Global Existence and Regularity Results for Strongly Coupled Nonregular Parabolic Systems via Iterative Methods

The global existence of classical solutions to strongly coupled parabolic systems is shown to be equivalent to the availability of an iterative scheme producing a sequence of solutions with uniform continuity in the BMO norms. Amann's results on global existence of classical solutions still hold under much weaker condition that their BMO norms do not blow up in finite time. The proof makes use of some new global and local weighted Gagliardo-Nirenberg inequalities involving BMO norms.


Introduction
Among the long standing questions in the theory of strongly coupled parabolic systems and its applications are the global existence and regularity properties of their solutions. We consider in this paper the following system x ∈ Ω Boundary conditions for u on ∂Ω × (0, T ). (1.1) Here, Ω is a bounded domain with smooth boundary ∂Ω in IR n , n > 1, and u : Ω → IR m , f : IR m → IR m are vector valued functions. A(u) is a full matrix m × m. Thus, the above is a system of m equations. The vector valued solution u satisfies either Dirichlet or Neumann boundary condition on ∂Ω × (0, T ).
The system (1.1) arises in many mathematical biology and ecology applications as well as in differential geometry theory. In the last few decades, papers concerning such strongly coupled parabolic systems usually assumed that the solutions under consideration were bounded, a very hard property to check as maximum principles had been unavailable for systems in general. In addition, past results usually relied on the following local existence result of Amann.
Theorem 1.1 ( [1,2]) Suppose Ω ⊂ IR n , n ≥ 2, with ∂Ω being smooth. Assume that (1.1) is normally elliptic. Let p 0 ∈ (n, ∞) and U 0 be in W 1,p 0 (Ω). Then there exists a maximal time T 0 ∈ (0, ∞] such that the system (1.1) has a unique classical solution in (0, T 0 ) with u ∈ C([0, T 0 ), W 1,p 0 (Ω)) ∩ C 1,2 ((0, T 0 ) ×Ω) Moreover, if T 0 < ∞ then lim t→T − 0 u(·, t) W 1,p 0 (Ω) = ∞. (1. 2) The proof of the above result worked directly with the system and based on semigroup and interpolation of functional spaces theories. We refer the readers to [1] for the definition of normal ellipticity. The checking of (1.2) is the most difficult one as known techniques for the regularity of solutions to scalar equations could not be extended to systems and counterexamples were available.
In this paper, we propose a different approach using iterative techniques and depart from the boundedness assumptions. Namely, we consider the following schemes (u k ) t = div(A(u k−1 )Du k ) + f (u k−1 , Du k ) k ≥ 1.
Under very weak assumptions on the uniform boundedness and continuity of the BMO norms of the solutions to the above systems, we will show the global existence of a classical solution to (1.1). Thus, global existence and regularity of solutions are established at once. Furthermore, without the boundedness assumptions the systems are no longer regular elliptic, we will only assume that the matrix A(u) in (1.1) is uniformly elliptic.
We also improve Theorem 1.1 by replacing the condition (1.2) with a weaker ones using the BMO or W 1,p 0 norms of u with p 0 = n. In a forthcoming work, we will show that the results in this paper can apply to a class of generalized Shigesada-Kawasaki-Teramoto models ( [13]) consisting of more than 2 equations. Namely, we will establish the global existence of classical solutions to the following system u t = ∆(P (u)) + f (u), (1.3) where P (u), f (u) are vector valued functions whose components have quadratic (or even polynomial) growth in u.
In the proof we make use of some new local Gagliardo-Nirenberg inequalities involving BMO norms and weights in A p classes. These are the generalizations of the inequalities by Strzelecki and Rivière in [12].

