global existence and uniqueness of solutions for

GLOBAL EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR MULTIDIMENSIONAL WEAKLY PARABOLIC SYSTEMS ARISING IN CHEMISTRY AND BIOLOGY F. R. GUARGUAGLINI† AND R. NATALINI Abstract. In this paper we establish general well-posedeness results for a wide class of weakly parabolic 2 × 2 systems in a bounded domain of RN . Our results cover examples arising in sulphation of marbles and chemotaxis, when the density of one chemical component is not diffusing. We show that, under quite general assumptions, uniform L∞ estimates are sufficient to establish the global existence and stability of solutions, even if in general the nonlinear terms in the equations depend also on the gradient of the solutions. Applications are presented and discussed.

1. Introduction In this paper we study a class of reaction diffusion systems of the form   ∂t (ϕ(c)s) = div(ϕ(c)∇s) + F (s, c), (1.1)  ∂t c = G(s, c),

for (x, t) ∈ QT := Ω × (0, T ] (T > 0), where Ω is a bounded C 2 subset of RN and ϕ, F, G are given smooth functions, which verify some assumptions which will be specified later on. We complement the system (1.1) with the initial conditions (1.2)

s(x, 0) = s0 (x),

c(x, 0) = c0 (x) ,

and the Dirichlet boundary conditions for the unknown s (1.3)

s(x, t) = ψ(x, t)

for (x, t) ∈ ∂Ω × (0, T ].

It is possible to deal with Neumann boundary conditions by using similar arguments. Degenerate parabolic systems in the form (1.1), for particular choices of functions F, G and ϕ, have been studied by several authors because of their large applicability to chemical and biological phenomena: sulphation in calcium carbonate stones [11, 12], penetration of radiolabeled antibodies into tumor tissue [13], chemotaxis and angiogenesis processes [15, 7, 4, 5, 3]. For this last application, we remark that some Keller-Segel type models for chemotaxis can be rewritten in the form (1.1) by a simple change of unknown; see Section 6. The local (in time) existence of solutions for this problems can be easily deduced by fixed point or compactness techniques, together with classical results for linear 1991 Mathematics Subject Classification. Primary 65M06; Secondary 76M20, 76R, 82C40. Key words and phrases. Reaction diffusion systems, global existence of solutions, nonlinear parabolic equations, porous media, sulphation, chemotaxis. † Dipartimento di Matematica Pura e Applicata, Universit´ a degli Studi di L’Aquila, Via Vetoio, I–67100 Coppito (L’Aquila), Italy. E–mail: [email protected] Istituto per le Applicazioni del Calcolo “M. Picone”, Consiglio Nazionale delle Ricerche, Viale del Policlinico 137, I–00161 Rome, Italy. E–mail: [email protected]. 1

2

F. R. GUARGUAGLINI AND R. NATALINI

and semilinear parabolic equations; however, global existence results, in several space dimensions, are still a question of large interest, strictly connected to the stucture of the reaction terms F and G and to the function ϕ. One has to taking into account the degeneracy of the system and the nonlinearity of the divergence term; so, classical techniques involving the global boundedness of Holder norms are not useful in this context and, due to the presence of terms in the form ∇s · ∇c in the first equation, methods based on the uniform boundedness of ksk L∞ are not directly applicable (see [18]). Moreover system (1.1) does not verify the coupling conditions of Shizuta-Kawashima for hyperbolic-parabolic problems [16]. In the present paper we establish a result of global existence and uniqueness for weak solutions to problems (1.1), under suitable general assumptions on F, G and ϕ. Results for some specific examples, following a similar approach, can be found in [15, 11, 12] in one space dimension and in [3, 5] in several dimensions. Our approach consists in a general sharp continuation principle: the growth in time of a natural weak norm, i.e. the norm C([0, T ]; W 2,q (Ω)) ∩ C 1 ([0, T ]; Lq (Ω)), can be controlled in terms of the data and of the norms ksk L∞ (QT ) and kckL∞ (QT ) . Moreover the solutions are proven to verify a stability result with respect to the data and then a uniqueness result. These results generalize those obtained for standard reaction-diffusion systems, see for instance [17, 18] and references therein. It is easy to give sufficient conditions to guarantee a priori L ∞ bounds on s and c, and some examples will be given in Section 6. However, it is well known that, for some choice of F , G and ϕ, it could be hard to find this kind of bounds. Actually, in some special cases, the blow up in finite time of these quantities has been even proved [7, 8, 19]. For this reason it would be important to obtain global existence results without assuming a priori L∞ bounds, possibly assuming some restrictions on the data. In [3] the authors study the Cauchy problem for a chemotaxis model in several space dimension and treat the global existence question without any informations about L∞ norm of s; they assume small initial data for s and prove that the Lp (RN ) norms of solutions are globally bounded. Here, in the same spirit, but in the case of a bounded domain with Dirichlet boundary conditions, we obtain a global estimate for L p -norms of the solutions, under smallness conditions on the data, which holds for a class of functions F, G not ensuring the global boundedness of s in L∞ norm; in fact the reaction term in the first equation of (1.1) is allowed to grow possibly with a quadratic rate. Boundary conditions introduce some complications in the proof, which are solved by a direct estimate. We conclude this section with a brief description of the plan of the paper. In the remainder of this section we state the main assumptions on the data and on the functions F, G and ϕ. Section 2 is devoted to prove a priori estimates for solutions in C([0, T ]; W 2,q (Ω)) ∩ C 1 ([0, T ]; Lq (Ω)), for large q; these estimates are written in details in order to explicit their dependence on the data, on T and on the norms kckL∞ (QT ) and kskL∞ (QT ) . In fact they will be used not only in Section 3, to prove a precised local existence result by compacteness method, but mainly in Section 4 where we show how to control their growth in time and prove global existence, stability and uniqueness for weak solutions corresponding to initial and boundary data satisfying the assumptions introduced below. In Section 5 we weaken the assumptions on F, G and ϕ and prove a global estimate for L p norms of solutions in the case of small data. Finally, in Section 6 we apply our results to some reactiondiffusion systems modeling chemical and biological phenomena. Assumptions Here we write in details the assumptions to be verified by the data and by the functions ϕ, F, G.

MULTIDIMENSIONAL WEAKLY PARABOLIC SYSTEMS

3

Let P = N if N > 2, P > 2 if N = 2, P = 2 if N = 1. The data c0 , s0 , ψ are nonnegative functions such that (1.4) (1.5)

P

s0 ∈ W 2, 2 (Ω) ∩ L∞ (Ω) , c0 ∈ W 1,P (Ω) ∩ L∞ (Ω) ; P

P

P

ψ ∈ C([0, T ]; W 2, 2 (Ω)) ∩ C 1 ([0, T ]; L 2 (Ω)) ∩ W 2, 2 (QT ) for all T > 0 ;

the trace of the function ψ verifies (1.6)

ψ ∈ L∞ (∂Ω × (0, T )) for all T > 0 .

Moreover we assume that: a) G(s, c) is a continuos function defined over (R+ )2 with its first and second derivatives defined over (R+ )2 and bounded over bounded intervals I ⊂ (R+ )2 ; b) F (s, c) is a continuos function defined over (R+ )2 ; moreover its first and second derivatives are defined over (R+ )2 and bounded over bounded intervals I ⊂ (R+ )2 ; c) the functions ϕ(c), ϕ0 (c), ϕ00 (c) are defined over (R+ )2 and bounded over bounded intervals I ⊂ R+ ; d) for all T > 0 the functions F, G allow to determine a priori three quantities, T T cTm ≥ 0 and S∞ , C∞ > 0, depending on the L∞ norms of the data, such T that for 0 ≤ s ≤ S∞ the solutions the second equation in (1.1) satisfy the T bounds cTm ≤ c ≤ C T ∞ and for cTm ≤ c ≤ C∞ , the solutions of the first T equation in (1.1) satisfy 0 ≤ s ≤ S∞ ; T ]. e) for all T > 0 there exists ϕTm > 0 such that ϕTm ≤ ϕ(c) in [cTm , C∞

Now we define the notion of weak solution to problem (1.1)–(1.6).

Definition 1.1. A pair (s, c) is a weak solution to system (1.1)–(1.6) in Ω × [0, T ] if

Z

QT

T a) s ∈ C([0, T ]; L2 (Ω)) ∩ L2 ((0, T ); H 1 (Ω)) ∩ L∞ loc (Q ); 1 b) for all γ ∈ C0 (Ω × [0, T )) Z +∞ ϕ(c0 (x))s0 (x)γ(x, 0) dx = 0 ; (ϕ(c)sγt − ϕ(c)∇s∇γ + F (s, c)sγ) dx dt + 0

c) s(x, t) = ψ(x, t) for a.e. (x, t) ∈ ∂Ω × (0, T ]; T d) c ∈ C([0, T ]; L2 (Ω)) ∩ L2 ((0, T ); H 1 (Ω)) ∩ L∞ loc (Q ) and for a.e. x ∈ Ω and for all t ∈ [0, T ] Z t G(s(x, τ ), c(x, τ ))dτ . c(x, t) = c0 (x) + 0

2. A priori estimates We will obtain our existence and uniqueness results by an iterative prcedure, which involves the study of the problem described below. We introduce the following sets of functions (2.1)

W q (T ) = {f ∈ C([0, T ]; W 2,q (Ω)) ∩ C 1 ([0, T ]; Lq (Ω))} ,

and (2.2)

T . X q = f ∈ W q (T ) : f ≥ 0, kf kL∞ (QT ) ≤ S∞

T Here S∞ is the quantity introduced in assumption d) of Section 1 .

