Global Existence for some Cross Diffusion Systems with Equal Cross Diffusion/Reaction Rates
aa r X i v : . [ m a t h . A P ] O c t Global Existence for some Cross Diffusion Systems withEqual Cross Diffusion/Reaction Rates.
Dung Le Abstract
We consider some cross diffusion systems which is inspired by models in mathemat-ical biology/ecology, in particular the Shigesada-Kawasaki-Teramoto (SKT) model inpopulation biology. We establish the global existence of strong solutions to systems formultiple species having equal either diffusion or reaction rates. The systems are givenon bounded domains of arbitrary dimension.
In this paper, we study the global existence of following strongly coupled parabolic systemof m equations ( m ≥
2) for the unknown vector u = [ u i ] mi =1 ( u i ) t = ∆( u i p i ( u )) + u i g i ( u ) , ( x, t ) ∈ Ω × (0 , ∞ ) . (1.1)Here, p i , g i : IR m → IR are sufficienly smooth functions. Namely, p i ∈ C (IR m ) and g i ∈ C (IR m ). Ω is a bounded domain with smooth boundary in IR N , N ≥ (cid:26) u i = 0 on ∂ Ω × (0 , ∞ ) ,u i ( x,
0) = u i, ( x ) , x ∈ Ω . (1.2)The consideration of (1.1) is motivated by the extensively studied model in populationbiology introduced by Shigesada et al. in [9] (cid:26) u t = ∆( d u + α u + α uv ) + k u + β u + β uv,v t = ∆( d v + α uv + α v ) + k v + β uv + β v . (1.3)Here, d i , α ij , β ij and k i are constants with d i >
0. Dirichlet or Neumann boundary con-ditions were usually assumed for (1.3). This model was used to describe the populationdynamics of two species densities u, v which move and interact with each other under theinfluence of their population pressures.Of course, (1.3) is a special case of (1.1) with m = 2 and p i ( u, v ) = d i + α i u + α i v, g i ( u, v ) = k i + β i u + β i v. We will refer to the functions p i ’s (respectively, g i ’s) as the diffusion (respectively, raction)rates (see [8] for further discussions). Department of Mathematics, University of Texas at San Antonio, One UTSA Circle, San Antonio, TX78249.
Email: [email protected]
Mathematics Subject Classifications:
Key words:
Cross diffusion systems, H¨older regularity, global existence. α ij ’s, β ij ’s and that Ω is a planardomain ( N = 2), Yagi proved in [11] the global existence of (strong) positive solutions, withpositive initial data. In this paper, we will extend this investigation to multi-species versionsof (1.3) for more than two species on bounded domains of arbitrary dimension N .The global existence problem of (1.1), a fundamental problem in the theory of pdes. Wecan write (1.1) in its general divergence form u t = div( A ( u ) Du ) + f ( u ) . (1.4)This a strongly coupled parabolic system with the diffusion matrix A ( u ), the Jacobian of[ u i p i ( u )] m , being a full matrix. We say that the system is weakly coupled if A ( u ) is diagonal(i.e., p i depends only on u i ).The key point in the proof of global existence of strong solutions of (1.4) is the a prioriestimate of their spatial derivatives. 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 − k Du k L p (Ω) < ∞ . Thus, we need only prove that sup t ∈ (0 ,T ) k Du k L p (Ω) < ∞ for all T ∈ (0 , ∞ ) and some p > N . With this a priori estimate, one can alternatively use the homotopy or fixedpoint approaches in [5, 6, 7], instead of semigroup theories in [1], to obtain the local/globalexistence of strong solutions.The derivation of such estimates for (1.1) is a difficult issue when A ( u ) is full because theknown 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 appliedtogether 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 classicmethods can apply and yield new affirmative answers to the problem.Precisely, we study the case when either the diffusion or reaction rates are identical.Being inspired by the standard (SKT) system (1.3) where p i are a linear function in u , weconsider a function Ψ on IR, a linear combination L ( u ) of u i ’s, L ( u ) = P i a i u i , and assumethat for i = 1 , . . . , m p i ( u ) = λ + Ψ( L ( u )) . (1.5)We also assume that the reaction rates g i ’s satisfy the control growth | g i ( u ) | ≤ C + c Ψ( | L ( u ) | ) for some positive constants C, c . We will establish the global existence ofnonnegative strong solutions to (1.1) with nonnegative initial data.Clearly, (1.3) is the case when d = d , α i = α j , α i = α j and Ψ( s ) = s .On the other hand, we can relax the assumption that the diffusion rates are identical asin (1.5). The trade off is that the reaction rates g i ’s are identical and satisfying the abovecontrol growth.The paper is organized as follows. In Section 2, we discuss some regularity positivityresults for strong solutions to scalar parabolic equations. Our main results on the system(1.1) will be presented and proved in Section 3.2 Some facts on scalar equations
In this section we consider the following scalar equation v t = ∆( P ( v )) + div( vb ( v )) + vg ( v ) (2.1)in Q = Ω × (0 , T ) and and study the smoothness, uniform boundedness and positivity ofits strong solution v under some special conditions on P, g which will serve our purpose indiscussing cross diffusion systems later.To proceed, we first need the following parabolic Sobolev imbedding inequality.