Preliminaries and Main Results
Throughout this paper Ω is a bounded domain with smooth boundary in IR n , n > 1. To describe our assumptions we recall the definitions of BMO spaces and A p classes.
For any locally integrable vector valued function u ∈ L 1 loc (Ω, IR m ) and measurable set B ⊂ Ω with its Lebesgue measure |B| = 0, we denote For a smooth function u defined on Ω×(0, T ), T > 0, its temporal and spatial derivatives are denoted by u t , Du respectively. If A is a function in u then we also abbreviate ∂A ∂u by A u .
We will frequently work with a ball B(z, R) centered at z ∈ IR n with radius R. If z is understood we will write B R for B(z, R) For any x 0 ∈ Ω and R > 0, we also denote Ω(x 0 , R) = Ω ∩ B(x 0 , R). For any locally measurable function u on Ω and x 0 ∈ Ω and R > 0 we write u x 0 ,R for the average of u over Ω(x 0 , R). If x 0 is understood, we simply write u R for u x 0 ,R .
We say that a locally integrable vector valued function u : Ω → IR m is BMO (Bounded Mean Oscillation) if the seminorm where the supremum is taken over all balls B ⊂ Ω. The space BM O(Ω) is the Banach space of BMO functions on Ω with norm We recall the following well known fact (e.g., see [7]).
There is a connection between BMO functions and the so called A γ weights, which are defined as follows. Let Ψ be a measurable nonnegative function on Ω and γ > 1. We say that Ψ belongs to the class A γ or Ψ is an A γ weight if the quantity For more details on these classes we refer the readers to [5,11,14].
We also recall the following result from [7] on the connection between BMO functions and weights. We also recall the definition of the Campanato spaces L p,γ (Ω, IR m ). For any p ≥ 1 and γ > 0 and u ∈ L p (Ω, IR m ), we define Then L p,γ (Ω, IR m ) is the Banach space of such functions with finite norm Clearly, L p,n (Ω, IR m ) = BM O(Ω, IR m ). Moreover, it is well known that ([6, Theorem 2.9, p.52]) L p,γ (Ω, IR m ) is isomorphic to C 0,α (Ω) if α = γ−n p > 0. As usual, W 1,p (Ω, IR m ), p ≥ 1, will denote the standard Sobolev spaces whose elements are vector valued functions u : Ω → IR m with finite norm where Du is the the derivative of u.
We now state our structural conditions on the system (1.1).
On the other hand, it is clear from the definition of weights that so that if 1 < k < 2 then Φ α β is bounded from above and Φ − α β ∼ (1 + |u|) α β (1−k/2) . Thus, we can find α > 2/3 and β < 1/3 such that Φ − α β is Hölder in u and therefore BMO if u is. Again, this gives that Φ − α β is a weight and belongs to ∩ γ>1 A γ . Concerning global existence of classical solutions, we also assume that the ellipticity constants λ, Λ in A.1) are not too far apart. R) (The ratio condition) There is δ ∈ [0, 1) such that One should note that there are examples in [3] of blow up solutions to (1.1) if the condition R) is violated.
We assume the following growth conditions on the nonlinearity f . F) There are positive constants C, b such that for any vector valued functions u ∈ C 1 (Ω, IR m ) and p ∈ C 1 (Ω, IR mn ) |f (u, p)| ≤ C|p| + C|u| b + C.
To solve (1.1), we can make use of the following iterative scheme. We start with any smooth vector valued function u 0 on Q and define a sequence {u k } of solutions to the following linear systems (2.6) The initial and boundary conditions for the above systems are those of u in (1.1). Note that the global existence of the strong solutions to the above systems is not generally available by standard theories (see [4]) because of the presence of Du k in f . However, this is the case if we assume R) and the linear growth of f in Du of F) and make use of the results in [3].
Concerning the approximation sequence {u k }, we assume the following uniform bound and continuity of their BMO norms.
V) Let {u k } and Φ be defined by (2.6) and A.2). There exists a continuous function C on (0, ∞) such that for any T > 0 Moreover, for any ε > 0 and (x, t) ∈ Q, there exists R = R(ε, T ) > 0 such that In addition, for all (x, t) ∈ Q and integer k ≥ 1 we assume that The uniform boundedness assumption on the BMO norm of u k is of course much weaker than the L ∞ boundedness assumptions in literature. Moreover, the uniform continuity assumption (2.7) on the BMO norms is somehow necessary for the regularity of the limit solution u.
The assumption (2.8) seems to be technical at first glance but it is clearly necessary if we would like to produce a the sequence {u k } that converges in L ∞ (Q) to a solution of (1.1). In paticular, if λ(u) behaves like λ(u) ∼ (1 + |u|) k for some k ≥ 0 then, as discussed earlier, we see that Φ(u) ∼ (1 + |u|) (k/2−1) . Thus, Hence, if k ∈ [0, 2] then (2.8) is clearly verified. The generalized Shigesada-Kawasaki-Teramoto model (1.3) clearly is a typical example. We then have our main result of this paper as follows.
A simple consequence of the above theorem is the following.
3), R), F) and V). Then the system (1.1) has a classical solution u that exists globally on Ω × (0, ∞) if and only if there is u 0 and a sequence {u k } satisfying V) such that {u k } has a subsequence that converges weakly to u in L 2 (Ω × (0, T )) for each T > 0.
The necessary part is trivial as we can take u k = u for all k ≥ 0, the solution sequence is then a constant one. The sufficient part comes from Theorem 2.3 and the uniqueness of weak limits in L 2 (Ω × (0, T )).
In the next theorem we discuss the global existence of classical solutions when their local existence can be achieved by other methods (e.g., Theorem 1.1). (2.9) By Poincaré's inequality, it is easy to see that if Du(·, t) ∈ L n (Ω) then u(·, t) is BMO and u(·, t) BM O(B R ) is small if R is small. Therefore, as a simple consequence of the above theorem, we have the following improvisation of Theorem 1.1.