4


Let q > P and fix f ∈ X q . We take the solution c˜ of the ordinary differential problem   ∂t c˜ = G(f, c˜), (2.3)  c˜(x, 0) = c0 (x) ,

where c0 ≥ 0, c0 ∈ W 2,q (Ω) and the solution s˜ of the initial-boundary problem  ϕ(˜ c)∂t s˜ = ∇(ϕ(˜ c)∇˜ s) + F (˜ s, c˜) − s˜ϕ0 (˜ c)G(˜ s, c˜),      s˜(x, 0) = s0 (x), x ∈ Ω (2.4)      s˜(x, t) = ψ(x, t), (x, t) ∈ ∂Ω × (0, T ) ,

where s0 , ψ ≥ 0, s0 ∈ W 2,q (Ω) and ψ ∈ C([0, T ]; W 2,q (Ω)) ∩ C 1 ([0, T ]; Lq (Ω)) ∩ q W 2, 2 (QT ). We will obtain some a priori estimates for the solutions of the above problems. Our first step is to prove some a priori estimates for the solution of (2.3). Proposition 2.1. Let f ∈ X q and let c˜ be the solution of (2.3) in Ω × [0, T ]. Then for all t ∈ [0, T ], for all p such that 1 < p ≤ q, for j = 1, ..., N , we have Z tZ p p p |Gf ||fxj | eA1p t , (2.5) k˜ cxj kLp (Ω) ≤ kc0xj kLp (Ω) + 0

(2.6) k˜ cxj xj kpLp (Ω)

≤

(kc0xj xj kpLp (Ω) +

Z tZ 0

Ω

Ω

(|Gc˜c˜||˜ cxj |2p +|Gf f ||fxj |2p +|Gf ||fxj xj |p )eA2p t ,

where A1p = pkGc˜k∞ +(p−1)kGf k∞ and A2p = pkGc˜k∞ +(p−1)(kGf k∞ kGf f k∞ + kGc˜c˜k∞ Proof. First we derive the equation in (2.3) with respect to x j and we multiply it by |˜ cxj |p−2 c˜xj , obtaining (|˜ c xj | p ) t . = Gc˜|˜ cxj |p + Gf fxj c˜p−1 xj p Now, integrating over Ω × (0, t) and using Young’s inequality we have Z tZ Z Z Z Z p−1 1 t p p p |Gf | |˜ c xj | p |Gf ||fxj | + p |Gc˜| + |c0xj | + |˜ c xj | ≤ p 0 Ω p 0 Ω Ω Ω . Inequality (2.6) follows by the same technique.

Thanks to the previous proposition and the results in [14] we can assert that problem (2.4) has a solution s˜ ∈ C([0, T ]; L2 (Ω)) ∩ L2 ((0, T ); H 1 (Ω)). Moreover we remark that, if the data are smooth enough and satisfy standard compatibility conditions on the set {(x, t) : x ∈ ∂Ω , t = 0}, s˜ is a classical solution belonging to α the space C 2+α,1+ 2 (QT ) Now, we are going to prove that, if f ∈ X q then s˜ ∈ X q . All the computations in the proofs of the following results are made for classical solutions and then, by density arguments on the data, extended to solutions in X q . Notice that in the following estimates we are going to stress the dependence of all the constants on the data of the problem. First we obtain the estimate of ∇˜ s in L 2 (QT ).


5

Lemma 2.1. Let T > 0, f ∈ X q and let (˜ s, c˜) be the solution of (2.3)-(2.4) in QT . Then Z Z 2 ϕTm |∇˜ s|2 dxdt ≤ 2 ϕ(c0 (x)) s20 (x) + ψ E (x, 0) dx QT Ω (2.7) +K1 kψk2L2 (QT ) + 4ϕT k∇ψk2L2 (QT ) + Kψ + K2 T |Ω| , where ϕTm is the quantity introduced in assumption e) of Section 1, ϕT =

K1 =

Kψ = and K2 =

  

  2k5ϕ0 G + F k∞ + 2ϕT 

2k5ϕ0 G + F k∞

ϕ(c),

if N > 1 if N = 1 ,

2ϕT kψt k2L2 (QT )

if N > 1

T 4ϕT S∞ + kψkL∞ (Ω) kψkL1 (Ω)

if N = 1 ,

 T T 2 + 1) + (S∞ ) (4kϕ0 Gk∞ + ϕT )  2 kF k∞ (2S∞ 

sup T c∈[cT m ,C∞ ]

T 2 T ) kϕ0 Gk∞ + 1) + 4(S∞ 2 kF k∞ (2S∞

if N > 1 if N = 1 .

Proof. We introduce the function s = s˜ − ψ and we multiply the equation in (2.4) by s. Integrating by parts over QT we obtain Z Z Z s2 (x, t) s2 (x) ϕ(c0 (x)) 0 ϕ(˜ c(x, t)) ϕ(˜ c)|∇s|2 dx dτ = dx + dx 2 2 Ω Ω QT +

+

Z

Z

ϕ0 (˜ c) QT

QT

Z G(f, c˜) − G(˜ s, c˜) s2 dx dτ − ϕ(˜ c)∇ψ∇s dx dτ 2 QT

F (˜ s, c˜)s dx dτ −

Z

QT

ϕ0 (˜ c)G(f, c˜)ψs dx dτ −

Z

ϕ(˜ c)ψt s dx dτ . QT

Then, in the case N > 1, by Cauchy inequality, Z Z Z 2 2 0 T ϕ(c0 (x))s0 (x)dx + (4kϕ Gk∞ + ϕ ) ϕ(˜ c)|∇s| dxdτ ≤ QT

+2kF k∞ +ϕ

T

Z

Z

0

QT

s dx dτ + kϕ Gk∞ 2

QT

|ψt | dx dτ + ϕ

s2 (x, τ ) dx dτ

QT

Ω

T

Z

QT

Z

ψ 2 dx dτ QT

|∇ψ|2 dx dτ

and the claim follows. In the case N = 1 we use the estimate Z T + kψkL∞ (QT ) kψt kL1 (QT ) ϕ(˜ c)ψt s dxdτ ≤ ϕT S∞ QT

and we obtain the claim in similar way.

The following lemma gives an estimate for the Lp − norm of s˜t in terms of the data Lp − norm of ∇˜ s.

6


Lemma 2.2. Let T > 0, f ∈ X q and let (˜ s, c˜) be the solution to problem (2.3)-(2.4) in QT . Then, for suitable small and 1 < p ≤ q, there holds

sup t∈[0,T ]

+

Z

Ω

|˜ st |p dx ≤ Ap

ϕT ϕTm

Z

Ω

sup t∈[0,T ]

|ψt (x, 0)|p dx +

Z

Ω

|ψt |p dx + eK1 (p)T K2 (p)

p−1 0 kϕ G + ϕk∞ ϕTm

Z

QT

!

+ Bp eK1 (p)T K0 (p)

(|∇˜ s|p + |∇ψ|p ) dxdt ,

where Ap and Bp are positive constants depending on p and

ϕT K0 (p) = T ϕm

Z

|ϕ0 |∞ |∆s0 | dx + 2ϕTm Ω p

Z

Ω

(|∇c0 |

2p

kF + ϕ0 Gskp∞ + |∇s0 | )dx + p ϕTm 2p

,

p − 1p − 2 0 kϕ G + ϕk∞ + kϕ0 (2G + s˜Gs˜) − Fs˜ + ϕk∞ ϕTm 2

0 2p − 1 T 00 2 0

G + Gs˜s˜ − Fs˜ +S∞ kϕ G + ϕ Gc˜G + Fc˜Gk∞ + p ϕ

, p ∞

K1 (p) =

p−1 K2 (p) = T kϕ0 G + ϕk∞ ϕm

Z

QT

(|∇ψ|p + |∇ψt |p ) dxdt

0

+kϕ (2G + s˜Gs˜) − Fs˜ + ϕk∞

Z

QT

(|ψt |p + |ψtt |p ) dxdt

T +S∞ kϕ00 G2 + ϕ0 Gc˜G + Fc˜Gk∞ |Ω|T .