Lemma 2.1
Let r ∗ = p/N if N > p and r ∗ be any number in (0 , if N ≤ p . For anysufficiently nonegative smooth functions g, G and any time interval I there is a constant C such that ZZ Ω × I g r ∗ G p dz ≤ C sup I Z Ω ×{ t } g dx ! r ∗ ZZ Ω × I ( | DG | p + G p ) dz (2.2) If G = 0 on the parabolic boundary ∂ Ω × I then the integral of G p over Ω × I on the righthand side can be dropped.Furthermore, if r < r ∗ then for any ε > we can find a constant C ( ε ) such that ZZ Ω × I g r G p dz ≤ C sup I Z Ω ×{ t } g dx ! r ZZ Ω × I ( ε | DG | p + C ( ε ) G p ) dz (2.3) Proof:
For any r ∈ (0 ,
1) and t ∈ I we have via H¨older’s inequality Z Ω g r G p dx ≤ (cid:18)Z Ω g dx (cid:19) r (cid:18)Z Ω G p − r dx (cid:19) − r . (2.4)If r = r ∗ then p/ (1 − r ) = N ∗ = pN/ ( N − p ), the Sobolev conjugate of p if N > p (the case N ≤ p is obvious), so that the Sobolev inequality gives (cid:18)Z Ω G p − r dx (cid:19) − r ≤ Z Ω ( | DG | p + G p ) dx. Using the above in (2.4) and integrating over I , we easily obtain (2.2). On the otherhand, if r < r ∗ , then p/ (1 − r ) < N ∗ . A simple contradiction argument and the compactnessof the imbedding of W ,p (Ω) into L p/ (1 − r ) (Ω) imply that for any ε > C ( ε ) suchthat (cid:18)Z Ω G p − r dx (cid:19) − r ≤ ε Z Ω | DG | p dx + C ( ε ) Z Ω G p dx. We then obtain (2.3).We now have the following a priori boundedness of solution of (2.1).3 heorem 2.2
Consider a (weak or strong) solution v to (2.1) in Q = Ω × (0 , T ) . Assumethat there are a function λ ( v ) and a number λ such that λ ( v ) ≥ λ > and P v ( v ) ≥ λ ( v ) , (2.5) | b ( v ) | ≤ g λ ( v ) , (2.6) | g ( v ) | ≤ g λ ( v ) , (2.7) where g , g are functions such that g + g ∈ L q ( Q ) for some q > N/ .For v ∈ IR and p ≥ consider the function F ( v, p ) = Z v λ ( s ) s p − ds, (2.8) and assume that | F ( v, p ) | ∼ Cpλ ( v ) | v | p for all p and v ∈ IR . (2.9) If k vλ ( v ) k L ( Q ) is finite then v, Dv are bounded and H¨older continuous in Ω × ( τ, T ) forany τ ∈ (0 , T ) . Their norms depend on k vλ ( v ) k L ( Q ) . The condition (2.9) is clearly verified if λ ( v ) has a polynomial growth in | v | . Proof:
We test the equation with | v | p − v and use integration by parts Z Ω ∆( P ( v )) | v | p − v dx = − Z Ω P v ( v ) DvD ( | v | p − v ) dx, Z Ω div( vb ( v )) | v | p − v dx = − Z Ω vb ( v ) D ( | v | p − v ) dx. Because D ( | v | p − v ) = (2 p − | v | p − Dv and the assumptions on Q v ( v ) and b ( v ) , g ( v ), weeasily get for all p ≥ (0 ,T ) p Z Ω | v | p dx + (2 p − ZZ Q λ ( v ) | v | p − | Dv | dz ≤ C ZZ Q g | λ ( v ) || v | p − | Dv | dz + C ZZ Q g | λ ( v ) || v | p dz. (2.10)Applying Young’s inequality g | λ ( v ) || v | p − | Dv | ≤ ε | v | p − | Dv | + C ( ε ) g | v | p for ε small,sup (0 ,T ) p Z Ω | v | p dx + (2 p − ZZ Q λ ( v ) | v | p − | Dv | dz ≤ C ZZ Q ( g + g ) | λ ( v ) || v | p dz. As λ ( v ) | v | p − = F v ( v, p ) by the definition (2.8), for g = g + g the above issup (0 ,T ) p Z Ω | v | p dx + (2 p − ZZ Q | D ( F ( v, p ) | dz ≤ C ZZ Q g | λ ( v ) || v | p dz. Thus, for p ≥ (0 ,T ) Z Ω | v | p dx, ZZ Q | D ( F ( v, p )) | dz ≤ Cp ZZ Q g | λ ( v ) || v | p dz. g = | v | p and G = F ( v, p ),the above estimate yields for r = 2 /N (cid:18)ZZ Q | v | pr | F ( v, p ) | dz (cid:19) r ≤ Cp r ZZ Q g | λ ( v ) || v | p dz. As F ( v ) ∼ Cp − λ ( v ) | v | p by (2.9), we then obtain for γ = 1 + 2 /N (cid:18)ZZ Q | v | pγ λ ( v ) dz (cid:19) γ ≤ Cp γ ZZ Q g λ ( v ) | v | p dz. H¨older’s inequality yields ZZ Q g λ ( v ) | v | p dz ≤ C (cid:18)ZZ Q g q λ ( v ) dz (cid:19) q (cid:18)ZZ Q | v | pq ′ λ ( v ) dz (cid:19) q ′ . Let dµ = λ ( v ) dz . As we assume that g , g ∈ L q ( Q, dµ ), g ∈ L q ( Q, dµ ) and the first factoron the right hand side is finite. The above inequality is k v k L pγ ( Q,dµ ) ≤ (2 Cp ) (1+ γ ) p k v k L pq ′ ( Q,dµ ) . (2.11)Because q > N/ q ′ < γ = 1 + 2 /N . Replacing p by pq ′ and defining γ = γ/q ′ > k v k L pγ ( Q,dµ ) ≤ (2 Cp ) ( q ′ + γ ) p k v k L p ( Q,dµ ) . (2.12)Because γ >