Corollary 2.6
In addition to the assumptions of Theorem 1.1, we assume R). Then there exists a maximal time T 0 ∈ (0, ∞] such that the system (1.1) has a unique classical solution Again, we remark that if the condition R) is violated then there are counterexamples for finite time blow up solutions to (1.1).

Weighted Gagliardo-Nirenberg inequalities
In this section we will establish global and local weighted Gagliardo-Nirenberg interpolation inequalities which allow us to control the L 2p+2 norm of Du k in the proof of our main theorems.
Since the inequalities can be useful for other applications, we will prove them under a set of the following independent assumptions.
. Assume that the following quantities are finite In the rest of this paper we will slightly abuse our notations and write the dot product u, v as uv for any two vectors u, v because its meaning should be clear in the context. Similarly, when there is no ambiguity C will denote a universal constant that can change from line to line in our argument. Furthermore, C(· · ·) is used to denote quantities which are bounded in terms of theirs parameters. Lemma 3.1 Assume GN.1) and GN.2). Let u, U : Ω → IR m be vector valued functions with U ∈ C 2 (Ω), u ∈ C 1 (Ω). Suppose further that either U or Φ 2 (u) ∂U ∂ν vanish on the boundary ∂Ω of Ω. We set

2)
and Before we go to the proof of this lemma, let us recall the following facts from Harmonic Analysis. We first recall the definition of the maximal function of a function F ∈ L 1 loc (Ω) We also note here the Hardy-Littlewood theorem for any F ∈ L q (Ω) In addition, let us recall the definition of the Hardy space H 1 . For any y ∈ Ω and ε > 0. Let φ be any function in Furthermore, concerning the A γ classes, it is well known (e.g. [ A simple use of Hölder's inequality also gives Proof: Integrating by parts, we have We will show that g = div(Φ 2 (u)|DU | 2p DU ) belongs to the Hardy space H 1 and Once this is established, (3.8) and the duality of the BMO and Hardy spaces give (3.4). We write g = g 1 + g 2 with g i = divV i , setting Let us consider g 1 first and define h = Φ(u)|DU | p−1 DU . For any y ∈ Ω and B ε = B ε (y) ⊂ Ω, we use integration by parts, the property of φ ε and then Hölder's inequality for any s > 1 to have the following By Sobolev-Poincaré's inequality, with s * = ns/(n + s), we have the following estimate in noting that |Dh| ≤ |DΦ||DU Using the above estimates in (3.10), we get Putting these estimates together we thus have sup ε>0, Because 2 > 2n/(n + 1) = s * , we can use Young's inequality and then (3.5) to get Therefore, by Holder's inequality and the above estimates We now turn to g 2 and note that |divV 2 | ≤ C(J 1 + J 2 ) for some constant C and We consider J 1 . For any r > 1/(p + 1) we denote r * = 1 − 1 r(p+1) . We also write F = Φ|DU | p+1 andF = Φ|Du| p+1 . For any s > 0, Hölder's inequality yields Similarly, if r 1 > 1/(p + 1) we have the following estimate for J 3 . Using Hölder's inequality for the integrand Φ|DU | p+1 |Du| = F |Du| in J 1 and the above estimates, we obtain (3.14) By the definition of weights, it is clear that .
In order to make use of the continuity in BMO norms in the asumption V) to obtain the regularity results , we will need the following local version of the above lemma. Lemma 3.3 Assume as in Lemma 3.1 and let B s , B t be two concentric balls in Ω with radii t > s > 0. We set