Proof. We use again the function s defined in the previous proof. We derive with respect to t the equation (2.4), then we multiply it by |s t |p−2 st and, writing the time derivatives of ϕ in terms of G(f, c˜) using (2.3), we have c) (G(f, c˜) + Gs˜(˜ s, c˜)˜ s + G(˜ s, c˜)) − Fs˜) ϕ(˜ c)|st |p−2 st stt + |st |p (ϕ0 (˜ +|st |p−2 st (ψt ϕ0 (˜ c)( G(f, c˜) + s˜Gs˜(˜ s, c˜) + G(˜ s, c˜) )−Fs˜ψt + ϕ(˜ c)ψtt ) s(ϕ00 (˜ c)G(f, c˜)G(˜ s, c˜) + ϕ0 (˜ c)Gc˜(˜ s, c˜)G(f, c˜)) +|st |p−2 st (˜ +Fc˜(˜ s, c˜)G(f, c˜)) =

P

ϕ(˜ c)(s + ψ)xj

xj t

|st |p−2 st .


7

Now we integrate over QT and taking into account that st is zero on ∂Ω × [0, T ], we have Z |st |p ϕ(˜ c) dxdt p QT t Z

+

+

+

Z

Z

p−1 G(f, c˜) + Gs˜(˜ s, c˜)˜ s + G(˜ s, c˜) − Fs˜ dxdt |st |p ϕ0 (˜ c) p

QT

QT

|st |p−2 st (ψt ϕ0 (˜ c) (G(f, c˜) + s˜Gs˜(˜ s, c˜) + G(˜ s, c˜)) − Fs˜ψt + ϕ(˜ c)ψtt ) dxdt

QT

s(ϕ00 (˜ c)G(f, c˜)G(˜ s, c˜) + ϕ0 (˜ c)Gc˜(˜ s, c˜)G(f, c˜)) + Fc˜G(f, c˜)) dxdt |st |p−2 st (˜

= − (p − 1)

Z

QT

N X

ϕ(˜ c)(s + ψ)xj

j=1

t

|st |p−2 stxj dxdt .

So, using Cauchy inequality to treat the term on the right hand side, we obtain Z Z ϕ(˜ c)|st |p T T dxdt + (p − 1) ϕm − kϕt k∞ − ϕ |st |p−2 |∇st |2 dxdt p 2 QT QT t p−1 ≤ 2 +

+

+

Z

Z

Z

Z

QT

|st |p−2 |ϕ0 (˜ c)G(f, c˜)| |∇s|2 + |∇ψ|2 + ϕ(˜ c)|∇ψt |2 dxdt

0 p−1 G(f, c˜) + Gs˜(˜ s, c˜)˜ s + G(˜ s, c˜) − Fs˜(˜ s, c˜) dxdt |st | ϕ (˜ c) p p

QT

QT

QT

E |st |p−1 |ψt ϕ0 (˜ dxdt c)( G(f, c˜) + s˜Gs˜(˜ s, c˜)G(˜ s, c˜) +Fs˜(˜ s, c˜)ψt + ϕ(˜ c)ψtt

s(ϕ00 (˜ c)G(f, c˜)G(˜ s, c˜) + ϕ0 (˜ c)Gc˜(˜ s, c˜)G(f, c˜)) + Fc˜(˜ s, c˜)G(f, c˜)| dxdt. |st |p−1 |˜

Now we use Young’s inequality to replace the terms with (p − 1) and (p − 2) powers of |st | on the right hand side of the above inequality by p powers for the quantities st , ∇s, ψ, ψt , ψtt , ∇ψt : Z Z ϕ(˜ c)|st |p dxdt + (p − 1) ϕTm − kϕt k∞ − ϕT |st |p−2 |∇st |2 dxdt p 2 QT QT t ≤

p−1 2

kϕ0 G + ϕk∞

Z

QT

p−2 p

R

|st |p +

2 (|∇s|p + |∇ψ|p + |∇ψt |p ) p

Z

0 p−1

+ ϕ |st |p dxdt G + G s ˜ + G − F s˜ s˜

p T ∞ Q + kϕ0 (2G + s˜Gs˜) − Fs˜ + ϕk∞

QT

T 00 2 (ϕ G + ϕ0 Gc˜G + Fc˜G ∞ +S∞

Z

p−1 p p |st |

QT

+

1 p

(|ψt |p + |ψtt |p )

p−1 p 1 |st | + p p

dxdt .

dxdt

dxdt

8


It follows that Z |st |p dx Ω

Z

ϕT ≤ T ϕm

p−1 0 kϕ G + ϕk∞ |st (x, 0)| dx + ϕTm Ω p

+K2 (p) + K1 (p)

Z

QT

Z

QT

|∇s|p dxdt

spt dxdt .

So, by Gronwall lemma we obtain Z Z p |ψt |p |˜ st | dx ≤ Ap sup sup t∈[0,T ]

+Ap

t∈[0,T ]

Ω

ϕT ϕTm

Z

Ω

|st (x, 0)|p +

Ω

p−1 0 p kϕ G + ϕk k∇sk + K (p) eK1 (p)T . ∞ 2 Lp (QT ) ϕTm

In the next lemma we estimate sup[0,T ] k∇˜ skL2p (Ω) and sup[0,T ] k∆˜ skLp (Ω) in terms of sup[0,T ] k˜ st kLp (Ω) and of the data.

Lemma 2.3. Let T > 0, f ∈ X q and let (˜ s, c˜) be the solution of problem (2.3)-(2.4) in QT . Then there exist two positive constants N∇ , N∆ , depending on p and on Ω, such that, for all 1 < p ≤ q, " sup

[0,T ]

T k∇˜ sk2p L2p (Ω) ≤ N∇ S∞

p ϕ0

p

T )2p sup k∇˜ ck2p ( ϕM S∞ L2p (Ω) m [0,T ]

p

+

T ) 2 ϕ0 pM ( S∞ p ϕT m

sup [0,T ]

k∇˜ ckpL2p (Ω)

p 2

p 2

sup k˜ skLp (Ω) + sup kψkW 2,p (Ω) [0,T ]

[0,T ]

!

# |Ω| + sup k˜ st kpLp (Ω) + T p kF − ϕ0 G˜ skp∞ + sup kψkpW 2,p (Ω) , ϕm [0,T ] [0,T ] and sup [0,T ]

k∆˜ skpLp (Ω)

"

T ≤ N ∆ S∞

p

ϕ0M ϕm

2p

sup k∇˜ ck2p L2p (Ω)

[0,T ]

p

(S T ) 2 ϕ0 p + ∞ϕT p M m

+ sup [0,T ]

where

ϕ0M

sup [0,T ]

k˜ st kpLp (Ω)

k∇˜ ckpL2p (Ω)

p 2

p 2

sup k˜ skLp (Ω) + sup kψkW 2,p (Ω)

[0,T ]

[0,T ]

!

# |Ω| 0 p + T p kF − ϕ G˜ sk∞ , ϕm

= max ϕ0 (c). T [cT m ,C∞ ]

Proof. Let p ≥ 1. For suitable N3 depending on p, we have p Z Z Z Z F (˜ s, c˜) − ϕ0 (˜ c)G(˜ s, c˜)˜ s |ϕ0 (˜ c)|p |∇˜ c|p ∇˜ s|p p p . + |˜ st | + |∆˜ s| ≤ N3 |ϕ(˜ c)|p ϕ(˜ c) Ω Ω Ω Ω By using Holder’s inequality, we obtain Z Z p ϕ0M p p p p p |˜ st | + T p k∇˜ |∆˜ s| ≤ N3 (2.8) ckL2p (Ω) k∇˜ skL2p (Ω) + |Ω|Σ , ϕm Ω Ω setting Σ =

kF − ϕ0 G˜ skL∞ (Ω) . T ϕm


9

Now, by using Gagliardo-Niremberg’s interpolation inequality we obtain the following estimate k∇˜ skpL2p (Ω)

p

p

T 2 2 ≤ CGN k˜ skW 2,p (Ω) (S∞ ) p

T 2 ≤ CGN (S∞ ) k˜ s − ψkW 2,p (Ω) + kψkW 2,p (Ω)

(2.9)

p2

p p p T p 2 ≤ N2 (S∞ ) 2 k˜ skL2 p (Ω) + k∆˜ , skL2 p (Ω) + kψkW 2,p (Ω)

where CGN and N2 are suitable constants depending on p and Ω. Using the above estimate in (2.8) we have st kpLp (Ω) + |Ω|Σp k∆˜ skpLp (Ω) ≤ N3 k˜ p

+N3 N2

T 2 0 p ) ϕM (S∞ k∇˜ ckpL2p (Ω) p ϕT m p

p p p 2 skL2 p (Ω) + kψkW . k∆˜ skL2 p (Ω) + k˜ 2,p (Ω)

2 therefore the quantity k∆˜ skp,Ω can be estimate as follows q p 1 k∆˜ skL2 p (Ω) ≤ A1 + A21 + 4A2 , 2

where

p

p

T 2 0 N2 N3 (S∞ ) ϕM k∇˜ ckpL2p (Ω) A1 := p ϕTm

and

p

A2

:=

T 2 0 p ) ϕM N2 N3 (S∞ k∇˜ ckpL2p (Ω) p ϕT m

+N3 k˜ st kpLp (Ω) + |Ω|Σp .

p

p

2 k˜ skL2 p (Ω) + kψkW 2,p (Ω)

Now we can use the previous estimates in (2.9) to obtain ! p 2 + 4A p p A A + 2 1 p 1 T p 2 skL2 p (Ω) + , k∇˜ skL2p (Ω) ≤ N2 (S∞ ) 2 k˜ + kψkW 2,p (Ω) 2

which proves the claims.

