Memoirs on Differential Equations and Mathematical Physics

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1 Memoirs on Differential Equations and Mathematical Physics Volume 51, 010, Said Kouachi and Belgacem Rebiai INVARIANT REGIONS AND THE GLOBAL EXISTENCE FOR REACTION-DIFFUSION SYSTEMS WITH A TRIDIAGONAL MATRIX OF DIFFUSION COEFFICIENTS

2 Abstract. The aim of this study is to prove the global existence of solutions for reaction-diffusion systems with a tridiagonal matrix of diffusion coefficients and nonhomogeneous boundary conditions. In so doing, we make use of the appropriate techniques which are based on invariant domains and Lyapunov functional methods. The nonlinear reaction term has been supposed to be of polynomial growth. This result is a continuation of that by Kouachi [1]. 010 Mathematics Subject Classification. 35K45, 35K57. Key words and phrases. reaction diffusion systems, invariant domains, Lyapunov functionals, global existence. îâäæñéâ. êžöîëéæï éæäžêæž ážéðçæùâijñè æóêâï ŽîŽâîåàãŽîëãŽêæ ïžïžäôãîë ŽéëùŽêâIJæï ŽéëêŽýïêåŽ àèëijžèñîæ ŽîïâIJëIJŽ æïâåæ îâžóùæñè-áæòñäæñîæ ïæïðâéâijæïžåãæï, îëéâèåž áæòñäææï çëâòæùæâêðâijæ óéêæžê ðîæáæ- ŽàëêŽèñî (æžçëijæï) éžðîæùï. ŽéæïŽåãæï àžéëõâêâijñèæž öâïžijžéæïæ ðâóêæçž, îëéâèæù âòñúêâijž æêãžîæžêðñèæ ŽîââIJæï áž èæžìñêëãæï òñêóùæëêžèæï éâåëáâijï. ŽîŽûîòæãæ îâžóùææï ûâãîäâ ážáâijñèæž ìëèæêëéæžèñîæ äîáæï ìæîëijž. êžöîëéöæ éëõãžêæèæ öâáâàæ ûžîéëžáàâêï çñžöæï [1] âîåæ öâáâàæï àžêãîùëijžï.

3 Reaction-Diffusion Systems Introduction We consider the reaction-diffusion system u t a 11 u a 1 v a 3 w = f(u, v, w) in R +, (1.1) v t a 1 u a v a 3 w = g(u, v, w) in R +, (1.) w t a 1 u a 3 v a 11 w = h(u, v, w) in R +, (1.3) with the boundary conditions and the initial data where: λu + (1 λ) u η = β 1, λv + (1 λ) v η = β λw + (1 λ) w η = β 3, on R +, (1.4) u(0, x) = u 0 (x), v(0, x) = v 0 (x), w(0, x) = w 0 (x) in, (1.5) (i) 0 < λ < 1 and β i R, i = 1,, 3, for nonhomogeneous Robin boundary conditions. (ii) λ = β i = 0, i = 1,, 3, for homogeneous Neumann boundary conditions. (iii) 1 λ = β i = 0, i = 1,, 3, for homogeneous Dirichlet boundary conditions. is an open bounded domain of the class C 1 in R N with boundary, and η denotes the outward normal derivative on. The diffusion terms a ij (i, j = 1,, 3 and (i, j) (1, 3), (3, 1)) are supposed to be positive constants with a 11 = a 33 and (a 1 + a 1 ) + (a 3 + a 3 ) < 4a 11 a, which reflects the parabolicity of the system and implies at the same time that the matrix of diffusion A = a 11 a 1 0 a 1 a a 3 0 a 3 a 11 is positive definite. The eigenvalues λ 1, λ and λ 3 (λ 1 < λ, λ 3 = a 11 ) of A are positive. If we put a = min {a 11, a } and a = max {a 11, a }, then the positivity of a ij s implies that λ 1 < a λ 3 a < λ.

