ∈(0,T). Indeed, we can adapt the freezing coefficient method in [3] to establish this fact. ∎
Remark 2.3.
The conditions in the theorem and remarks need only hold for |v| large. This is easily to see if we make use of the cutoff function
v
¯
(
k
)
=
{
v
if
|
v
|
≥
k
,
k
if
0
<
v
<
k
,
-
k
if
-
k
<
v
≤
0
,
with k sufficiently large and observe that Dv¯k=0 on the set |v|<k.
Remark 2.4.
In connection with the systems considered in the next section, we also consider the scalar equation
(2.7)
v
t
=
λ
0
Δ
v
+
Δ
(
Ψ
(
v
)
v
)
+
v
g
(
v
)
,
where λ0>0 and Ψ:ℝ→ℝ is a C1 function and satisfies for |v| large
(2.8)
Ψ
(
v
)
,
Ψ
′
(
v
)
v
≥
0
.
Assume also that for v∈ℝ and p≥1 the function
(2.9)
F
^
(
v
,
p
)
=
∫
0
v
Ψ
1
2
(
s
)
s
p
-
1
𝑑
s
satisfies for all p and v∈ℝ and some universal constants c1,c2,
(2.10)
c
1
p
-
1
Ψ
1
2
(
v
)
|
v
|
p
|
F
^
(
v
,
p
)
|
≤
c
2
p
-
1
Ψ
1
2
(
v
)
|
v
|
p
.
This condition allows us to apply Theorem 2.2 with P(v)=λ0v+Ψ(v)v and λ(v)=Ψ(v)+Ψ′(v)v+λ0. Thanks to (2.8), λ(v) satisfies (2.6). Also, (2.9) and (2.10) imply that the function F defined by (2.5) satisfies (2.6). We then apply Theorem 2.2 to (2.7) and obtain that v,Dv are bounded in Ω×(τ,T) for any τ∈(0,T) and their norms are bounded in term of ∥v∥L1(Q) and ∥v2Ψ(v)∥L1(Q)).
We can also consider the scalar equation
(2.11)
v
t
=
λ
0
Δ
v
+
Δ
(
Ψ
(
|
v
|
)
v
)
+
v
g
(
v
)
,
and let Ψ:ℝ→ℝ be a C1 function and satisfying for v≥0 and large
(2.12)
Ψ
(
v
)
,
Ψ
′
(
v
)
≥
0
.
Indeed, we now define ψ(v)=Ψ(|v|). We then have ψ′(v)v=Ψ′(|v|)signvv=Ψ′(|v|)|v|≥0 because of (2.12) and |v|≥0. Thus, ψ satisfies (2.8) and the theorem applies.
In applications we usually prefer that v is nonnegative if the initial is. The following result serves this purpose. It is well known but we include its proof here for the sake of convenience.
Theorem 2.5.
Let a,g be C1 functions on R×Q and let b be a bounded C1 map from Q into RN. Assume that a(w)≥λ0 for w≥0 and λ0 is a positive constant. Also suppose that a,g are bounded by a constant depending on w in (x,t)∈Q. Let w be the strong solution to
(2.13)
{
w
t
=
div
(
a
(
w
,
x
,
t
)
D
w
)
+
div
(
w
b
)
+
w
g
(
w
,
x
,
t
)
in
Q
,
w
(
x
,
0
)
=
w
0
(
x
)
on
Ω
.
If the data w0≥0, then w≥0 on Q.
Proof.
Because w is a strong solution, there is a constant M>0 such that |w|≤M. We then truncate a,g to C1 function a^,g^ which are constants for v outside [-M-1,M+1] and consider the equation
v
t
=
div
(
a
^
(
|
v
|
,
x
,
t
)
D
v
)
+
div
(
v
b
(
x
,
t
)
)
+
v
g
^
(
v
,
x
,
t
)
,
with initial data w0.
We have a^(|v|,x,t)≥λ0 and is bounded from above and |vg^(v,x,t)|≤C|v| for some constant C. These facts and the classical theory of scalar parabolic equation with smooth bounded coefficients (see [9]) show that (2.11) has a strong solution v.