Estimate (2.7) provides the starting point for a recurrency procedure which, by using Lemmas 2.2, 2.3 and interpolation results in L p spaces, proves that if f ∈ W q (T ), for q > P , then s˜ ∈ W q (T ). The proof is straighforward in the case q = 2M , for some integer M > 0. Otherwise 2M < q < 2M +1 , for some integer M > 0; in this case the recurrency argument gives a bound for sup k˜ s t kp,Ω for p ≤ 2M , in particular for p = 2q . Using Lemma 2.3 and then Lemma 2.2 for p = 2q we obtain that s˜ ∈ W q (T ). The results of this section imply that if f ∈ X q then s˜ ∈ X q . 3. Local existence We introduce the maps Φc and Φs which associate to f ∈ X q , q > P , respectively the solution c˜ to problem (2.3) and the solution s˜ ∈ X q to problem (2.4); moreover, starting from s0 ∈ X q , we define by recurrency the sequences cn = Φc (cn−1 ) and sn = Φs (sn−1 ) for n ∈ N. Thanks to the results of the previous section, we are going to prove local uniform estimates for them. Thanks to the assumption d) in Section 2, we know that, for all T > 0, the sequences are uniformly bounded in L∞ (QT ) .

10


We remark that the estimate in Lemma 2.1 depends on f and c˜ only through T T the quantities S∞ , C∞ , cTm ; hence we can write, for all n ∈ N, k∇sn kL2 (QT ) ≤ D2Q (T )

(3.1)

T T where D2Q (T ) is a positive constant determined by the data S ∞ , C∞ , cTm , Ω and T , non decreasing with T , independent on n. Moreover, by using Lemma 2.2, for 1 < p ≤ q, we have p n+1 p eK1 (p)T (3.2) sup ksn+1 k ≤ D (p, T ) + D (p, T ) 1 + k∇s k p 1 2 p T t L (Ω) L (Q ) [0,T ]

T T where Di (p) , i = 1, 2 are positive quantities depending on the data, S ∞ , C∞ , cTm , Ω and T , non decreasing with T , independent on n, and K 1 (p) is the quantity defined in Lemma 2.2; by using Lemma 2.3, for 1 ≤ p ≤ q, we have ∇ sup k∇sn+1 k2p L2p (Ω) ≤ D1 (p, T )

[0,T ]

sup k∇cn+1 k2p L2p (Ω)

[0,T ]

(3.3) + sup [0,T ]

k∇cn+1 kpL2p (Ω)

+ sup [0,T ]

ksn+1 kpLp (Ω) t

+1

!

and sup k∆sn+1 kpLp (Ω) ≤ D1∆ (p, T )

sup k∇cn+1 k2p L2p (Ω)

[0,T ]

[0,T ]

(3.4) + sup [0,T ]

k∇cn+1 kpL2p (Ω)

+ sup [0,T ]

ksn+1 kpLp (Ω) t

+1

!

,

T T where D1∇ (p, T ) , D1∆ (p, T ) are positive quantities depending on the data, S ∞ , C∞ , cTm , Ω and T , non decreasing with T , independent on n. By using Proposition 2.1 we are able to estimate the norms of c n+1 by means of n s !

(3.5)

sup k∇cn+1 kpLp (Ω) ≤

[0,T ]

D1c (p) + D2c (p)T sup k∇sn kpLp (Ω)

c

eD3 (p)T ,

[0,T ]

T T where Dic are positive quantities depending on the data, S ∞ , C∞ , cTm , Ω and T , non decreasing with T , independent on n. Therefore the inequalities (3.3) and (3.4) can be rewritten as follows " n+1 p ∇ sup k∇sn+1 k2p kLp (Ω) L2p (Ω) ≤ D2 (p, T ) 1 + sup kst

[0,T ]

[0,T ]

(3.6) +e

D c (p)T

(1 + T sup [0,T ]

k∇sn k2p L2p (Ω)

and sup [0,T ]

k∆sn+1 kpLp (Ω)

≤

D2∆ (p, T )

"

√ + T sup k∇sn kpL2p (Ω) ) [0,T ]

1 + sup ksn+1 kpLp (Ω) t [0,T ]

(3.7) +eD

c

(p)T

1 + T sup [0,T ]

#

k∇sn k2p L2p (Ω)

√ + T sup k∇sn kpL2 p(Ω) [0,T ]

!#

.


11

Let us emphasize that constants D2∇ , D2∆ , D c only depend on the norms of boundary T T and initial data, on S∞ , C∞ , cTm , Ω and on T ; moreover they are non decreasing when the previous quantities increase and are independent on n. Proposition 3.1. Let T > 0, s0 ∈ X q , q > P and q =

sup

2m . There exist

m∈N:2m ≤q

δ > 0 and 0 < Tq ≤ T , such that, for all p = 2m ≤ q, m ∈ N, sup ksnt (·, t)kpLp (Ω) ≤ K(p)

(3.8)

[0,Tq ]

(3.9)

∇ sup k∇sn (·, t)k2p L2p (Ω) ≤ Kδ (p)

for all n ∈ N ,

sup k∆sn (·, t)kpLp (Ω) ≤ Kδ∆ (p)

for all n ∈ N ,

[0,Tq ]

(3.10)

[0,Tq ]

where

for all n ∈ N ,

K(p) = D1 (p, T ) + D2 (p, T ) 1 + D2Q (T ) eK1 (p)T , c Kδ∇ (p) = D2∇ (p, T ) 1 + K(p, T ) + eD (p)T (1 + δ) , c Kδ∆ (p) = D2∆ 1 + K(p, T ) + eD (p)T (1 + δ) .

Proof. Let δ be large enough to have the inequality (3.9) satisfied for s 0 over the interval [0, T ]; we shall show that there exists Tq , 0 < Tq ≤ T , depending only on T T the data, S∞ , C∞ , cTm , Ω, Kδ∇ (p), such that s1 verifies inequalities (3.8)-(3.10). Let us fix T2 such that q (3.11) T2 Kδ∇ (2)) + T2 Kδ∇ (2) < δ Taking into account estimates (3.1),(3.2), (3.6), (3.7), this choice of T 2 implies that sup ks1t (·, t)k2L2 (Ω) ≤ K(2) , sup k∆s1 (·, t)k2L2 (Ω) ≤ Kδ∆ (2)

[0,T2 ]

[0,T2 ]

and sup k∇s1 (·, t)k4L4 (Ω) ≤ Kδ∇ (2) .

[0,T2 ]

Now we use a recursive argument based on inequalities (3.2), (3.6), (3.7), which allows to obtain L2p estimates provided Lp estimates: provided T2m we choose T2m+1 ≤ T2m such that (3.12) (3.13)

T2m+1 Kδ∇ (2m+1 ) < D2Q (T ) , q T2m+1 Kδ∇ (2m+1 ) + T2m+1 Kδ∇ (2m+1 ) < δ

which implies, thanks to (3.2), (3.6), (3.7), that sup [0,T2m+1 ]

m+1

ks1t (·, t)k2L2m+1 (Ω) ≤ K(2m+1 ) ,

and sup [0,T2m+1 ]

sup [0,T2m+1 ]

m+1

k∆s1 (·, t)k2L2m+1 (Ω) ≤ Kδ∆ (2m+1 )

m+2

k∇s1 (·, t)k2L2m+2 (Ω) ≤ Kδ∇ (2m+1 ) .

In a finite number of steps we determine Tq such that inequalities (3.8)-(3.10) are T T , C∞ , cTm , Ω, Kδ∇ (p), by satisfied by s1 . Since Tq depends only on the data, T, S∞ inductive argument we prove the claim. Proposition 3.2. Let q > P , s0 ∈ X q and Tq be the quantity introduced in the previous proposition. The sequences {sn } and {cn } are uniformly bounded in W q (Tq ).