4 96 Said Kouachi and Belgacem Rebiai The initial data are assumed to be in the domain } {(u 0, v 0, w 0 ) R 3 : µ v 0 a 1 u 0 + a 3 w 0 µ 1 v 0, a 3 u 0 a 1 w 0 if µ β a 1 β 1 + a 3 β 3 µ 1 β, a 3 β 1 a 1 β 3, {(u 0, v 0, w 0 ) R 3 : µ v 0 a 1 u 0 + a 3 w 0 µ 1 v 0, a 1 w 0 a 3 u 0 } if µ β a 1 β 1 + a 3 β 3 µ 1 β, a 1 β 3 a 3 β 1, (u 0, v 0, w 0 ) R 3 : 1 Σ = (a 1 u 0 +a 3 w 0 ) v 0 1 (a 1 u 0 +a 3 w 0 ), a 3 u 0 a 1 w 0 µ µ 1 1 if (a 1 β 1 +a 3 β 3 ) β 1 (a 1 β 1 +a 3 β 3 ), a 3 β 1 a 1 β 3, µ µ 1 (u 0, v 0, w 0 ) R 3 : 1 (a 1 u 0 +a 3 w 0 ) v 0 1 (a 1 u 0 +a 3 w 0 ), a 3 u 0 a 1 w 0 µ µ 1 1 if (a 1 β 1 +a 3 β 3 ) β 1 (a 1 β 1 +a 3 β 3 ), a 3 β 1 a 1 β 3, µ µ 1 where µ 1 = a λ 1 > 0 > µ = a λ. Since we use the same methods to treat all the cases, we will tackle only with the first one. We suppose that the reaction terms f, g and h are continuously differentiable, polynomially bounded on Σ, ( ) f (r 1, r, r 3 ), g (r 1, r, r 3 ), h (r 1, r, r 3 ) is in Σ for all (r 1, r, r 3 ) in Σ (we say that (f, g, h) points into Σ on Σ), i.e., a 1 f (r 1, r, r 3 ) + a 3 h (r 1, r, r 3 ) µ 1 g (r 1, r, r 3 ) (1.6) for all r 1, r and r 3 such that µ r a 1 r 1 +a 3 r 3 = µ 1 r and a 3 r 1 a 1 r 3, and µ g (r 1, r, r 3 ) a 1 f (r 1, r, r 3 ) + a 3 h (r 1, r, r 3 ) (1.6a) for all r 1, r and r 3 such that µ r = a 1 r 1 +a 3 r 3 µ 1 r and a 3 r 1 a 1 r 3, and a 3 f (r 1, r, r 3 ) a 1 h (r 1, r, r 3 ) (1.6b) for all r 1, r and r 3 such that µ r a 1 r 1 +a 3 r 3 µ 1 r and a 3 r 1 = a 1 r 3, and for positive constants E and D, we have (Ef + Dg + h) (u, v, w) C 1 (u + v + w + 1), (1.7) for all (u, v, w) in Σ, where C 1 is a positive constant. In the two-component case, where a 1 = 0, Kouachi and Youkana [13] generalized the method of Haraux and Youkana [4] with the reaction terms f (u, v) = λf (u, v) and g (u, v) = µf (u, v) with F (u, v) 0, requiring the condition [ ] ln (1 + F (r, s)) lim < α for any r 0, s + s

5 Reaction-Diffusion Systems 97 with α = { a 11 a λ n (a 11 a ) min u 0 µ, a } 11 a, a 1 where the positive diffusion coefficients a 11, a satisfy a 11 > a, and a 1, λ, µ are positive constants. This condition reflects the weak exponential growth of the reaction term F. Kanel and Kirane [6] proved the global existence in the case where g (u, v) = f (u, v) = uv n and n is an odd integer, under the embarrassing condition a 1 a 1 < C p, where C p contains a constant from Solonnikov s estimate [18]. Later they improved their results in [7] to obtain the global existence under the restrictions and H 1. a < a 11 + a 1, a 11 a (a 11 + a 1 a ) H. a 1 < ε 0 = a 11 a + a 1 (a 11 + a 1 a ) if a 11 a < a 11 + a 1, { } 1 H 3. a 1 < min (a 11 + a 1 ), ε 0 if a < a 11, F (v) C F (1 + v 1 ε), vf (v) 0 for all v R, where ε and C F are positive constants with ε < 1 and g (u, v) = f (u, v) = uf (v). Kouachi [1] has proved global existence for solutions of two-component reaction-diffusion systems with a general full matrix of diffusion coefficients and nonhomgeneous boundary conditions. Many chemical and biological operations are described by reaction-diffusion systems with a tridiagonal matrix of diffusion coefficients. The components u (t, x), v (t, x) and w (t, x) can represent either chemical concentrations or biological population densities (see, e.g., Cussler [1] and []). We note that the case of strongly coupled systems which are not triangular in the diffusion part is more difficult. As a consequence of the blow-up of the solutions found in [16], we can indeed prove that there is a blow-up of the solutions in finite time for such nontriangular systems even though the initial data are regular, the solutions are positive and the nonlinear terms are negative, a structure that ensured the global existence in the diagonal case. For this purpose, we construct invariant domains in which we can demonstrate that for any initial data in these domains, the problem (1.1) (1.5) is equivalent to the problem for which the global existence follows from the usual techniques based on Lyapunov functionals (see Kirane and Kouachi [8], Kouachi and Youkana [13] and Kouachi [1]).