1, we can apply the Moser iteration agument to show that v is bounded.Indeed, by taking 2 p = γ i with i = 0 , , . . . . to the above estimate implies k v k L γi ( Q,dµ ) ≤ (2 C ) γ γ γ k v k L ( Q,dµ ) , with γ = ( q ′ + γ ) P ∞ i =0 γ − i , γ = ( q ′ + γ ) P ∞ i =0 iγ − i . Letting i → ∞ and using the factthat lim p → ∞k v k L p ( Q,dµ ) = k v k L ∞ ( Q,dµ ) (we will show that dµ is finite below) we obtainfor some constant C that k v k L ∞ ( Q,dµ ) ≤ C k v k L ( Q,dµ ) .As λ ( v ) is bounded below by a positive constant, this implies that v is bounded if v ∈ L ( Q, dµ ) is bounded. Furthermore, we now show that dµ is finite. Because ZZ | v |≥ λ ( v ) dz ≤ k v k L ( Q,dµ ) , and λ ( u ) is bounded on the set | v | <
1, we see that dµ is finite.Once we show that v is bounded, we obtain the local Harnack inequality (using bothposive and negative power p and cutoff functions) and so that v is H¨older continuous. Theargument is now classical and we refer the readers to the classical books [3, 4] for details.It also follows that Dv is bounded and H¨older continuous in Ω × ( τ, T ) for any τ ∈ (0 , T ).Indeed, we can adapt the freezing coefficient method in [2] to establish this fact.5 emark 2.3 The conditions in the theorem and remarks need only hold only 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 ≤ k sufficiently large and observe that D ¯ v k = 0 on the set | v | < k . Remark 2.4
In connection with the systems considered in the next section, we considerthe scalar equation v t = λ ∆ v + ∆(Ψ( v ) v ) + vg ( v ) , (2.14)where λ > → IR be a C function and satisfying for | v | largeΨ( v ) , Ψ ′ ( v ) v ≥ . (2.15)Asume also that for v ∈ IR and p ≥ F ( v, p ) = Z v Ψ ( s ) s p − ds (2.16)satisfies | ˆ F ( v, p ) | ∼ Cp Ψ ( v ) | v | p for all p and v ∈ IR. (2.17)This condition allows us to apply Theorem 2.2 with P ( v ) = λ v + Ψ( v ) v and λ ( v ) =Ψ( v ) + Ψ ′ ( v ) v + λ . Thanks to (2.15), λ ( v ) satisfies (2.9). Also, (2.16) and (2.17) implythat the function F defined by (2.8) satisfies (2.9). We then apply Theorem 2.2 to (2.14)and obtain that v, Dv are bounded in Ω × ( τ, T ) for any τ ∈ (0 , T ) and their norms arebounded in term of k v k L ( Q ) and k v Ψ( v ) k L ( Q ) ).We can also consider the scalar equation v t = λ ∆ v + ∆(Ψ( | v | ) v ) + vg ( v ) , (2.18)and Ψ : IR → IR be a C function and satisfying for v ≥ v ) , Ψ ′ ( v ) ≥ . (2.19)Indeed, we now define ψ ( v ) = Ψ( | v | ). We then have ψ ′ ( v ) v = Ψ ′ ( | v | )sign vv = Ψ ′ ( | v | ) | v | ≥ | v | ≥
0. Thus, ψ satisfies (2.15) and the theorem applies.In applications we usually prefer that v is nonnegative if the initial is. The followingresult serves this purpose. Theorem 2.5
Let a, g be C functions on IR × Q and b be a bounded C map from Q into IR N . Assume that a ( w ) ≥ λ for w ≥ and λ 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 (cid:26) w t = div ( a ( w, x, t ) Dw ) + div ( wb ) + wg ( w, x, t ) , in Qw ( x,