12


Proof. By (3.1), Proposition 3.1 and interpolation results in L p spaces we obtain uniform estimates in C([0, Tq ]; W 2,p (Ω)) ∩ C 1 ([0, Tq ]; Lp (Ω)) for 2 ≤ p ≤ 2M where 2M ≤ q < 2M +1 . If q = 2M the proof is complete. Otherwise 2q < 2M , then we have q q uniform estimates in C([0, Tq ]; W 2, 2 (Ω))∩C 1 ([0, Tq ]; L 2 (Ω)); hence we can perform one step of the recursive argument introduced in the proof of Proposition 3.1 which allows to obtain the uniform estimates in C([0, Tq ]; W 2,q (Ω)) ∩ C 1 ([0, Tq ]; Lq (Ω)). As regard to the sequence {cn }, the claim follows by inequality (3.5), (2.6) and (3.7). Now , by compactness techniques, we are able to prove the following existence theorem. Theorem 3.1. Take q > P , and let c0 , s0 , ψ ≥ 0, with s0 , c0 ∈ W 2,q (Ω) and ψ ∈ q T C([0, T ); W 2,q (Ω)) ∩ C 1 ([0, T ); Lq (Ω)) ∩ W 2, 2 (Q ) for all T > 0; let assumptions a)d) of Section 1 be satisfied. Then there exists a weak solution (s, c) to problem (1.1)(1.3) in Ω × (0, Tq ); moreover (s, c) ∈ (C([0, Tq ]; W 2,q (Ω)) ∩ C 1 ([0, Tq ]; Lq (Ω)))2 . Proof. Thanks to Proposition 3.2, we can can use compactness arguments to prove that a subsequence of {sn } converges weakly∗ in the space C([0, Tq ]; W 2,q (Ω)) ∩ C 1 ([0, Tq ]; Lq (Ω)) to a limit function s; the same arguments prove that a subsequence of {cn } converges weakly∗ in C([0, Tq ]; W 2,q (Ω)) ∩ C 1 ([0, Tq ]; Lq (Ω)) to a limit function c. For all z ∈ C01 (Ω×[0, Tq )), the functions of the sequence {sn } satisfy the following problem Z Z Z n n ϕ(c )s zt dxdt = − ϕ(c0 )s0 z(x, 0) dx + ϕ(cn )∇sn ∇z dxdt QT

(3.14)

−

Z

QT

Ω

QT

F (sn , cn ) − ϕ0 (cn )(G(sn , cn ) − G(sn−1 , cn )sn z dxdt ,

sn (x, t) = ψ(x, t)

(x, t) ∈ ∂Ω × (0, Tq )

and the functions of the sequence {cn } can be written Z t n G(sn−1 (x, τ ), cn (x, τ ))dτ . (3.15) c (x, t) = c0 (x) + 0

Since the sequences s and c are relatively compact in L2 (ΩTq ) and ∇sn is bounded in the same space, we can pass to the limit for n → ∞ , along subsequences, and show that the pair (s, c) is a local weak solution of problem (1.1)-(1.6). n

n

4. Global existence, stability and uniqueness In this sections we first show that the local solution obtained in Section 4 can be extended over every time interval [0, T ); then we state a stability theorem, with respect to the data, for weak solutions belonging to C([0, T ); W 1,P (Ω)), where P is defined in Section 1. The consequence of this result is the existence and uniqueness of global weak solutions belonging to the above space. Theorem 4.1. Let the assumption a)-d) of Section 1 be satified and let s 0 , c0 , ψ ≥ q T 0, s0 , c0 ∈ W 2,q (Ω), ψ ∈ C([0, T ]; W 2,q (Ω)) ∩ C 1 ([0, T ]; Lq (Ω))) ∩ W 2, 2 (Q ) for all T > 0 and q > P . Then there exits a global weak solution to problem (1.1)-(1.3), (s, c) ∈ (C([0, T ]; W 2,q (Ω)) ∩ C 1 ([0, T ]; Lq (Ω)))2 .


13

Proof. Let (s, c) be the local weak solution obtained in Theorem 3.1. Let T > Tq T T and let S∞ , C∞ , cTm the quantities introduced in assumption d) in Section 1. We T T know that kskL∞ (QTq ) ≤ S∞ and cTm ≤ c(x, t) ≤ C∞ for (x, t) ∈ (Ω × (0, Tq )); moreover, the functions s and c are bounded in W (Tq ). For this reason, the same arguments used to obtain the local solution allow to extend it over a time interval T T , C∞ , cTm and the W (Tq )-norms of s and c [0, Tq + τ ], where τ depends on Ω, T , S∞ . On the other hand, the estimates obtained for the sequences {s n }, {cn } in the previous sections can be derived, by the same thecniques, for the solution of problem (1.1)-(1.3). First of all, in analogy with (3.1) and (3.2), we have k∇skL2 (Ω×(0,Tq )) ≤ D2Q (T ) ,

(4.1)

(4.2)

sup kst kpLp (Ω) ≤ D1 (p, T ) + D2 (p, T ) 1 + k∇skpLp (Ω×(0,Tq )) eK1 (p)T ,

[0,Tq ]

where D2Q (T ), D1 (p, T ), D2 (p, T ), K1 (p) are the same quantities introduced in (3.1) T T and (3.2), determined by the norms of the data, S ∞ , C∞ , cTm , Ω, T . Then, proceeding as in the proof of of Proposition 2.1, we obtain c (4.3) sup k∇ckpLp (Ω) ≤ D1c (p) + D2c (p)k∇skpLp (Ω×(0,Tq )) eD3 (p)T , [0,Tq ]

D1c (p), D2c (p), D3c (p)

where are the quantities introduced in (3.5); moreover, arguing as in Lemma 2.3, thanks to the above estimate, we obtain the following inequalities p ∇ k∇s(·, t)k2p L2p (Ω) ≤ D2 (p, T ) 1 + sup kst kLp (Ω) [0,Tq ]

(4.4) +eD

c

(p)T

Z t 1+T k∇s(·, τ )k2p dτ , 2p L (Ω) 0

sup k∆skpLp (Ω) ≤ D2∆ (p, T ) 1 + sup kst kpLp (Ω)

[0,Tq ]

[0,Tq ]

(4.5) +e

D c (p)T

1 + T sup [0,Tq ]

k∇sk2p L2p (Ω)

+

√

T sup [0,Tq ]

k∇skp2p(Ω)

!!

,

where D2∇ (p, T ), D2∆ (p, T D c (p) are the quantities introduced in (3.6) and (3.7). Using the Gronwall lemma in (4.4), we obtain the estimate of sup k∇sk2p L2p (Ω) in [0,Tq ]

T T terms of the norms of the data, S∞ , C∞ , cTm , Ω, T and kst kL∞ ((0,Tq );Lp (Ω)) ; thanks p to (4.5) we are able to control sup k∆skLp (Ω) by means the same quantities. Using [0,Tq ]

the recursive procedure introduced in the previous sections we estimate the W (Tq )T T norms of s and c in terms of the norms of the data, S∞ , C∞ , cTm , Ω, T ; it follows that τ depends only on these quantities and the extenction procedure can be repeated until reaching T . The final part of this section is devoted to establish a stability result in the norm |f |X := sup kf (·, t)kL2 (Ω) + k∇f kL2 (Ω×(0,T )) . First we need to prove the following [0,T ]

lemma, P being defined as in Section 1.

14


Lemma 4.1. Let the assumption a)-d) of Section 1 be satisfied, let T > 0 and let (s0 , c0 , ψ), (s∗0 , c∗0 , ψ ∗ ) nonnegative data verifying the assumptions (1.4)-(1.6). If there exist solutions (s, c), (s∗ , c∗ ) ∈ (C([0, T ]; W 1,P (Ω)))2 to problem (1.1), corresponding respectively to initial-boudary data (s0 , c0 , ψ), (s∗0 , c∗0 , ψ ∗ ) , then there exists T T a costant C, depending on the data, P, T, S∞ , C∞ , cTm and the C([0, T ]; W 1,P (Ω))norms of the solutions, such that ! Z T ∗ 2 ∗ 2 ∗ 2 |c − c |X ≤ C kc0 − c0 kH 1 (Ω + |s − s |X . 0

Proof. Taking into account that Z t Z t G(s∗ (x, τ )c∗ (x, τ )) dτ , G(s(x, τ )c(x, τ )) dτ , c∗ (x, t) = c∗0 (x)+ c(x, t) = c0 (x)+ 0

0

∗

we easily estimate the term |c − c | as follows

|c(x, t) − c∗ (x, t)| ≤ etkGc k∞ (|c0 (x) − c∗0 (x)| +kGs k∞

Rt 0

|s(x, τ ) − s∗ (x, τ )| dτ ekGc k∞ t .

Concerning the term k∇c − ∇c∗ kL2 (QT ) , we easily obtain (4.6) k∇c(·, t) − ∇c∗ (·, t) k2L2 (Ω) ≤ C1 k∇c0 − ∇c∗0 k2L2 (Ω) +kGs k2∞ t +kGc k2∞ t

Z

Z

+kGss k2∞ t +kGcc k2∞ t +kGsc k2∞ t +kGsc k2∞ t

t

0 t

0

k∇s(·, τ ) − ∇s∗ (·, τ )k2L2 (Ω) dτ k∇c(·, τ ) − ∇c∗ (·, τ )k2L2 (Ω) dτ

Z tZ

Ω

|∇s∗ (x, τ )|2 |s(x, τ ) − s∗ (x, τ )|2 dxdτ

Ω

|∇c∗ (x, τ )|2 |c(x, τ ) − c∗ (x, τ )|2 dxdτ

0

Z tZ 0

Z tZ 0

Ω

Z tZ 0

Ω

|∇s∗ (x, τ )|2 |c(x, τ ) − c∗ (x, τ )|2 dxdτ |∇c∗ (x, τ )|2 |s(x, τ ) − s∗ (x, τ )|2 dxdτ

Now we treat the fourth integral on the right hand side in the following manner Z tZ 2 2 |∇s∗ |2 |s − s∗ |2 ≤ k∇s∗ k 2p , 2r T ks − s∗ kL2p,2r (QT ) 0

L p−1

Ω

r−1

(Q )

with

r = 1,

p=

P P −2

when N > 2 (P = N ),

P 2

, p=

P P −2

when N = 2 (P > 2),

r=

r = P,

p=∞

when N = 1 (P = 2).