6 98 Said Kouachi and Belgacem Rebiai. Local Existence and Invariant Regions This section is devoted to proving that if (f, g, h) points into Σ on Σ, then Σ is an invariant domain for the problem (1.1) (1.5), i.e., the solution remains in Σ for any initial data in Σ. Once the invariant domains are constructed, both problems of the local and global existence become easier to be established. For the first problem we demonstrate that the system (1.1) (1.3) with the boundary conditions (1.4) and the initial data in Σ is equivalent to a problem for which the local existence throughout the time interval [0, T [ can be obtained by the known procedure, and for the second one we need invariant domains as explained in the preceeding section. The main result of this section is Proposition 1. Suppose that (f, g, h) points into Σ on Σ. Then for any (u 0, v 0, w 0 ) in Σ the solution (u, v, w) of the problem (1.1) (1.5) remains in Σ for all t s in [0, T [. Proof. Let (x i1, x i, x i3 ) t, i = 1,, 3, be the eigenvectors of the matrix A t associated with its eigenvalues λ i, i = 1,, 3 (λ 1 < λ 3 < λ ). Multiplying the equations (1.1), (1.) and (1.3) of the given reaction-diffusion system by x i1, x i and x i3, respectively, and summing the resulting equations, we get t z 1 λ 1 z 1 = F 1 (z 1, z, z 3 ) in ]0, T [, (.1) t z λ z = F (z 1, z, z 3 ) in ]0, T [, (.) t z 3 λ 3 z 3 = F 3 (z 1, z, z 3 ) in ]0, T [, (.3) with the boundary conditions and the initial data where and λz i + (1 λ) z i η = ρ i, i = 1,, 3, on ]0, T [, (.4) z i (0, x) = z 0 i (x), i = 1,, 3, in, (.5) z i = x i1 u + x i v + x i3 w, i = 1,, 3, in ]0, T [, (.6) ρ i = x i1 β 1 + x i β + x i3 β 3, i = 1,, 3, F i (z 1, z, z 3 ) = x i1 f + x i g + x i3 h, i = 1,, 3, (.7) for all (u, v, w) in Σ. First, as has been mentioned above, note that the condition of the parabolicity of the system (1.1) (1.3) implies the parabolicity of the system

7 Reaction-Diffusion Systems 99 (.1) (.3) since (a 1 + a 1 ) + (a 3 + a 3 ) < 4a 11 a = = (det A > 0 and a 11 a a 3 a 3 > 0). Since λ 1, λ and λ 3 (λ 1 < λ 3 < λ ) are the eigenvalues of the matrix A t, the problem (1.1) (1.5) is equivalent to the problem (.1) (.5) and to prove that Σ is an invariant domain for the system (1.1) (1.3) it suffices to prove that the domain { ( z 0 1, z 0, z 0 3) R 3 : z 0 i 0, i = 1,, 3 } = ( R +) 3 (.8) is invariant for the system (.1) (.3) and that { } Σ= (u 0, v 0, w 0 ) R 3 : zi 0 =x i1 u 0 +x i v 0 +x i3 w 0 0, i=1,, 3. (.9) Since (x i1, x i, x i3 ) t is an eigenvector of the matrix A t associated to the eigenvalue λ i, i = 1,, 3, we have (a 11 λ i ) x i1 + a 1 x i = 0, a 1 x i1 + (a λ i ) x i + a 3 x i3 = 0, i = 1,, 3, a 3 x i + (a 11 λ i ) x i3 = 0. If we assume, without loss of generality, that a 11 a and choose x 1 = µ 1, x = µ and x 33 = a 1, then we have a 1 u 0 +µ 1 v 0 a 3 w 0 0, x i1 u 0 +x i v 0 +x i3 w 0 0, i=1,, 3 a 1 u 0 µ v 0 +a 3 w 0 0, a 3 u 0 + a 1 w 0 0. µ v 0 a 1 u 0 + a 3 w 0 µ 1 v 0, a 3 u 0 a 1 w 0. Thus (.9) is proved and (.6) can be written as z 1 = a 1 u + µ 1 v a 3 w, z = a 1 u µ v + a 3 w, z 3 = a 3 u + a 1 w. (.6a) Now, to prove that the domain (R + ) 3 is invariant for the system (.1) (.3), it suffices to show that F i (z 1, z, z 3 ) 0 for all (z 1, z, z 3 ) such that z i = 0 and z j 0, j = 1,, 3 (j i), i = 1,, 3, thanks to the invariant domain method (see Smoller [17]). Using the expressions (.7), we get F 1 = a 1 f + µ 1 g a 3 h, F = a 1 f µ g + a 3 h, (.7a) F 3 = a 3 f + a 1 h for all (u, v, w) in Σ. Since from (1.6), (1.6a) and (1.6b) we have F i (z 1, z, z 3 ) 0 for all (z 1, z, z 3 ) such that z i = 0 and z j 0, j = 1,, 3 (j i), i = 1,, 3, we obtain z i (t, x) 0, i = 1,, 3, for all (t, x) [0, T [. Then Σ is an invariant domain for the system (1.1) (1.3).