Let v+,v- be the positive and negative parts of v. We test the equation with v-. Using the facts that |v|=v++v-, |v|=v- on the set v->0, v+Dv-=Dv+Dv-=0 on the set v->0, we obtain
-
d
d
t
∫
Ω
(
v
-
)
2
𝑑
x
-
∫
Ω
a
^
|
D
v
-
|
2
𝑑
x
=
∫
Ω
[
-
b
v
-
D
v
-
+
(
v
-
)
2
g
^
]
𝑑
x
.
Because b are bounded by a constant C(M), applying Young’s inequality
∫
Ω
|
b
v
-
D
v
-
|
𝑑
x
≤
ε
∫
Ω
|
D
v
-
|
2
𝑑
x
+
C
(
ε
,
M
)
∫
Ω
(
v
-
)
2
𝑑
x
.
Because g^ is bounded by a constant C depending on M and a(v)≥λ0, we can choose ε sufficiently small in the above inequality to arrive at
d
d
t
∫
Ω
(
v
-
)
2
𝑑
x
+
∫
Ω
|
D
v
-
|
2
𝑑
x
≤
C
(
M
)
∫
Ω
(
v
-
)
2
𝑑
x
.
Thus, we see that the function
z
(
t
)
=
∫
Ω
(
v
-
)
2
𝑑
x
satisfies the differential inequality z′≤C1z and z(0)=0 because the initial data v0≥0. We then apply comparison theorem to the equation y′=Cy with y(0)=0 which has the solution y(t)=0. We then have z(t)=0 for all t∈(0,T). Hence, v-=0 on Q so that v≥0. It follows that the solution v of (2.11) also solves (2.13). By the uniqueness of strong solutions, w=v≥0 in Q. ∎
3 Cross Diffusion System with Equal Diffusion/Reaction Rates
In this section, we consider system (1.1) and assume either that the diffusion rates pi or the reaction rates are equal. We will always assume nonnegative initial data ui,0.
Throughout this section we will consider a nonnegative C1 function Ψ on ℝ satisfying
Ψ
′
(
s
)
≥
0
for
s
≥
0
.
3.1 Equal Diffusion Rates
We first consider the following system of m equations for u=[ui]1m:
(3.1)
{
(
u
i
)
t
=
Δ
(
λ
0
u
i
+
Ψ
(
L
(
u
)
)
u
i
)
+
u
i
g
i
(
u
)
in
Ω
×
(
0
,
∞
)
,
u
i
(
x
,
0
)
=
u
i
,
0
(
x
)
on
Ω
,
where λ0>0 and L(u) is a linear combination of ui. That is, L(u)=Σi=1maiui with ai>0. Besides the main condition Ψ,Ψ′≥0, we recall here assumptions (1.5) and (1.6) of our main result stated in the introduction for convenience.
Assume that there are constants Cij,cij≥0 such that
(3.2)
|
g
i
(
u
)
|
≤
∑
j
(
C
i
j
+
c
i
j
Ψ
(
|
u
i
|
)
)
,
and
(3.3)
F
(
v
,
p
)
:=
∫
0
v
Ψ
1
2
(
s
)
s
p
-
1
𝑑
s
∼
p
-
1
Ψ
1
2
(
v
)
|
v
|
2
p
for all
p
and
v
≥
0
.
We will prove Theorem 1.1, our first main results, which states
Theorem 3.1.
Assume (3.2) and (3.3). If c0=maxcij is sufficiently small, then (3.1) has a unique nonnegative strong solution.
As we explained in the introduction, we need only establish a priori the finiteness of sup(0,Tmax)∥Du∥Lp(Ω), with some p>N, for any strong solution u=[ui]1m of (3.1) on Ω×(0,Tmax) for any Tmax∈(0,∞). We will do this for p=∞ via several lemmas.
First, we show that u is nonnegative.
Lemma 3.2.
Assume that u=[ui]1m is a bounded strong solution on (0,Tmax). Then ui≥0 on Ω×(0,Tmax).
Proof.