.


15

For these choices of the parameters (p, r), we have the following embedding (see [14]) k · k2L2p,2r (QT ) ≤ γ| · |2X ,

for suitable γ > 0.

This yields Z tZ

(4.7)

0

Ω

|∇s∗ |2 |s − s∗ |2 ≤ γk∇s∗ k2 P, L

2r r−1

(Ω)

|s − s∗ |X ,

and, by arguing as above, we estimate in the same manner the other three integrals which follow. Hence, we use these estmates in (4.6) to obtain (4.8) k∇c(·, t) − ∇c∗ (·, t)k2L2 (Ω) ≤ C2 k∇c0 − ∇c∗0 k2L2 (Ω) + |s − s∗ |2X + |c − c∗ |2X T T where C2 depends on P , T , Ω, S∞ , C∞ , cTm and the norms sup(0,T ) k∇s∗ kLP (Ω) ∗ and sup(0,T ) k∇c kLP (Ω) . Now the claim follows putting together (4.6) and (4.8), since we obtain ! Z T Z T ∗ 2 ∗ 2 ∗ 2 ∗ 2 |c − c |X , |s − s |X + |c − c |X ≤ C3 kc0 − c0 kH 1 (Ω) + 0

0

T T where C3 depends on P , T , Ω, S∞ , C∞ , cTm and the norms sup(0,T ) k∇s∗ kLP (Ω) ∗ and sup(0,T ) k∇c kLP (Ω) .

Theorem 4.2. Let the assumption a)-d) of Section 1 be satisfied, let T > 0 and let (s0 , c0 , ψ), (s∗0 , c∗0 , ψ ∗ ) nonnegative data verifying the assumptions (1.4)- (1.6). P If there exist solutions (s, c), (s∗ , c∗ ) ∈ (C([0, T ]; W 1,P (Ω)) ∩ C 1 ([0, T ]; L 2 (Ω)))2 to problem (1.1), corresponding respectively to initial-boudary data (s0 , c0 , ψ) and T T (s∗0 , c∗0 , ψ ∗ ), then there exists a costant K, depending on the data, P, T, S∞ , C∞ , cTm P end the C([0, T ]; W 1,P (Ω)) ∩ C 1 ([0, T ]; L 2 (Ω))- norms the solutions, such that |s − s∗ |2X ≤ K ks0 − s∗0 k2L2 (Ω) + kψt − ψt∗ kpLp (QT ) + kc0 − c∗0 kH 1 (Ω) + |ψ − ψ ∗ |X , where p = 2 if N ≥ 2 and p = 1 if N = 1.

Proof. We introduce the functions s = s − ψ and s∗ = s∗ − ψ ∗ . We subtracting the equation satisfied by s∗ by the one satisfied by s, we multiply by (s − s∗ ) and, integrating over QT , we obtain Z Z (ϕs − ϕ∗ s∗ )t (s − s∗ ) dxdt = div(ϕ∇s − ϕ∗ ∇s∗ )(s − s∗ ) dxdt QT

(4.9)

+

+

Z

Z

QT

∗

div(ϕ∇ψ − ϕ∗ ∇ψ )(s − s∗ ) dxdt −

QT

ΩT

Z

QT

(ϕψ − ϕ∗ ψ ∗ )t (s − s∗ ) dxdt

(F − F ∗ )(s − s∗ ) dxdt ,

∗

where ϕ = ϕ(c∗ ), F ∗ = F (s∗ , c∗ ). For the term on the left hand side we have T Z Z Z ∗ 2 |s − s∗ |2 0 (s − s ) ∗ ∗ ∗ ϕ dxdt dx + ϕ G (ϕs − ϕ s )t (s − s ) dxdt = 2 2 Ω QT QT 0 −

Z

QT

s∗t (ϕ − ϕ∗ )(s − s∗ ) dxdt −

Z

QT

s∗ (ϕ − ϕ∗ )t (s − s∗ ) dxdt

16


and for the third term on the right hand side Z ∗ ∗ ∗ T (ϕψ − ϕ ψ )t (s − s ) dx dt Q

≤ C1 +

Z

Z

QT

QT

(|ψ − ψ ∗ | + |ψt − ψt∗ |) |s − s∗ | dxdt

(|ψt∗ |

∗

∗

∗

∗

+ |ψ |) (|s − s | + |c − c |) |s − s | dxdt

,

T where C1 depends on S T , C∞ , cTm . Now, by using Cauchy inequality and applying the divergence theorem to the first and the second term on the right hand side of (4.9), we obtain Z Z Z ϕ |s − s∗ |2 dx + |∇s − ∇s∗ |2 dxdt ≤ C2 |s0 − s∗0 |2 dx 2 T Ω Q Ω

+

+

+

+

Z

Z

Z

QT

(|s − s∗ | + |c − c∗ | + 1)|s − s∗ | dxdt

QT

(|s∗t | + |s∗ | + |ψt∗ | + |ψ ∗ |)(|s − s∗ |2 + |c − c∗ |2 ) dxdt

QT

(|∇s∗ ||c − c∗ | + |∇ψ − ∇ψ ∗ |) |∇s − ∇s∗ | dxdt

Z

QT

(|ψ − ψ ∗ | + |ψt − ψt∗ |)|s − s∗ | dxdt

,

T where C2 depends on S T , C∞ , cTm . Using again the Cauchy inequality and arguing as in the proof of the previous lemma to treat the third and fourth integral on the right hand side, we have

|s − s∗ |2X ≤ K1 |c − c∗ |2X +C3

+

Z

Z

Ω

|s0 −

s∗0 |2

dx +

Z

∗ 2

QT

QT

|s − s∗ |2 + |c − c∗ |2 + |s − s∗ |2 dxdt ∗ 2

(|ψ − ψ | + |∇ψ − ∇ψ | + |ψt −

ψt∗ |

∗

|s − s |) dxdt

,

T where C3 depends on the data, P, Ω, S T , C∞ , cTm and K1 depends also on the P C([0, T ]; W 1,P (Ω)) ∩ C 1 ([0, T ]; L 2 (Ω))-norms of s∗ , ψ ∗ . By using Lemma 4.1 and the expressions for s, s∗ we obtain Z T |s − s∗ |2X + |ψ − ψ ∗ |2X dt |s − s∗ |2X ≤ K2 0

+ks0 − s∗0 k2L2 (Ω) + kc0 − c∗0 k2H 1 (Ω) + kψt − ψt∗ kpLp (QT ) ,

T where the constant K2 depends on T, Ω, S∞ , cT∞ , cTm , and on the C([0, T ]; W 1,P (Ω))P norms of c∗ and the C([0, T ]; W 1,P (Ω)) ∩ C 1 ([0, T ]; L 2 (Ω))-norms of s∗ , ψ ∗ ; here p = 2 if N ≥ 2 and p = 1 if N = 1 .


17

We recall that, thanks to the results in Lemmas 2.2 and 2.3, we can bound the P C([0, T ]; W 1,P (Ω)) ∩ C 1 ([0, T ]; L 2 (Ω))-norms of solutions to problem (1.1)-(1.6) by P T T means of T, Ω, S∞ , C∞ , cTm , the W 2, 2 (Ω)-norm of s0 , the W 1,P (Ω)-norm of c0 and P P the C([0, T ); W 2, 2 (Ω))∩C 1 ([0, T ); L 2 (Ω))-norm of ψ. This fact , together with the previous stability result, imply the main theorem for the existence and uniqueness of weak solutions to problem (1.1)-(1.6). Theorem 4.3. Let assumptions a)-d) of Section 1 be satisfied. Then there exists a unique global weak solution to problem (1.1)-(1.6). 5. A uniform Lp estimate The results obtained until now are strongly based on the assumption that a priori L∞ bounds are available for the solutions of system (1.1). On the other hand, it is known that for several systems in the class (1.1) such estimates can be hardly obtained and, actually, in some case we have even the blow up in finite time of the L∞ norm of solutions (see [19, 3]). In [3] the authors proved a uniform L p -estimate for the solutions of a reaction-diffusion system, related to chemotaxis modeling, which can be written in the form (1.1) with F (c, s) = 0 and G(c, s) = −ϕ(c)c γ s (see next section), without using the global boundedness of L ∞ norm; they prove their result for the Cauchy problem, in several space dimension, by controlling the N growth in time of ks(·, t)kLp (RN ) , for small data in L 2 (RN ). In the same spirit, here we show that it is possible to obtain the same kind of global Lp estimate for our initial-boundary value problems, again without any assumption about global boundedness of the L∞ norm of the solution. Let us assume that, for all T > 0, the function G allows to determine a priori T two quantities, cTm ≥ 0 and C∞ > 0 such that , for all s ≥ 0, the second equation T in (1.1) has solutions satisfying the bounds cTm ≤ c ≤ C∞ and we assume that conditions a)-c) and e) of Section 2 are satisfied; finally we assume that there exist some constants ϕm , ϕM , ϕ0M > 0 such that T ϕm ≤ ϕ(c) ≤ ϕM , ϕ0 (c) ≤ ϕ0M for all c ∈ [ inf cTm , sup C∞ ];

(5.1)

T ≥0

T ≥0

there exists a constant k such that T |F (s, c)| ≤ ks2 , |G(s, c)| ≤ ks , for all c ∈ [ inf cTm , sup C∞ ].