0) = w ( x ) on Ω . (2.20) If w ≥ then w ≥ on Q . roof: Because w is a strong solution, there is a constant M > | w | ≤ M .We then truncate a, g to C function ˆ a, ˆ g which are constants for v outside [ − M − , M + 1]and consider the equation v t = div(ˆ a ( | v | , x, t ) Dv ) + div( vb ( x, t )) + v ˆ g ( v, x, t ) , (2.21)with initial data w .We have ˆ a ( | v | , x, t ) ≥ λ and is bounded from above and | v ˆ g ( v, x, t ) | ≤ C | v | for someconstant C . These facts and the classical theory of scalar parabolic equation show that(2.18) has a strong solution v .Let v + , v − be the positive and negative parts of v . We test the equation with v − . Usingthe facts that | v | = v + + v − , | v | = v − on the set v − > v + Dv − = Dv + Dv − = 0 on theset v − >
0, we obtain − ddt Z Ω ( v − ) dx − Z Ω ˆ a | Dv − | dx = Z Ω [ − bv − Dv − + ( v − ) ˆ g ] dx. Because b are bounded by a constant C ( M ), applying Young’s inequality Z Ω | bv − Dv − | dx ≤ ε Z Ω | Dv − | dx + C ( ε, M ) Z Ω ( v − ) dx. Because ˆ g is bounded by a constant C depending on M and a ( v ) ≥ λ , we can choose ε sufficiently small in the above inequality to arrive at ddt Z Ω ( v − ) dx + Z Ω | Dv − | dx ≤ C ( M ) Z Ω ( v − ) dx. Thus, we see that the function z ( t ) = Z Ω ( v − ) dx satisfies the differential inequality z ′ ≤ C z and z (0) = 0 because the initial data v ≥ y ′ = Cy with y (0) = 0 which has thesolution 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 (2.18) also solves (2.20). By the uniqueness of strongsolutions, w = v ≥ Q . In this section, we consider the system (1.1) and assume either that the diffusion rates p i ’sor the reaction rates are equal. We will always assume nonngative initial data u i, .Throughout this section we will consider a nonnegative C function Ψ on IR satisfyingΨ ′ ( s ) ≥ s ≥ . (3.1)7 .1 Equal diffusion rates: We first consider the following system of m equations for u = [ u i ] m (cid:26) ( u i ) t = ∆( λ u i + Ψ( L ( u )) u i ) + u i g i ( u ) in Ω × (0 , ∞ ) ,u i ( x,
0) = u i, ( x ) on Ω , (3.2)where λ > L ( u ) is a linear combination of u i . That is, L ( u ) = Σ mi =1 a i u i with a i > C ij , c ij ≥ | g i ( u ) | ≤ X j ( C ij + c ij Ψ( | u i | )) . (3.3)We have Theorem 3.1 If c = max c ij is sufficiently then (3.2) has a unique nonnegative strongsolution. As we explained in the introduction, we need only establish a priopi the finiteness ofsup (0 ,T max ) k Du k L p (Ω) , with some p > N , for any strong solution u = [ u i ] m of (3.2) onΩ × (0 , T max ) for any T max ∈ (0 , ∞ ). We will do this for p = ∞ via several lemmas. Lemma 3.2 u i ≥ on Ω × (0 , T max ) . Proof:
We can use Theorem 2.5 to show first that u i ≥ Q = Ω × [0 , T ] for any0 < T < T max and all i . We rewrite the equation of u i as (cid:26) ( u i ) t = div( a i ( u i , x, t ) Du i ) + div( u i b i ( x, t )) + u i g i ( u ) in Q,u i ( x,
0) = u i, ( x ) on Ω , (3.4)where a i ( u i , x, t ) = λ + Ψ( L ( u )) + ∂ u i Ψ( L ( u )) u i , b i ( x, t ) = X j = i ∂ u j Ψ( L ( u ( x, t ))) . Following the proof of Theorem 2.5, because u bounded on Q , | L ( u ) | ≤ M for someconstant M . We truncate the function Ψ outside the interval [ − M − , M + 1] to obtain abounded C function ψ satisfying: ψ ( s ) , ψ ′ ( s ) ≥ ψ ( s ) is a constant when | s | ≥ M + 1.Denoting ˆ v = [ | v i | ] m for any vector v = [ v i ] m . We consider the system (cid:26) ( v i ) t = div(ˆ a i ( v, x, t ) Dv i ) + div( v i b i ( x, t )) + v i g i ( u ) in Q,v i ( x,