8 100 Said Kouachi and Belgacem Rebiai In addition, the system (1.1) (1.3) with the boundary conditions (1.4) and initial data in Σ is equivalent to the system (.1) (.3) with the boundary conditions (.4) and positive initial data (.5). As has been mentioned at the beginning of this section and since ρ i, i = 1,, 3, given by ρ 1 = a 1 β 1 + µ 1 β a 3 β 3, ρ = a 1 β 1 µ β + a 3 β 3, ρ 3 = a 3 β 1 + a 1 β 3, are positive, we have for any initial data in C ( ) or L p (), p ]1, + ], the local existence and uniqueness of solutions to the initial value problem (.1) (.5) and consequently those of the problem (1.1) (1.5) follow from the basic existence theory for abstract semilinear differential equations (see Friedman [3], Henry [5] and Pazy [15]). These solutions are classical on [0, T [, where T denotes the eventual blow up time in L (). A local solution is continued globally by a priori estimates. Once invariant domains are constructed, one can apply the Lyapunov technique and establish the global existence of unique solutions for (1.1) (1.5). 3. Global Existence As the determinant of the linear algebraic system (.6), with respect to the variables u, v and w, is different from zero, to prove the global existence of solutions of the problem (1.1) (1.5) one needs to prove it for the problem (.1) (.5). To this end, it suffices (see Henry [5]) to derive a uniform estimate of F i (z 1, z, z 3 ) p, i = 1,, 3 on [0, T ], T < T, for some p > N/, where p denotes the usual norms in spaces L p () defined by u p p = 1 u(x) p dx, 1 p <, and u = esssup x Let θ and σ be two positive constants such that u(x). θ > A 1, (3.1) ( θ A ) ( 1 σ A ) 3 > (A13 A 1 A 3 ), (3.) where and let A ij = λ i + λ j λ i λ j, i, j = 1,, 3 (i < j), θ q =θ (p q+1) and σ p =σ p, for q =0, 1,..., p and p=0, 1,..., n, (3.3) where n is a positive integer. The main result of this section is

9 Reaction-Diffusion Systems 101 Theorem 1. Let (z 1 (t, ), z (t, ), z 3 (t, )) be any positive solution of (.1) (.5). Introduce the functional ( t L(t) = H n z1 (t, x), z (t, x), z 3 (t, x) ) dx, (3.4) where H n (z 1, z, z 3 ) = n CnC p pθ q q σ p z q 1 zp q z n p 3, (3.5) with n being a positive integer and C p n = n! (n p)!p!. Then the functional L is uniformly bounded on the interval [0, T ], T < T. For the proof of Theorem 1 we need some preparatory Lemmas. Lemma 1. Let H n be the homogeneous polynomial defined by (3.5). Then = n C p z Cq pθ q+1 σ p+1 z q 1 zp q z () p 3, (3.6) 1 z z 3 = n = n C p Cq pθ q σ p+1 z q 1 zp q z () p 3, (3.7) C p Cq pθ q σ p z q 1 zp q z () p 3. (3.8) Proof. Differentiating H n with respect to z 1 and using the fact that qc q p = pc q 1 p 1 and pcp n = nc p 1 (3.9) for q = 1,,..., p, p = 1,,..., n, we get n = n C p 1 z Cq 1 p 1 θ qσ p z q 1 1 z p q z n p 3. 1 p=1 q=1 Replacing in the sums the indexes q 1 by q and p 1 by p, we deduce (3.6). For the formula (3.7), differentiating H n with respect to z, taking into account C q p = C p q p, q = 0, 1,..., p 1 and p = 1,,..., n, (3.10) using (3.9) and replacing the index p 1 by p, we get (3.7). Finally, we have = z 3 Since (n p) C p n = (n p) C n p n (n p) CnC p pθ q q σ p z q 1 zp q z n p 1 3. = nc n p 1 = nc p, we get (3.8).