(5.2)

T ≥0

T ≥0

Then we are going to prove the following theorem. Theorem 5.1. Assume condtion a)–c) and e) of Section 1, and the new condtions (5.1) and (5.2). Let N > 2 and q ≥ max{1, N2 − 1}. Let s0 , c0 , ψ0 nonnegative functions, c0 ∈ L∞ (Ω). There exist constants K1 = K1 (N, Ω, ϕm , ϕM , ϕ0M , k, q) , kψ(·, 0)k N2 ) such that, if and K2 = K2 (N, Ω, ϕm , ϕM , ϕ0M , k, ks0 k N2 L

ks0 k and sup [0,+∞)

Z

L

N 2

N

Ω

|ψ 2 +1 | dx , sup

[0,+∞)

(Ω)

Z

+ kψ(·, 0)k N +1 2 2

Ω

(Ω)

|ψt

L

L

N 2

(Ω)

| dx , sup

(Ω)

≤ K1 , ,

[0,+∞)

Z

3

Ω

N

|∇ψ 2 ( 2 +1) | dx ≤ K2 ,

then sup ks(·, t)kLq (Ω) is bounded by a quantity depending on ks0 kLq (Ω) , [0,T ] Z Z Z q +1 2 q 3 q +1 2 2 |∇ψ 2 ( 2 +1) | dx. |ψt | dx , sup | dx , sup |ψ sup kψ(·, 0)kLq (Ω) , sup [0,T ]

[0,T ]

Ω

[0,T ]

Ω

[0,T ]

Ω

18


Proof. By proceeding in similar way as in the proof of Proposition 2.1, we multiply the first equation in (1.1) by s|s|p−2 , and integrate over Ω to obtain Z Z |s|p (x, t) |s|p 0 ϕ(˜ c(x, t)) dx − ϕ (c)G(s, c) dx p Ω Ω p t = −(p − 1) +

Z

ϕ(c) Ω

+(p − 1)

Z

Z

ϕ(c)|s|

p−2

Ω

2

|∇s| dx +

Z

Ω

p+1 2 p − 1 p+1 + |s| |ψ E | 2 p+1 p+1 t

s|s|p−2 (F (s, c) − ϕ0 (c)sG(s, c)) dx

dx 2

ϕ(c)|s|

1 (p − 2)|s|p+1 + 3|∇ψ E | 3 (p+1) k∇s|2 + 2 2(p + 1)

p−2

Ω

!

dx.

By using the new assumptions (5.1) and (5.2) on the functions F, G, we have Z Z 2(p − 1) ϕ(c)|s|p−2 |∇s|2 dx (ϕ(˜ c(x, t))|s|p (x, t))t dx + p Ω Ω (5.3) Z ϕ(c)|s|p+1 dx + D(p, t) ≤ µ2 Ω

where µ2 =

2ϕ0M (1 + 2p) + ϕm p2 (p − 1) + 4kp2 2ϕm (p + 1)

and D(p, t) = (5.4) M + 2pϕ p+1

R

k(ϕ0M + 2p) p+1

|ψt | Ω

p+1 2

+

Z

Ω

|ψ|(p+1)

3p(p−1)ϕM 2

R

2

|∇ψ| 3 (p+1) Ω

o

.

Now we treat the first term in the right hand side arguing as in [3]; by using standard interpolation and the Gagliardo-Niremberg- Sobolev inequality , for all p ≥ max{1, N2 − 1}, we have Z p ϕ(c)sp+1 ≤ C(N, p)ϕM k∇(s 2 )k2L2 (Ω) ksk N2 L

Ω

(5.5)

p

N

≤ C(N, p) ϕM2 k∇(s 2 )k2L2 (Ω) kϕ 2 sk N ϕm

L

N 2

(Ω)

(Ω)

.

By using the above inequality in (5.3) we have Z (ϕ|s|p )t dx ≤ D(p, t)

(5.6)

Ω

p 2 +k∇(s 2 )k2L2 (Ω) µ2 C(N, p) ϕM2 kϕ N sk N ϕm

Next, we use this relation for p =

N 2

N L2

(Ω)

2

2

N L2

(Ω)

≤ kϕ0N s0 k

L

N 2

L

N 2

(Ω)

(Ω)

2 N

of the term D(p, t), here it would be sufficient to assume kϕ 0 s0 k ensure that that ∂t kϕ sk

.

and prove that the assumptions of the theorem,

for suitable constants K1 , K2 , imply that kϕ N sk 2 N

2(p−1)ϕm p

−

L

N 2

. In absence

(Ω)

small, to

is negative (see [3]). In our case we have to use a


19

supplementary argument. We shall show that if K1 and K2 are small enough such that 4 N2 − 1 ϕm 2 µ2 C(N )ϕM >0 δ := − kϕ N s0 k N2 2 L (Ω) N N ϕm and sup D [0,+∞)

2 1+ N

ϕm
0, Ω ⊂ RN ). In (6.1), s stands for the porous concentration of SO 2 , namely the concentration taken with respect to the volume of the pores, c for the local density of CaCO 3 and the function ϕ(c) is the porosity. This model has been introduced in [2] to describe the transformation in time of CaCO3 (calcium carbonate) stones under the chemical aggression due to the action of SO2 (sulphur dioxide). This reaction converts the calcium carbonate on the surface of a stone, in a thin crust of calcium sulphate (gypsum). Global existence and uniqueness results and analisys of the macroscopic behavior in time of the crust of calcium sulphate are obtained in [11] and [12] in the case of one space dimension. The interested reader can look at [2, 1, 9] and the comprehensive book [10] for more details about the chemical background as well as for other related references. One of the main features of the model is the fact that the porosity function ϕ is assumed to depend on the local density of the calcite c, actually as a linear function ϕ(c) = A+Bc, which is strictly positive on the interval [0, kc0 k∞ ]. It is easy to see that assumptions a),b) and c) of Section 2 are satisfied by system (6.1). As regard to assumption d),e) we first observe that the function G is nonpositive and G(s, 0) = 0 for all value of s ∈ R+ , so we immediately obtain the bound 0 ≤ c ≤ kc0 k∞ . This means that there are two strictly positive constants ϕm < ϕM , such that ϕm ≤ ϕ(c) ≤ ϕM ,

for all c ∈ [0, kc0 k∞ ].

In particular, this implies that min{A, A + Bkc0 k∞ } ≥ ϕm > 0. In order to obtain the a priori L∞ bounds for s, we explicitely write the source term of the first equation in (6.1) F (s, c) − ϕ0 (c)G(s, c)s = −cs(1 − ϕ0 (c)s) . ϕ(c) Since this term vanishes when s vanishes, for all value of c , we easily obtain the nonnegativity of s; on the other hand, if ϕ0 (c) < 0, then −cs(1 − ϕ0 (c)s) ≤ 0 for s, c > 0, which implies that s remains bounded by ks 0 k∞ , while , if ϕ0 (c) > 0 , the same bound is ensured by assuming that ks0 k∞ < ϕ01(c) (see [11] and [12] for more details). As a consequence of the above considerations, the results of the present paper provide global existence and uniqueness result for weak solutions to system (6.1) when the data satisfies (1.2)-(1.6) and, in the case of increasing ϕ, under a further smallness assumption on the data. Chemotaxis phenomena. A second example of reaction-diffusion system where our results apply, is given by the following class of Keller-Segel type models of chemotaxis

(6.2)

  ∂t u 

∂t c

= µ∆u − ∇ · (uχ(c)∇c) + f (u, c), = g(u, c),

where u represents the density of some motile living species and c represents the concentration of chemical species. The coefficient µ > 0 is the motility coefficient