0) = u i, ( x ) on Ω , (3.5)where ˆ a i ( v, x, t ) = λ + ψ ( L (ˆ v )) + ∂ v i ψ ( L (ˆ v )) v. Because ψ ′ ( s ) ≥ s ≥ L (ˆ v ) ≥
0, we have ∂ u i ψ ( L (ˆ v )) v i = ψ ′ ( L (ˆ v )) a i sign( v i ) v i = ψ ′ ( L (ˆ v )) a i | v i | ≥
0. We also have ψ ( L (ˆ v )) ≥
0. Thus ˆ a i ( v, x, t ) ≥ λ and bounded fromabove. The system (3.5) is a diagonal system with bounded continuous coefficients and hasa unique strong solution v according to the classical theory (e.g., see [3, Chapter 7]).8pplying the argument in the proof of Theorem 2.5 to each equation in (3.5), the system(3.5) has a nonnegative strong solution v , so that ψ (ˆ v ) = ψ ( v ), which also solves (3.4) bythe definition of ψ , an extension of Ψ. By the uniqueness of strong solutions, u i = v i ≥ Q for all i .Next, define W = L ( u ). The following lemma provides bounds of W, DW that areindependent of the number M , which was used only in establishing that u i ≥ Lemma 3.3
Let W = L ( u ) ≥ . Assume that F ( v, p ) := Z v Ψ ( s ) s p − ds ∼ Cp Ψ ( v ) | v | p for all p and v ≥ . (3.6) Then
W, DW are bounded in Ω × ( τ, T ) for any τ ∈ (0 , T ) by a constant depending onlyon k W k L ( Q ) , k W Ψ( W ) k L ( Q ) . Proof:
Taking a linear combination of the equations, we obtain W t = λ ∆ W + ∆(Ψ( W ) W ) + f ( u ) , (3.7)where f ( u ) = P i a i u i g i ( u ). Because u i ≥ a i > W is nonnegative and | u i | ≤ W .Since Ψ( s ) is increasing for s ≥
0, the assumption on g i ’s (3.3) implies | g i ( u ) | ≤ X j ( C ij + c ij Ψ( | u i | )) ≤ X j ( C ij + c ij Ψ( W )) . Hence, f satisfies for some positive constants C and c = max c ij | f ( u ) | ≤ C | W | (1 + c Ψ( W )) . (3.8)We then apply Theorem 2.2 (to be precise, its Remark 2.4 and the equation (2.14))with v = W , noting that v = W ≥
0. The assumption (3.6) on Ψ guarantees that (2.17) issatisfied. We see that the norms of
W, DW are bounded in Ω × ( τ, T ) for any τ ∈ (0 , T ) byconstants independent of M but on k W k L ( Q ) and k W Ψ( W ) k L ( Q ) ). The lemma follows. Remark 3.4
If the constant c in (3.8) is sufficiently small then the norms k W k L ( Q ) and k W Ψ( W ) k L ( Q ) ) are bounded by a constant. Indeed, testing the equation of W by W andusing (3.8)sup t ∈ (0 ,T ) Z Ω ×{ t } W dx + ZZ Ω × (0 ,t ) Ψ( W ) | DW | dz ≤ C ZZ Ω × (0 ,t ) [1 + c Ψ( W )] W dz. (3.9)Applying the Sobolev inequality to the function F ( W,
1) (see (3.6)) we find a constant C ( N ) such that Z Ω ×{ t } Ψ( W ) W dx ≤ C ( N ) Z Ω ×{ t } Ψ( W ) | DW | dx. c is sufficiently small then the integral of Cc Ψ( W ) W inthe inequality (3.9) can be absorbed to the left and we getsup t ∈ (0 ,T ) Z Ω ×{ t } W dx + ZZ Ω × (0 ,t ) Ψ( W ) | DW | dz ≤ C ZZ Ω × (0 ,t ) W dz. This yields an integral Gr¨onwall inequality for y ( t ) = k W k L (Ω ×{ t } ) on (0 , T ) and shows thatthis norm is bounded by a universal constant on (0 , T ). This fact and the above inequalityshow that the left hand side quantities are bounded. We then make use of the parabolicSobolev inequality to see that k W γ Ψ( W ) k L ( Q,dµ ) is bounded by a constant. This implies k W Ψ( W ) k L ( Q ) ) is bounded because 2 γ > Proof of Theorem 3.1:
We write the equation of u i in its divergence form( u i ) t = div( aDu i ) + div( u i b )) + u i g i ( u ) , where a = λ + Ψ( W ) and b = D (Ψ( W )).Using the facts that Ψ( W ) ≥ W ≥