19)
and Let ψ be a C 1 function such that ψ = 1 in B s and ψ = 0 outside B t . For any ε > 0 there is a constant C(ε) such that

(3.21)
Proof: We revisit the proof of the previous lemma. Integrating by parts, we have Again, we will show that g = div(Φ 2 ψ 2 |DU | 2p DU ) belongs to the Hardy space H 1 . We write g = g 1 + g 2 with g i = divV i , setting In estimating V 1 we follow the proof of Lemma 3.1 and replace Φ(u), Φ 0 (u) respectively by Φ(u)ψ(x) and Φ 0 (u)ψ. There will be the following extra term in estimating Dh in the right hand side of (3.10) and it can be estimated as follows We then use the the following in the right hand side of (3.12) (with Ω = B t ) sup |Dψ| The last term can be bounded via (3.5) by Using the fact that |ψ| ≤ 1 and taking Ω to be B t , the previous proof can go on and (3.13) now becomes Similarly, in considering g 2 = divV 2 , we will have an extra term Φ(u)|Dψ||DU | p+1 J 3 in J 1 . We then use the following estimate

and via Young's inequality
Therefore the estimate (3.18) is now (3.22) with g 1 being replaced by g 2 . Combining the estimates for g 1 , g 2 and using Young's inequality, we get (3.23) The above gives an estimate for the H 1 nowm of g. By FS theorem, we obtain Since ψ = 1 in B s , the above yields (3.21) and the proof is complete.
Finally, the following lemma will be crucial in obtaining uniform estimates for the approximation sequence {u k }.

Lemma 3.4 Assume as in Lemma 3.3 and let B ρ , B R be two concentric balls in
for any s, t such that 0 < s < t < R.
We are now ready to give the proof of Lemma 3.4.
Proof: For any s, t, ρ such that 0 < ρ < s < t < R, let ψ be a cutoff function for . By a simple use of Young's inequality to the last product in (3.21) of Lemma 3.3 and our assumption (3.24), we can see easily that if ε, ε 0 are sufficiently small then for some ν 0 ∈ (0, 1) }. The above yields It is clear that we can Lemma 3.5 to f = I 1 to get The constant C(ν 0 ) can be taken to be (1 − ν) −2 (1 − ν −2 ν 0 ) −1 for any ν satisfying ν −2 ν 0 < 1. We can take C(ν 0 ) to be a fixed constant for ν 0 ∈ (0, 1 2 ). Obviously, the above also holds for ρ, R being replaced by s, t and we proved the lemma.

Proof of the main theorems
We now go back to the iterative scheme (2.6) and prove our main theorems in this section.
The following lemma is the main vehicle of the proof of Theorem 2.3.