21

which here is assumed to be constant and the term uχ(c) is the chemotactic sensitivity function which here is assumed to be linear in the species u. The function χ is usually assumed nonnegative and non increasing. This class of systems was largely studied when the second equation contains an additional linear diffusion term (see [6] for a survey of results). A global existence result for system (6.2) with f (s, c) = 0 and g(s, c) = −sc α can be found in [3]; in [19] the authors studied the system with χ(c) = 1c , f (s, c) = 0, g(s, c) = αs − βc or g(s, c) = c(αs − β) and prove global existence of solutions in the first case and a blow up result, for large data, in the second one. Introduce a function ϕ(c) related to the function χ(c) by the equality µϕ0 (c) = ϕ(c)χ(c). u , and setting F (s, c) = f (ϕ(c)s, c) and G(s, c) = Using the new unknown s = ϕ(c) g(ϕ(c)s, c), we obtain   ∂t (ϕ(c)s) = µ div(ϕ(c)∇s) + F (s, c) (6.3)  ∂t c = G(s, c),

which belongs to the class studied in this paper, provided that the functions ϕ, G, F satisfy the assumptions in Section 2 or 5; notice that, by definition, ϕ is nonnegative and then the same holds for ϕ0 (c). In particular, global existence and uniqueness of solutions to system (6.2) follow in the cases when a priori the L∞ bounds in assumption d) of Section 2 can be derived for system (6.3) and then Theorem 4.3 holds. Here we will show some examples for which such a priori estimates can be proved. All the other assumptions stated in Section 2 are easily verified. We treat the case f (u, c) = 0 and g(u, c) = αu − βc, α, β > 0 and for semplicity we assume µ = 1; then system (6.3) has the form  ϕ0 (c)  ∂t s = ∆s + ∇s·∇c ϕ(c) − ϕ(c) s(αϕ(c)s − βc) (6.4)  ∂t c = αϕs − βc,

For some classes of functions ϕ satisfying the assumptions in Section 2, we are T T > 0 such able to prove that for all T > 0 there exist two positive quantities S ∞ , C∞ T that for smooth s such that 0 ≤ s ≤ S∞ the second equation in (6.4) has solutions T satisfying the bounds 0 ≤ c ≤ C T ∞ and for smooth c such that 0 ≤ c ≤ C∞ , T smooth solutions of the first equation in (6.4) satisfies 0 ≤ s ≤ S ∞ . 1) ϕ(c) satisfies assumptions c), e) in Section 2 and c (6.5) sup ≤L, cϕ(c) ≤ K2 c2 + K1 , K1 , K2 > 0 . c∈[0,+∞) ϕ(c) T The proof of the nonnegativity of s, c is immediate. Let T > 0, S ∞ = β T max{ks0 kL∞ (Ω) , kψkL∞ (∂Ω×(0,T )) , L α } and S > S∞ ; let 0 < τ ≤ T, y ∈ Ω such that s(y, τ ) = S and s(x, t) < S for x ∈ Ω and t < τ . Then in the point (y, τ ) the following inequality holds

S≤L

β ; α

T by contraddiction, it follows that s(x, t) ≤ S∞ for (x, t) ∈ Ω × [0, T ]. Moreover, since ct ≤ αϕ(c)s ,

the second condition in (6.5) implies the L∞ bound for c.

22


The class of functions ϕ we are dealing with, contains obviously linear functions, strictly positive for c ∈ R+ , which correspond to chemiotactic A functions of the form χ(c) = Ac+B , for suitable coefficients A, B. Notice that, if c0 (x) ≥ c˜ > 0, then it is easily possible to prove the following lower bound for c c(x, t) ≥ c˜e−βt ;

(6.6)

in this case it sufficient to ask that, for all T > 0, c ≤ L(T ) , sup c∈[˜ ce−βT ,+∞) ϕ(c) 2) ϕ(c) satisfies assumptions c), e) in Section 2 and 0 < ϕm ≤ ϕ(c) ≤ ϕM

(6.7)

Let T > 0 such that

for c ∈ R+ .

1 ϕM 1 − e−βT = ϕm 2

and let

T S∞ = max{ks0 kL∞ (Ω) , kψkL∞ (∂Ω×(0,T )) , 2

βkc0 k∞ }; αϕm

in Ω × [0, T ] holds

(6.8)

ct ≤ αϕM ksk∞ − βc and, by comparison results, we have α α T c ≤ ϕM ksk∞ (1 − e−βt ) + c0 e−βt ≤ ϕM ksk∞ + kc0 k∞ =: C∞ . β β

T On the other hand, if ksk∞ > S∞ arguing as in the previous example and taking into account the estimate (6.8) it should be β ϕ(c)ksk∞ ≤ ϕM ksk∞ (1 − e−βt ) + c0 e−βt ; α it follows that βkc0 k∞ ϕM ksk∞ (1 − e−βT ) + , ksk∞ ≤ ϕm αϕm T then the positions on T, S∞ , and ksk∞ lead to a contraddiction. Since T depends only on ϕm and ϕM , we can repeat the above argument over a number of time intervals of width T and obtain a priori L ∞ estimates over every time interval. 1 − 1 (α−1)(c+δ)α−1 The choice χ(c) = (c+δ) α , α > 1, δ > 0, leads to ϕ(c) = e which belongs to the class of functions verifying (6.7). As in the previous example, if c0 (x) ≥ c˜ > 0, then the bound (6.6) −

1

holds, in such a way that, if ϕ(0) = 0 (e.g. ϕ(c) = e (α−1)cα−1 , α > 1), then the above argument can be applied over every interval [0, T ], with ϕm = inf [˜ce−βT ,+∞] ϕ(c). Acknowledgments A special thanks to Benoit Perthame and Lucilla Corrias to pointing out to us the connection between sulphate models and chemotaxis. The research activity reported in this paper has been also partially conducted within the European Union RTN project FRONTS-SINGULARITIES: HPRN-CT2002-0027.


23

References [1] Al`ı , G., V. Furuholt, R. Natalini, I. Torcicollo, A mathematical model of sulphite chemical aggression of limestones with high permeability. Part I: Presentation of the model and qualitative analysis; IAC Report (2004); Al`ı , G., V. Furuholt, R. Natalini, I. Torcicollo, A mathematical model of sulphite chemical aggression of limestones with high permeability. Part II: Numerical approximation; IAC Report (2004) [2] D. Aregba-Driollet, F. Diele, R. Natalini, A mathematical model for the SO 2 aggression to calcium carbonate stones: numerical approximation and asymptotic analysis, SIAM J.Appl.Math., 64, (2004), 1636-1667. [3] L. Corrias, B. Perthame, H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, preprint DMA-03-15 (2003); A chemotaxis model motivated by angiogenesis, C. R., Math., Acad. Sci. Paris 336, No.2, (2003), 141–146. [4] M.A. Fontelos, A. Friedman, B. Hu, Mathematical Analysis of a model for the initiatio of angiogenesis, SIAM J.Math.Anal., 33, 1330-1355 (2002). [5] A. Friedman, I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks, J.Math.Anal.Appl., 272, 138-163 (2002). [6] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, Teil I, Jahresbericht der DMV Vol. 105 (3), Seiten 103 - 165, 2003, Teil II, Jahresbericht der DMV Vol. 106 (2), Seiten 51 - 69, 2004. [7] H.A. Levine, B.D.Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks, SIam J.Appl.Math.,57, 683-730 (1997). [8] H.A. Levine, B.D.Sleeman, Partial differential equations of chemotaxis and angiogenesis, Math.Meth. Appl.Sci., 24, 405-426 (2001). [9] E. Borrelli, C. Giavarini, M. Incitti, M. L. Santarelli, R. Natalini, A material model for the evolution of gypsum crusts: numerical and experimental results, in Proceedings of the “10th International Congress on Deterioration and Conservation of Stone”, Eds. D. Kwiatkowski and R. L¨ ofvendahl, p. 35-42, Vol. I, Stockholm, Sweden (2004) [10] K.L.Gauri, J.K. Bandyopadhyay, Carbonate stone : chemichal behavior, durability and conservation, John Wiley & Sons, New York (1999). [11] F.R. Guarguaglini, R. Natalini, Global existence of solutions to a nonlinear model of sulphation phenomena in calcium carbonate stones,Nonlinear Analysis: Real World Applications, Volume 6, Issue 3, July 2005, Pages 477-494 . [12] F.R. Guarguaglini, R.Natalini, Fast reaction limit and large time behavior of solutions to a nonlinear model of sulphation phenomena, to appear on Commun. in Partial Differential Equations. [13] D. Hilhorst, R. van der Hout, L.A. Peletier, The fast reaction limit for a reaction-diffusion system, J. of Math. Anal. and Appl., 199, (1996), 349-373. [14] O.A.Ladyzenskaja, N.N. Solonnikov, V.A.Ural’ceva, Linear and Quasi-linear Equations of Parabolic Type, AMS, 1968. [15] M. Rascle Sur une ´ equation int´ egro-differenti´ elle non lin´ eaire issue de la biologie, J.Diffential Equation., 32, 420-453 (1979). [16] Y. Shizuta, S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14, 249-275, (1985). [17] J. Smoller, Shock waves and reaction-diffusion equations. 2nd ed. Grundlehren der Mathematischen Wissenschaften. 258. New York: Springer- Verlag (1994). [18] M.E. Taylor, Partial differential equations. 3: Nonlinear equations. Applied Mathematical Sciences. 117. Berlin: Springer-Verlag (1996). [19] Y. Yang, H. Chen, W. Liu, On existence of global solutions and blow-up to a system of reaction-diffusion equations modelling chemotaxis SIAM J. Math. Anal., 33, 763-785 (2001).