0) and W is bounded, we have a ≥ λ andbounded from above. Also, b = D (Ψ( W )) are bounded. In addition, u i g i ( u ) is boundedbecause 0 ≤ u i ≤ W/a i which is bounded. We then use the standard theory of scalarparabolic equation with bounded coefficients to show that Du i is bounded and H¨oldercontinuous in Ω × ( τ, T ) for any τ ∈ (0 , T ). We now present two examples which relax the assumption of equal diffusion rates p i ’s.However, we have to consider equal reaction rates g i ’s and restrict ourselves to the case ofsystems of two equations.In the sequel, we will always assume that Ψ is a C function on IR such thatΨ( s ) , Ψ ′ ( s ) ≥ s ) ≥ s for s ≥ . (3.10)We consider first the following system (cid:26) u t = ∆( λ u + u Ψ( L ( u, v )) + ε a ∆( uv ) + ug ( u, v ) ,v t = ∆( λ v + v Ψ( L ( u, v )) − ε b ∆( uv ) + vg ( u, v ) . (3.11)Here, L ( u, v ) = bu + av . λ , ε , a, b are positive constants. Regarding the reaction term, wealso assume that there are positive constants C, c such that (compare with (3.3)) | g ( u, v ) | ≤ C + c Ψ( | L ( u, v ) | ) for all u, v ∈ IR . (3.12)We consider nonnegative initial data u , v for u, v . Theorem 3.5 If ε , c are sufficiently small then the system (3.11) has a unique globalstrong solution ( u, v ) with u, v ≥ . We need the following proposition which will be useful later.10 roposition 3.6
We consider a strong solution ( u, v ) with nonnegative initial data u , v to the following system (cid:26) u t = ∆( λ u + u Ψ( L ( u, v )) + ε a ∆( u | v | ) + ug ( u, v ) ,v t = ∆( λ v + v Ψ( L ( u, v )) − ε b ∆( u | v | ) + vg ( u, v ) . (3.13) For any ε > we have that u, v and Du, Dv are bounded. Also u ≥ in Q .If ε are sufficiently small then v is also nonnegative in Q . Proof:
The proof will be divided into several steps. First of all, taking a linearcombination of the above two equations, we see that W = L ( u, v ) satisfying W t = λ ∆ W + ∆(Ψ( W ) W ) + W g ( u, v ) . (3.14) Step 1:
We show that
W, DW are bounded and W ≥ u, v ) of (3.13) we consider the the equation w t = λ ∆ w + ∆(Ψ( | w | ) w ) + wg ( u, v ) (3.15)and the initial data w = au + bv ≥
0. We proved in Theorem 2.2 that this equation has astrong solution w and, by Theorem 2.5, w ≥
0. By uniqueness of strong solutions, W = w so that W ≥ W, DW are bounded in Ω × ( τ, T ) for any τ ∈ (0 , T ) in terms of k W k L ( Q ) and k W Ψ( W ) k L ( Q ) The latter two norms can be boundedby a constant if c is sufficiently small (see Remark 3.4).We should note that because we already proved that W ≥
0, hence we do not needhere the fact that u, v ≥ s ) , Ψ ′ ( s ) ≥ s ≥ | g ( u, v ) | ≤ C + c Ψ( | W | ) in (3.12)(see equation (2.18) of Remark 2.4). Step 2:
We prove that u ≥
0. We write the equation of u in its divergence form u t = div( ADu ) + div( uB ) + ug ( u, v ) (3.16)with A = λ + Ψ( W ) + ε a | v | , B = − Ψ ′ ( W ) DW + ε aD ( | v | ).Again, we can assume that u, v are locally bounded as in the proof of Lemma 3.2.Because W, DW are bounded, we apply Theorem 2.5 to prove that u ≥ Step 3:
We now prove that u is bounded by using the iteration argument in Theo-rem 2.2.We multiply the above equation (3.16) by u p − , recall that u ≥
0, and follows the proofof Theorem 2.2 to get ddt Z Ω u p dx + (2 p − Z Ω Au p − | Du | dx ≤ Z Ω div( uB ) u p − dx + Z Ω g ( u, v ) u p dx (3.17)From the definition of B we need to study the following two terms on the right of (3.17) − Z Ω div( u Ψ ′ ( W ) D ( W )) u p − dx, Z Ω a div( uD ( | v | )) u p − dx. (3.18)11he first one can be treated easily, using the fact that W is bounded (see also below).We consider the second term. We have Z Ω a div( uD ( | v | )) u p − dx = − (2 p − Z Ω auD ( | v | ) u p − Du dx.