We can use Theorem 2.5 to show first that ui≥0 on Q=Ω×[0,T] for any 0<T<Tmax and all i. We rewrite the equation of ui as
(3.4)
{
(
u
i
)
t
=
div
(
a
i
(
u
i
,
x
,
t
)
D
u
i
)
+
div
(
u
i
b
i
(
x
,
t
)
)
+
u
i
g
i
(
u
)
in
Q
,
u
i
(
x
,
0
)
=
u
i
,
0
(
x
)
on
Ω
,
where
a
i
(
u
i
,
x
,
t
)
=
λ
0
+
Ψ
(
L
(
u
)
)
,
b
i
(
x
,
t
)
=
Ψ
′
(
L
(
u
(
x
,
t
)
)
)
D
(
L
(
x
,
t
)
)
.
Following the proof of Theorem 2.5, because u,Du bounded on Q, |L(u)|≤M for some constant M. We truncate the function Ψ outside the interval [-M-1,M+1] to obtain a bounded C1 function ψ satisfying the following: ψ(s)≥0 and ψ(s) is a constant when |s|≥M+1.
Denoting v^=[|vi|]1m for any vector v=[vi]1m, we consider the system
(3.5)
{
(
v
i
)
t
=
div
(
a
^
i
(
v
,
x
,
t
)
D
v
i
)
+
div
(
v
i
b
i
(
x
,
t
)
)
+
v
i
g
i
(
u
)
in
Q
,
v
i
(
x
,
0
)
=
u
i
,
0
(
x
)
on
Ω
,
where a^i(v,x,t)=λ0+ψ(L(v^)).
Because ψ(s)≥0 for s≥0 and L(v^)≥0, we have ψ(L(v^))≥0. Thus a^i(v,x,t)≥λ0 and bounded from above. System (3.5) is a diagonal system with bounded continuous coefficients (because Ψ∈C1(ℝ)) and has a unique strong solution v according to the classical theory (e.g., see [4, Chapter 7]).
Applying the argument in the proof of Theorem 2.5 to each equation in (3.5), system (3.5) has a nonnegative strong solution v, so that ψ(v^)=ψ(v), which also solves (3.4) by the definition of ψ, an extension of Ψ. By the uniqueness of strong solutions, ui=vi≥0 in Q for all i. ∎
Next, define W=L(u). The following lemma provides bounds of W,DW that are independent of the number M, which was used only in establishing that ui≥0.
Lemma 3.3.
Assume (3.3). Then W,DW are bounded in Ω×(τ,T) for any τ∈(0,T) by a constant depending only on ∥W∥L1(Q),∥W2Ψ(W)∥L1(Q).
Proof.
Taking a linear combination of the equations, we obtain
W
t
=
λ
0
Δ
W
+
Δ
(
Ψ
(
W
)
W
)
+
f
(
u
)
,
where f(u)=∑iaiuigi(u). Because ui≥0 and ai>0, W is nonnegative and |ui|≤W. Since Ψ(s) is increasing for s≥0, the assumption on gi (3.2) implies
|
g
i
(
u
)
|
≤
∑
j
(
C
i
j
+
c
i
j
Ψ
(
|
u
i
|
)
)
≤
∑
j
(
C
i
j
+
c
i
j
Ψ
(
W
)
)
.
Hence, f satisfies for some positive constants C and c0=maxcij,
(3.6)
|
f
(
u
)
|
≤
C
|
W
|
(
1
+
c
0
Ψ
(
W
)
)
.
We then apply Theorem 2.2 (to be precise, its Remark 2.4 and equation (2.7)) with v=W, noting that v=W≥0 and v=0 on the boundary. Assumption (3.3) on Ψ guarantees that (2.10) is satisfied. Of course, (2.8) is verified because W,Ψ′(W)≥0. We derive that the norms of W,DW are bounded in Ω×(τ,T) for any τ∈(0,T) by constants independent of M but on ∥W∥L1(Q) and ∥W2Ψ(W)∥L1(Q)). The lemma follows. ∎
Lemma 3.4.
If the constant c0 in (3.6) is sufficiently small, then the norms ∥W∥L1(Q) and ∥W2Ψ(W)∥L1(Q)) are bounded by a constant.
Proof.
Indeed, testing the equation of W by W and using (3.6), we get
(3.7)
sup
t
∈
(
0
,
T
)
∫
Ω
×
{
t
}
W
2
𝑑
x
+
∬
Ω
×
(
0
,
t
)
Ψ
(
W
)
|
D
W
|
2
𝑑
z
≤
C
∬
Ω
×
(
0
,
t
)
[
1
+
c
0
Ψ
(
W
)
]
W
2
𝑑
z
.