10 10 Said Kouachi and Belgacem Rebiai Lemma. The second partial derivatives of H n are given by H n z 1 n = n (n 1) n H n = n (n 1) z 1 z n H n = n (n 1) z 1 z 3 H n z n = n (n 1) n H n = n (n 1) z z 3 H n z 3 n = n (n 1) Proof. Differentiating z 1 H n z 1 = n p=1 q=1 C p n Cq pθ q+ σ p+ z q 1 zp q z (n ) p 3, (3.11) C p n Cq pθ q+1 σ p+ z q 1 zp q z (n ) p 3, (3.1) C p n Cq pθ q+1 σ p+1 z q 1 zp q z (n ) p 3, (3.13) C p n Cq pθ q σ p+ z q 1 zp q z (n ) p 3, (3.14) C p n Cq pθ q σ p+1 z q 1 zp q z (n ) p 3, (3.15) C p n Cq pθ q σ p z q 1 zp q z (n ) p 3. (3.16) given by (3.6) with respect to z 1 yields qc p Cq pθ q+1 σ q+1 z q 1 1 z p q z () p 3. Using (3.9), we get (3.11). H n p 1 = n (p q) C p z 1 z Cq pθ q+1 σ p+1 z q 1 zp q 1 z () p 3. p=1 q=0 Applying (3.10) and then (3.9), we get (3.1). n H n = n z 1 z 3 ((n 1) p) C p Cq pθ q+1 σ p+1 z q 1 zp q z (n ) p 3. Applying successively (3.10), (3.9) and (3.10) for the second time, we deduce (3.13). H n z p 1 = n p=1 q=0 (p q) C p Cq pθ q σ p+1 z q 1 zp q 1 z () p 3. The application of (3.10) and then of (3.9) yields (3.14). n H n = n z z 3 ((n 1) p) C p Cq pθ q σ p z q 1 zp q z (n ) p 3.

11 Reaction-Diffusion Systems 103 Applying (3.10) and then (3.9) yields (3.15). Finally we get (3.16) by differentiating with respect to z 3 and applying successively (3.10), (3.9) z 3 and (3.10) for the second time. Proof of Theorem 1. Differentiating L with respect to t yields ( L Hn z 1 (t) = z 1 t + z z t + H ) n z 3 dx = z 3 t ( ) = λ 1 z 1 + λ z + λ 3 z 3 dx+ z 1 z z 3 ( Hn + F 1 + F + H ) n F 3 dx = z 1 z z 3 =: I + J. Using Green s formula in I, we get I = I 1 + I, where ( ) z 1 I 1 = λ 1 z 1 η + λ z z η + λ z 3 3 ds, z 3 η where ds denotes the (n 1)-dimensional surface element, and [ H n I = λ 1 z1 z 1 + (λ 1 + λ ) H n z 1 z z 1 z + (λ 1 + λ 3 ) H n H n z 1 z 3 + λ z 1 z 3 z z + (λ + λ 3 ) H n ] H n z z 3 + λ 3 z z 3 z3 z 3 dx. We prove that there exists a positive constant C independent of t [0, T [ such that I 1 C for all t [0, T [, (3.17) and that I 0. (3.18) To see this, we follow the same reasoning as in [11]. (i) If 0 < λ < 1, using the boundary conditions (.4) we get ( ) I 1 = λ 1 (γ 1 αz 1 )+λ (γ αz )+λ 3 (γ 3 αz 3 ) ds, z 1 z z 3 where α = λ 1 λ and γ i = ρ i 1 λ, i = 1,, 3. Since H (z 1, z, z 3 ) = λ 1 (γ 1 αz 1 )+λ (γ αz )+λ 3 (γ 3 αz 3 ) z 1 z z 3 = P (z 1, z, z 3 ) Q n (z 1, z, z 3 ),