For each t > + ( t ) ∪ Ω − ( t ) where Ω + ( t ) = { x : v ( x, t ) ≥ } . Since aDv = DW − bDu and on Ω + ( t ), D ( | v | ) = Dv , we have that the integral over Ω + ( t ) of − auD ( | v | ) u p − Du is Z Ω + ( t ) u p − ( − DW + bDu ) Du dx = − Z Ω + ( t ) u p − DW Du dx + Z Ω + ( t ) bu p − | Du | dx. Because DW is bounded in Ω × ( τ, T ) for any τ ∈ (0 , t ), it follows that for any ε > c ( ε ) such that Z Ω u p − | DW Du | dx ≤ Z Ω ( εu p − | Du | + c ( ε ) u p ) dx. Choosing ε small, the integral of u p − | Du | can be absorbed to the integral of λ u p − | Du | in the left of (3.17). This argument also applies to the first integral in (3.18).Meanwhile, on the set v ≥
0, as W ≥ bu ≥ W ) ≥ W ≥ bu (by theassumption (3.10) on Ψ). Thus, the integral over Ω + ( t ) of bu p − | Du | can also be absorbedto the integral over Ω + ( t ) of Ψ( W ) u p − | Du | in Au p − | Du | of the left of (3.17).On Ω − ( t ), D ( | v | ) = − Dv , we have that the integral over Ω − ( t ) of − auD ( | v | ) u p − Du is Z Ω − ( t ) u p − ( DW − bDu ) Du dx = Z Ω − ( t ) u p − DW Du dx − Z Ω − ( t ) bu p − | Du | dx. The first integral on the right hand side can be handled as before. The second integral isnonnegative and can be dropped.Putting these togetter, we then obtain for all p ≥ ddt Z Ω u p dx + Z Ω u p − | Du | dx ≤ C Z Ω u p dx. This allows to obtain a bound for k u k L ∞ ( Q ) in terms of k u k L ( Q ) (see Theorem 2.2). Let p = 1 in the above inequality to get a Gr¨onwall inequality for k u k L (Ω) . We see that k u k L (Ω) , so is k u k L (Ω) , is bounded on (0 , T ).Once we prove that u and W, DW are bounded we then use a cutoff function andrepeat a similar argument to the above one in order to obtain local strong/weak Harnackinequalities. It follows that u is H¨older continuos. This is a standard procedure and thereaders are referred to the book [4]. It also follows that Du is bounded. Step 4:
We show that v is bounded. This is easy because v = ( W − bu ) /a and u, Du and W, DW are bounded. We should note that in the above steps we have not imposed anyassumptions on ε . 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 itsdivergence form v t = div( A Dv ) + div( | v | B ) + vg, A = λ + Ψ( W ) − ε bu sign( v ), B = Ψ ′ ( W ) DW + ε bD ( u ).Since u is bounded by a constant independent of ε and Ψ( W ) ≥
0, we can choose ε small such that A ≥ λ /
2. Also, B is bounded because W, DW and Du are. The proofof Theorem 2.5 applies and shows that v ≥ Proof of Theorem 3.5:
From Proposition 3.6 the system (3.13) has a strong solution( u, v ) which also solves (3.11). By uniqueness of strong solutions, we see that strong solution( u, v ), and its spatial derivatives, of (3.11) are bounded uniformly in terms of the data.Because k Du k L ∞ (Ω) , k Dv k L ∞ (Ω) do not blow up in any time interval (0 , T ), the solutionexists globally.We also consider the following system (cid:26) u t = ∆( λ u + u Ψ( L ( u, v )) + ε a ∆( uv ) + ug ( u, v ) ,v t = ∆( λ v + v Ψ( L ( u, v )) + ε b ∆( uv ) + vg ( u, v ) . (3.19)Here, L ( u, v ) = bu − av and ε , a, b are positive constants.We then have the following result similar to Theorem 3.5 without the assumption onthe smallness of ε . However, we have to strengthen the condition (3.10) by assuming inaddition that Ψ( s ) ≥ ∀ s ∈ IR . (3.20) Theorem 3.7 If c in the assumption (3.12) is sufficiently small then the system (3.11)has a unique global strong solution ( u, v ) with u, v ≥ . Proof:
Following the proof of Theorem 3.5, we consider a strong solution ( u, v ) withthe same initial data to the following system (cid:26) u t = ∆( λ u + u Ψ( L ( u, v )) + ε a ∆( u | v | ) + ug ( u, v ) ,v t = ∆( λ v + v Ψ( L ( u, v )) + ε b ∆( u | v | ) + vg ( u, v ) . (3.21)For any ε > u, v and Du, Dv are bounded. We also show that u, v ≥ 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 Step1 of the proof of Proposition 3.6 to show that W, DW are bounded. Note that we cannotprove that W ≥ bu − av is not nonnegative.Similarly, Step 2 also yields that u ≥
0. We need to change the argument in Step 3 ofthe proof to prove that u, Du are bounded. We test the equation of u by u p − . As in Step3, we need to consider the following term on the right hand side of (3.17) Z Ω a div( uD ( | v | )) u p − dx = − (2 p − Z Ω auD ( | v | ) u p − Du dx.