Applying the Sobolev inequality to the function F(W,1) (see (3.3)), we find a constant C(N) such that
∫
Ω
×
{
t
}
Ψ
(
W
)
W
2
𝑑
x
≤
C
(
N
)
∫
Ω
×
{
t
}
Ψ
(
W
)
|
D
W
|
2
𝑑
x
.
Thus, using this, we see that if c0 is sufficiently small, then the integral of Cc0Ψ(W)W2 in inequality (3.7) can be absorbed to the left and we get
sup
t
∈
(
0
,
T
)
∫
Ω
×
{
t
}
W
2
𝑑
x
+
∬
Ω
×
(
0
,
t
)
(
1
+
Ψ
(
W
)
)
|
D
W
|
2
𝑑
z
≤
C
∬
Ω
×
(
0
,
t
)
W
2
𝑑
z
.
This yields an integral Grönwall inequality for y(t)=∥W∥L2(Ω×{t}) on (0,T) and shows that this norm is bounded by a universal constant on (0,T). This fact and the above inequality show that the left-hand side quantities are bounded. We then make use of the parabolic Sobolev inequality with g=W and G=F(W,1) (one notes that |DG|2∼Ψ(W)|DW|2) to see that ∥W2Ψ(W)∥L1(Q) is bounded for some 2γ>1 by a constant. This implies ∥W2Ψ(W)∥L1(Q)) is bounded. ∎
Proof of Theorem 3.1.
We write the equation of ui in its divergence form
(
u
i
)
t
=
div
(
a
D
u
i
)
+
div
(
u
i
b
)
)
+
u
i
g
i
(
u
)
,
where a=λ+Ψ(W) and b=D(Ψ(W)).
As we are assuming c0 small, by Lemma 3.3 and Lemma 3.4, W and DW are bounded. Using the facts that Ψ(W)≥0 (because W≥0) and W is bounded, we have a≥λ0 and bounded from above. Also, b=D(Ψ(W)) are bounded. In addition, uigi(u) is bounded because 0≤ui≤Wai which is bounded. We then use the standard theory of scalar parabolic equation with bounded coefficients to show that Dui is bounded and Hölder continuous in Ω×(τ,T) for any τ∈(0,T). The existence of u follows from the theory in [1]. ∎
3.2 Equal Reaction Rates
We now present two examples which relax the assumption of equal diffusion rates pi. However, we have to consider equal reaction rates gi and restrict ourselves to the case of systems of two equations. We will consider first the following system:
(3.8)
{
u
t
=
Δ
(
λ
0
u
+
u
Ψ
(
L
(
u
,
v
)
)
+
ε
0
a
Δ
(
u
v
)
+
u
g
(
u
,
v
)
,
v
t
=
Δ
(
λ
0
v
+
v
Ψ
(
L
(
u
,
v
)
)
-
ε
0
b
Δ
(
u
v
)
+
v
g
(
u
,
v
)
.
Here, L(u,v)=bu+av and λ0,ε0,a,b are positive constants.
Again, we will repeat assumptions (1.7) and (1.9) on the system for the reader’s convenience: Ψ is a C1 function on ℝ such that
(3.9)
Ψ
(
s
)
,
Ψ
′
(
s
)
≥
0
and
Ψ
(
s
)
≥
s
for
s
≥
0
,
and there are positive constants C,c0 such that (compare with (3.2))
(3.10)
|
g
(
u
,
v
)
|
≤
C
+
c
0
Ψ
(
|
L
(
u
,
v
)
|
)
for all
u
,
v
∈
ℝ
.
We consider nonnegative initial data u0,v0 for u,v and prove the following (Theorem 1.2):
Theorem 3.5.
If ε0,c0 are sufficiently small, then system (3.8) has a unique global strong solution (u,v) with u,v≥0.
We need the following proposition which will be useful later.
Proposition 3.6.
We consider a strong solution (u,v) with nonnegative initial data u0,v0 to the system
(3.11)
{
u
t
=
Δ
(
λ
0
u
+
u
Ψ
(
L
(
u
,
v
)
)
+
ε
0
a
Δ
(
u
|
v
|
)
+
u
g
(
u
,
v
)
,
v
t
=
Δ
(
λ
0
v
+
v
Ψ
(
L
(
u
,
v
)
)
-
ε
0
b
Δ
(
u
|
v
|
)
+
v
g
(
u
,
v
)
.