Lemma 4.1 Assume A.1)-A.3), R)and V). Let p ≥ 1 be a number such that
If R is sufficiently small then for any two concentric balls B ρ ⊂ B R with center inΩ there is a constant C(T ) such that the following holds for all intergers k ≥ 1 sup t∈(0,T ) Bρ∩Ω Lemma 4.2 Assume the ellipticity condition A). Let α be a number such that there is δ α ∈ (0, 1) such that α 2+α = δ α λ Λ . We then have Furthermore, since u k−1 , u k are C 2 in x, we can differentiate (2.6) with respect to x to get Proof: (Proof of Lemma 4.1) We consider the interior case B ρ ⊂ B R ⊂ Ω and leave the boundary case, when the center of B R is on the boundary ∂Ω, to Remark 4.4 following the proof. For any s, t such that 0 ≤ s < t ≤ R let ψ be a cutoff function for B s , B t . That is, ψ ≡ 1 in B s and ψ ≡ 0 outside B t with |Dψ| ≤ 1/(t − s). Testing (4.4) with |Du k | 2p−2 Du k ψ 2 . The assumption (4.1) shows that α = 2p − 2 satisfies the condition of Lemma 4.2 so that we can find a positive constant C(p) such that For simplicity, we will assume in the sequel that f ≡ 0. The presence of f will be discussed in Remark 4.3 after the proof. Therefore, as in A.2). Applying Young's inequality to the integrand of the first integral on the right of the above and the following Here, we denoted Q t = B t × (0, T ). Again, a use of Young's inequality to the first integral on the right yields By the choice of ψ, we obtain from the above the following sup τ ∈(0,T ) Bs (4.5) Here, for any fixed integer k ≥ 1, we set We now apply Lemma 3.4 for u = u k−1 and U = u k . We will see that our assumptions A.2) and A.3) imply the assumptions GN.1) and GN.2) of Lemma 3.4 for any p ≥ 1. Indeed, by our assumption (2.2) on Φ 0 , Φ in A.2) the constants in of GN.1) are finite. Furthermore, since u k−1 is BMO with uniform bounded norm and the assumption A.3), Φ 2 3 (u k−1 ) belongs to the A 4 3 class. As 2 3 ≥ 2 p+2 and 4 3 ≤ p p+2 + 1, Φ 2 p+2 (u k−1 ) belongs to the A p p+2 +1 class. Thus, the quantity C(Φ, Φ 0 ) defined in (3.24) is finite. Also, our continuity assumption (2.7) on the BMO norm of u k implies the smallness of C(Φ, Φ 0 ) u k BM O(B R ) if R is small. Hence, for any given µ 1 > 0 if R = R(µ 1 ) > 0 is sufficiently small then we have from (3.21) of Lemma 3.4 the following estimate.
We now consider B 0 . Applying Lemma 3.4 with u = U = u k−1 , so that I 1 =Î 1 , and Φ(u) = Φ 0 (u), we see that if u k−1 BM O(B R ) , or R, is sufficiently small then there is a constant C 0 (Φ, Φ 0 ) such that for any s, t satisfying 0 < s < t < R Going in back to the notation Φ 0 (u) = λ 1 2 (u), by (2.8), we can split Q t into two disjoint sets Similarly, Using these estimates in (4.12), we obtain Using the above estimate for B 0 in (4.10) and (4.11) and adding the results, we can easily see that if u k−1 BM O(B R ) is sufficiently small then (4.14) for some C 5 depends on Φ, Φ 0 , k 1 , k 2 , T and We now define We then have from (4.14) that (4.15) As before, we can assume that R is sufficiently small such that µ 4 < 1. For any a ∈ (0, 1) such that µ 4 a −2 < 1 we define the sequences t 0 = ρ and t i+1 = t i + (1 − a)a i (R − ρ). Iterate the above k − 2 times to get This shows that the quantity can be bounded by Using this and (4.13) in (4.5), and the estimate for B 0 (t), we obtain (4.2) of the lemma.
Testing (4.4) with |Du k | 2p−2 Du k ψ 2 , we will have the extra term Df (u k−1 , Du k )|Du k | 2p−1 ψ 2 on the right of our estimates in the proof. For any positive ε > 0 we can use Young's inequality to have . Since u k , u k−1 are BMO and Φ is bounded from below, the integral of the first term on the right of the above inequalities is bounded. (4.8) now becomes Choosing ε small, we can see that the iteration arguments in the proof are still in force and the proof can continue. For any point x = (x 1 , . . . , x n ) we denote byx its reflection across the plane x n = 0, i.e.,x = (x 1 , . . . , −x n ). Accordingly, we denote by B − the reflection of B + . For u = u k we define the odd reflection of u byū, i.e.ū(x, t) = −u(x, t) for x ∈ B − . We then consider the odd extension It is easy to see thatū satisfies in B − a system similar to (4.4) for u in B + . As in the proof of the lemma, we test the system for u k with |Du k | 2p−2 Du k ψ 2 and the system for u with |Dū k | 2p−2 Dū k ψ 2 and then sum the results. The integration parts results the extra boundary terms along the flat boundary parts ∂B + and ∂B − . Using the facts that either D x i u = D x iū = 0 for i = n or D xn u = D xnū and the outward normal vectors of B + and B − are opposite we can easily see that those boundary terms are either zero or cancel each others in the summation. Thus, we can obtain (4.5) again with u k−1 , u k being replaced by U k−1 , U k . Since U k belong to W 2,∞ (B) the argument can continue and the lemma holds for U k and then u k . The same argument applies for the Neumann boundary condition if we we use the even extension for u k .