We again split Ω = Ω + ∪ Ω − where Ω + = { v ≥ } . 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 p to the right. Meanwhile, on Ω − we have W = bu − av ≥ bu ≥ W ) ≥ bu and13he integral over Ω − of bu p − | Du | now can be absorbed to the left hand side. The proofthen continues to prove that u, Du are bounded.Using v = ( bu − W ) /a , we see that v, Dv are bounded.We now show that v ≥
0, without the assumption that ε is small. We slightly modifyStep 5 of Corollary 3.6. We write the equation of v as v t = div( A Dv ) + div( ε uD ( | v | )) + div( vB ) + vg ( u, v ) . Here, A = λ + Ψ( W ), B = ε sign( v ) Du + D Ψ( W ). We follow the proof of Theorem 2.5and test the equation with v − . We need to consider the integral of div( ε uD ( | v | )) v − on theright hand side. Using integration by parts and the fact that D ( | v | ) = Dv + + Dv − , Z Ω div( ε uD ( | v | )) v − dx = − Z Ω ε uD ( | v | ) Dv − dx = − Z Ω ε u | Dv − | dx. Because u ≥
0, the last term provides a nonnegative term on the left hand side. Meanwhile,we have that A ≥ λ and A , B are bounded (as u, Du, W, DW are bounded). We obtainas in the proof of Theorem 2.5 a Gr¨onwall inequality of k v − k L (Ω) and conclude that v − = 0on Q . Thus, v is nonnegative. The proof is complete. References [1] H. Amann. Dynamic theory of quasilinear parabolic systems III. Global existence,
Math Z.
202 (1989), pp. 219-250.[2] M. Giaquinta and M. Struwe. On the partial regularity of weak solutions of nonlinearparabolic systems.
Math. Z. , 179(1982), 437–451.[3] O. A Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva],
Linear and Quasi-linearEquations of Parabolic Type,
Translations of Mathematical Monographs, AMS, 1968.[4] G.M. Lieberman Second Order Parabolic Differential Equations, World Scientific, 1996.[5] D. Le. Regularity of BMO weak solutions to nonlinear parabolic systems via homotopy.Trans. Amer. Math. Soc. 365 (2013), no. 5, 2723–2753.[6] D. Le. Weighted Gagliardo-Nirenberg Inequalities Involving BMO Norms and Solvabil-ity of Strongly Coupled Parabolic Systems.
Adv. Nonlinear Stud.
Vol. 16, No. 1(2016),125–146.[7] D. Le,
Strongly Coupled Parabolic and Elliptic Systems: Existence and Regularity ofStrong/Weak Solutions.
De Gruyter, 2018.[8] T. Lepoutre and A. Moussa. Entropic structure and duality for multiple species cross-diffusion systems,
Nonlinear Analysis
Vol. 159, (2017), 298–315.[9] N. Shigesada, K. Kawasaki and E. Teramoto.
Spatial segregation of interacting species .J. Theor. Biol., 79(1979), 83– 99. 1410] Y. Lou and M. Winkler. Global Existence and Uniform Boundedness of Smooth Solu-tions to a Cross-Diffusion System with Equal Diffusion Rates.
Comm. in PDes . 40(10),2015.[11] A. Yagi. Global solution to some quasilinear parabolic systems in population dynamics.