For any ε0>0 we have that u,v and Du,Dv are bounded. Also u≥0 in Q. If ε0 are sufficiently small, then v is also nonnegative in Q.
Proof.
The proof will be divided into several steps. First of all, taking a linear combination of the above two equations, we see that W=L(u,v) satisfying
W
t
=
λ
0
Δ
W
+
Δ
(
Ψ
(
W
)
W
)
+
W
g
(
u
,
v
)
.
Step 1: We show that W,DW are bounded and W≥0. For a given strong solution (u,v) of (3.11) we consider the equation
w
t
=
λ
0
Δ
w
+
Δ
(
Ψ
(
|
w
|
)
w
)
+
w
g
(
u
,
v
)
and the initial data w0=au0+bv0≥0. We proved in Theorem 2.2 that this equation has a strong solution w and, by Theorem 2.5, w≥0. By uniqueness of strong solutions, W=w so that W≥0.
Now, from the proof of Lemma 3.3 we see that W,DW are bounded in Ω×(τ,T) for any τ∈(0,T) in terms of 1τ, ∥W∥L1(Q) and ∥W2Ψ(W)∥L1(Q) The latter two norms can be bounded by a constant if c0 is sufficiently small (see Remark 3.4). Keep in mind that τ∈(0,T) can be arbitrary.
We should note that because we already proved that W≥0, hence we do not need here the fact that u,v≥0 (which will be established later) as before in Lemma 3.3 but the conditions Ψ(s),Ψ′(s)≥0 for s≥0 in (3.9) and that |g(u,v)|≤C+c0Ψ(|W|) in (3.10) (see equation (2.11) of Remark 2.4).
Step 2: We prove that u≥0. We write the equation of u in its divergence form
(3.12)
u
t
=
div
(
A
D
u
)
+
div
(
u
B
)
+
u
g
(
u
,
v
)
with A=λ0+Ψ(W)+ε0a|v|, B=-Ψ′(W)DW+ε0aD(|v|). As u,v are strong solutions, by [1], they are classical and their derivatives Du,Dv are bounded in some interval (0,τ) with some τ>0. We will fix τ from now on. Because W,DW are bounded, we apply Theorem 2.5 to prove that u≥0 as long as Du,Dv are bounded. We will show that this is the case for all T>0. This will leads us to the first assertion of the statement.
Step 3: We now prove that u is bounded by using the iteration argument in Theorem 2.2. We multiply the above equation (3.12) by |u|2p-2u and follow the proof of Theorem 2.2 to get
(3.13)
d
d
t
∫
Ω
|
u
|
2
p
𝑑
x
+
(
2
p
-
1
)
∫
Ω
A
|
u
|
2
p
-
2
|
D
u
|
2
𝑑
x
≤
∫
Ω
div
(
u
B
)
|
u
|
2
p
-
2
u
𝑑
x
+
∫
Ω
g
(
u
,
v
)
|
u
|
2
p
𝑑
x
.
From the definition of B we need to study the following two terms on the right of (3.13):
(3.14)
-
∫
Ω
div
(
u
Ψ
′
(
W
)
D
(
W
)
)
|
u
|
2
p
-
2
u
𝑑
x
,
∫
Ω
a
div
(
u
D
(
|
v
|
)
)
|
u
|
2
p
-
2
u
𝑑
x
.
The first one can be treated easily, using the fact that W,DW are bounded we can absorb the first integral in (3.14) to the integral of λ0|u|2p-2|Du|2 in the left of (3.13). We consider the second term. We have
∫
Ω
a
div
(
u
D
(
|
v
|
)
)
|
u
|
2
p
-
2
u
𝑑
x
=
-
(
2
p
-
1
)
∫
Ω
a
u
D
(
|
v
|
)
|
u
|
2
p
-
2
D
u
𝑑
x
.