We now give the proof of Theorem 2.3.
Proof: We test the systems (2.6) with u k and use Young's inequality to have The uniform bound assumption on the BMO norms of u k−1 yields that the right hand side is bounded uniformly for all k. Thus, there is a constant C such that Q λ(u k−1 )|Du k | 2 dz ≤ C ∀k. Now, for any 0 < ρ < R and concentric balls B ρ , B R with centers inΩ let us assume that there is some p ≥ 1 such that there is a constant C 0 (ρ, R, u 0 ) depending on ρ, R and sup t∈(0,T ) u 0 (·, t) C 1 (Ω) on such that ( It is well known that the C 1 norms of u 1 and u 2 can be bounded by that of u 0 . Now, if p satisfies (4.1) then Lemma 4.1 and (4.19) establish the existence of a constant C 1 (ρ, R) such that the following holds for all integers k sup t∈(0,T ) Bρ if 0 < ρ < R and R is sufficiently small.
Let χ 0 be any number such that 1 < χ 0 < 1 + 2 n . Denote V = |Du k | p and use Hölder's inequality to get where λ = λ(u k−1 ) and r is a number such that r ′ χ 0 = 1 + 2 n . Recall the Sobolev imbedding inequality and the fact that u k is BMO so that λ(u k−1 ) belongs to L r (Ω) for any r > 1 (see [6]). The above estimates for Q = Q ρ show that there is a constant C(ρ) such that From the ellipticity condition A) and (4.20) we see that the right hand side is bounded. Hence Therefore, (4.19) holds again with p now is pχ 0 . We already showed that (4.19) is valid for p = 1. Thus, we can repeat the argument k times until 2χ k 0 > n as long as the ratio condition (4.1) of Lemma 4.1 is verified for p = χ k 0 . The assumption R) shows that we can choose χ 0 , k such that 1 < χ 0 < 1 + 2 n , 2χ k 0 > n and the ratio condition (4.1) holds for p = χ k 0 . Therefore, (4.20) holds for 2p = 2χ k 0 . We now cover Ω with finitely many balls of radius R/2 to obtain sup t∈(0,T ) Ω |Du k | 2p dx + Q Φ 2 0 (u k−1 )|Du k | 2p−2 |D 2 u k | 2 dz ≤ C(Ω, T, u 0 ). (4.22) For each t ∈ (0, T ), (4.22) shows that the norms u k (·, t) W 1,2p (Ω) for some 2p > n are bounded uniformly in t by a constant depending only on the size of Ω, T and u 0 . By Sobolev's imbedding theorem {u k (·, t)} is a bounded sequence in C α (Ω) for some α > 0. From the system for u k , (4.22) with p = 1 also shows that (u k ) t L 2 (Q) is uniformly bounded. Together, these facts show that the solutions u k are uniformly Hölder continuous in (x, t) and that {u k } is bounded in C β (Q) for some β > 0 We then see that there is a relabeled subsequence {u k } converges in C 0 (Q) to some u. Using difference quotient in t we see that u t ∈ L 2 (Q). The above estimate (4.22) also shows that we can assume Du k+1 (·, t) converges weakly to Du(·, t) in L 2 (Ω) for each t ∈ (0, T )). By the continuity of A in its variable u, we see that u weakly solves (1.1).
By the semicontinuity of norms, (4.22) implies sup t∈(0,T ) Ω |Du| 2p dx + Q λ(u)|Du| 2p−2 |D 2 u| 2 dz ≤ C(Ω, u 0 ). (4.23) Since 2p > n, the above implies that u is Hölder continuous and its regularity in x. Since u t is in L 2 (Q). It is easy to derive from these facts that u is Hölder in (x, t). By [?], Du is Hölder in (x, t) and then u is a classical solution.
We now turn to the proof of our second theorem.
We now see that a similar argument in the proof of Theorem 2.3 with u k being u now gives sup t∈(0,T ) Bρ |Du| 2p dx + Qρ Φ 2 0 |Du| 2p−2 |D 2 u| 2 dz ≤ C 1 (ρ, R) (4.27) if 0 < ρ < R and R is sufficiently small and some p such that 2p > n. Finite covering Ω with balls B R/2 yields sup t∈(0,T ) Ω |Du| 2p dx + Q λ(u)|Du| 2p−2 |D 2 u| 2 dz ≤ C(Ω, R). (4.28) Hence u is Hölder continuous and its regularity in x. From the system for u and the above, with p = 1, we see that u t is in L 2 (Q). It is now standard to show that u is Hölder in (x, t) and Du is Hölder continuous. We now can refer to Amann's results to see that u exists globally.