For each t>0 we split Ω=Ω+(t)∪Ω-(t), where Ω+(t)={x:v(x,t)≥0}. Since aDv=DW-bDu and on Ω+(t), D(|v|)=Dv, we have that the integral over Ω+(t) of -auD(|v|)|u|2p-2Du is
∫
Ω
+
(
t
)
|
u
|
2
p
-
2
u
(
-
D
W
+
b
D
u
)
D
u
𝑑
x
=
-
∫
Ω
+
(
t
)
|
u
|
2
p
-
2
u
D
W
D
u
𝑑
x
+
∫
Ω
+
(
t
)
b
|
u
|
2
p
-
2
u
|
D
u
|
2
𝑑
x
.
Because DW is bounded in Ω×(τ,T), it follows that for any ε>0 there is c1(ε) such that
∫
Ω
|
u
|
2
p
-
2
|
u
D
W
D
u
|
𝑑
x
≤
∫
Ω
(
ε
|
u
|
2
p
-
2
|
D
u
|
2
+
c
1
(
ε
)
|
u
|
2
p
)
𝑑
x
.
By choosing ε small, the integral of |u|2p-2|Du|2 can be absorbed to the integral of λ0|u|2p-2|Du|2 in the left of (3.13).
Meanwhile, on the set v≥0, as W≥bu so that Ψ(W)≥W≥bu (by assumption (3.9) on Ψ). Thus, the integral over Ω+(t) of b|u|2p-2u|Du|2 can also be absorbed to the integral over Ω+(t) of Ψ(W)|u|2p-2|Du|2 in Au2p-2|Du|2 of the left of (3.13).
On Ω-(t), D(|v|)=-Dv, we have that the integral over Ω-(t) of -auD(|v|)u2p-2Du is
∫
Ω
-
(
t
)
|
u
|
2
p
-
2
u
(
D
W
-
b
D
u
)
D
u
𝑑
x
=
∫
Ω
-
(
t
)
|
u
|
2
p
-
2
u
D
W
D
u
𝑑
x
-
∫
Ω
-
(
t
)
b
|
u
|
2
p
-
2
u
|
D
u
|
2
𝑑
x
.
The first integral on the right-hand side can be handled as before. The second integral is nonnegative and can be dropped.
Putting these together, we then obtain for all p≥1,
d
d
t
∫
Ω
|
u
|
2
p
𝑑
x
+
∫
Ω
|
u
|
2
p
-
2
|
D
u
|
2
𝑑
x
≤
C
∫
Ω
|
u
|
2
p
𝑑
x
.
This allows to obtain a bound for ∥u∥L∞(Q) in terms of ∥u∥L1(Q) (see Theorem 2.2). Let p=1 in the above inequality to get a Grönwall inequality for ∥u∥L2(Ω)2. We see that ∥u∥L2(Ω)2, so is ∥u∥L1(Ω), is bounded on (0,T).
Once we prove that u and W,DW are bounded, we then use a cutoff function and repeat a similar argument to the above one in order to obtain local strong/weak Harnack inequalities. It follows that u is Hölder continuous. This is a standard procedure and the readers are referred to the book [9]. It also follows that Du is bounded.
Step 4: We show that v is bounded. This is easy because v=1a(W-bu) and u,Du and W,DW are bounded. We should note that in the above steps we have not imposed any assumptions on ε0. Thus, the first assertion of the proposition was proved.
Step 5: Finally, we prove that v≥0. First of all, we write the equation of v in its divergence form
v
t
=
div
(
A
1
D
v
)
+
div
(
|
v
|
B
1
)
+
v
g
,
where A1=λ0+Ψ(W)-ε0busign(v), B1=Ψ′(W)DW+ε0bD(u). Since u is bounded by a constant independent of ε0 and Ψ(W)≥0, we can choose ε0 small such that A1≥λ02. Also, B1 is bounded because W,DW and Du are. The proof of Theorem 2.5 applies and shows that v≥0. ∎
Proof of Theorem 3.5.
From Proposition 3.6 system (3.11) has a strong solution (u,v) which also solves (3.8). By uniqueness of strong solutions, we see that strong solution (u,v), and its spatial derivatives, of (3.8) are bounded uniformly in terms of the data. Because ∥Du∥L∞(Ω),∥Dv∥L∞(Ω) do not blow up in any time interval (0,T), the solution exists globally and is nonnegative. ∎
We also consider the following system:
{
u
t
=
Δ
(
λ
0
u
+
u
Ψ
(
L
(
u
,
v
)
)
+
ε
0
a
Δ
(
u
v
)
+
u
g
(
u
,
v
)
,
v
t
=
Δ
(
λ
0
v
+
v
Ψ
(
L
(
u
,
v
)
)
+
ε
0
b
Δ
(
u
v
)
+
v
g
(
u
,
v
)
.
Here, L(u,v)=bu-av and ε0,a,b are positive constants.
We then have the following result similar to Theorem 3.5 without the assumption on the smallness of ε0. However, we have to strengthen the condition (3.9) by assuming in addition that
Ψ
(
s
)
≥
0
for all
s
∈
ℝ
.
Theorem 3.7.
If c0 in assumption (3.10) is sufficiently small, then system (3.8) has a unique global strong solution (u,v) with u,v≥0.
Proof.
Following the proof of Theorem 3.5, we consider a strong solution (u,v) with the same initial data to the following system:
{
u
t
=
Δ
(
λ
0
u
+
u
Ψ
(
L
(
u
,
v
)
)
+
ε
0
a
Δ
(
u
|
v
|
)
+
u
g
(
u
,
v
)
,
v
t
=
Δ
(
λ
0
v
+
v
Ψ
(
L
(
u
,
v
)
)
+
ε
0
b
Δ
(
u
|
v
|
)
+
v
g
(
u
,
v
)
.
For any ε0>0 we will prove that u,v and Du,Dv are bounded. We also show that u,v≥0 in Q. We follow the proof of Proposition 3.6 and provide necessary modifications.
Let W=bu-av. Taking a linear combination of the two equations, we can follows Step 1 of the proof of Proposition 3.6 to show that W,DW are bounded. Note that we cannot prove that W≥0 as before because its initial data bu0-av0 is not nonnegative.
However, a similar argument as in Step 2 with some modifications still yields that u≥0. We need to change the argument in Step 3 of the proof to prove that u,Du are bounded. We test the equation of u by |u|2p-2u. As in Step 3, we need to consider the following term on the right-hand side of (3.13):
∫
Ω
a
div
(
u
D
(
|
v
|
)
)
|
u
|
2
p
-
2
u
𝑑
x
=
-
(
2
p
-
1
)
∫
Ω
a
u
D
(
|
v
|
)
|
u
|
2
p
-
2
D
u
𝑑
x
.
We again split Ω=Ω+∪Ω-, where Ω+={v≥0}. Because av=bu-W (instead of av=W-bu as before), we need to interchange Ω+,Ω- in the previous argument. Namely, the integral over Ω+ now contributes a nonnegative term to the left and an integral of |u|2p to the right. Meanwhile, on Ω- we have W=bu-av≥bu≥0 so that Ψ(W)≥bu and the integral over Ω- of b|u|2p-2u|Du|2 now can be absorbed to the left-hand side. The proof then continues to prove that u,Du are bounded.
Using v=1a(bu-W), we see that v,Dv are bounded.
We now show that v≥0, without the assumption that ε0 is small. We slightly modify Step 5 of Corollary 3.6. We write the equation of v as
v
t
=
div
(
A
2
D
v
)
+
div
(
ε
0
u
D
(
|
v
|
)
)
+
div
(
v
B
2
)
+
v
g
(
u
,
v
)
.
Here, A2=λ0+Ψ(W), B2=ε0sign(v)Du+DΨ(W). We follow the proof of Theorem 2.5 and test the equation with v-. We need to consider the integral of div(ε0uD(|v|))v- on the right-hand side. Using integration by parts and the fact that D(|v|)=Dv++Dv-,
∫
Ω
div
(
ε
0
u
D
(
|
v
|
)
)
v
-
𝑑
x
=
-
∫
Ω
ε
0
u
D
(
|
v
|
)
D
v
-
𝑑
x
=
-
∫
Ω
ε
0
u
|
D
v
-
|
2
𝑑
x
.
Because u≥0, the last term provides a nonnegative term on the left-hand side. Meanwhile, we have that A2≥λ0 and A2,B2 are bounded (as u,Du,W,DW are bounded). We obtain as in the proof of Theorem 2.5 a Grönwall inequality of ∥v-∥L2(Ω) and conclude that v-=0 on Q. Thus, v is nonnegative. The proof is complete. ∎