12 104 Said Kouachi and Belgacem Rebiai where P and Q n are polynomials with positive coefficients and respective degrees n 1 and n, and since the solution is positive, we obtain lim sup H (z 1, z, z 3 ) =, (3.19) ( z 1 + z + z 3 ) + which proves that H is uniformly bounded on (R + ) 3, and consequently (3.17). (ii) If λ = 0, then I 1 = 0 on [0, T [. (iii) The case of the homogeneous Dirichlet conditions is trivial since the positivity of the solution on [0, T [ implies z1 z z3 η 0, η 0 and η 0 on [0, T [. Consequently, one again gets (3.17) with C = 0. Now, we prove (3.18). Applying Lemma 1 and Lemma, we get I = n (n 1) n C p n p[ Cq (Bpq z) z ] dx, where λ 1 + λ λ 1 + λ 3 λ 1 θ q+ σ p+ θ q+1 σ p+ θ q+1 σ p+1 B pq = λ 1 + λ λ + λ 3 θ q+1 σ p+ λ θ q σ p+ θ q σ p+1 λ 1 + λ 3 λ + λ, 3 θ q+1 σ p+1 θ q σ p+1 λ 3 θ q σ p for q = 0, 1,..., p, p = 0, 1,..., n and z = ( z 1, z, z 3 ) t. The quadratic forms (with respect to z 1, z and z 3 ) associated with the matrices B pq, q = 0, 1,..., p, p = 0, 1,..., n, are positive since their main determinants 1, and 3 are positive too, according to the Sylvester criterion. To see this, we have 1. 1 = λ 1 θ q+ σ p+ > 0 for q = 0, 1,..., p and p = 0, 1,..., n. λ 1 + λ λ 1 θ q+ σ p+ θ q+1 σ p+. = λ 1 + λ θ q+1 σ p+ λ θ q σ p+ ( = λ 1 λ θq+1σ p+ θ A1), for q = 0, 1,..., p and p = 0, 1,..., n. Using (3.1), we get > 0. λ 1 + λ λ 1 + λ 3 λ 1 θ q+ σ p+ θ q+1 σ p+ θ q+1 σ p = λ 1 + λ λ + λ 3 θ q+1 σ p+ λ θ q σ p+ θ q σ p+1 λ 1 + λ 3 λ + λ 3 θ q+1 σ p+1 θ q σ p+1 λ 3 θ q σ p =λ 1 λ λ 3 θq+1θ q σ p+ σp+1[ (θ A 1)(σ A 3) (A 13 A 1 A 3 ) ], for q = 0, 1,..., p and p = 0, 1,..., n.

13 Reaction-Diffusion Systems 105 Using (3.), we get 3 > 0. Consequently we have (3.18). Substitution of the expressions of the partial derivatives given by Lemma 1 in the second integral yields [ ] J = n C p Cq pz q 1 zp q z () p 3 Using the expressions (.7a), we get θ q+1 σ p+1 F 1 + θ q σ p+1 F + θ q σ p F 3 = (θ q+1 σ p+1 F 1 + θ q σ p+1 F + θ q σ p F 3 ) dx. = ( θ q+1 σ p+1 a 1 +a 1 θ q σ p+1 a 3 θ q σ p ) f +(θq+1 σ p+1 µ 1 µ θ q σ p+1 ) g+ + ( ) θ q+1 σ p+1 a 3 + a 3 θ q σ p+1 + a 1 θ q σ p h = ( ) ( a 1 (θ q σ p+1 θ q+1 σ p+1 ) a 3 θ q σ p = a 3 (θ q σ p+1 θ q+1 σ p+1 )+a 1 θ q σ p f + a 3 (θ q σ p+1 θ q+1 σ p+1 )+a 1 θ q σ p ) θ q+1 σ p+1 µ 1 µ θ q σ p+1 + g + h = a 3 (θ q σ p+1 θ q+1 σ p+1 ) + a 1 θ q σ p ( ( ) ) σ p+1 θq θ q = θ q+1 σ p a 3 1 +a 1 σ p θ q+1 θ q+1 a 1 σ p+1 σ p a 3 σ p+1 σ p θ q θ q+1 ( θq θ q+1 1 ) θ a q 3 θ q+1 ( θq θ q+1 1 ) θ +a q f + 1 θ q+1 µ 1 σ p+1 a 3 σ p+1 σ p θ σ p µ q σ p+1 θ q+1 σ p ( θq θ q+1 1 ) θ + a q 1 g + h. θ q+1 Since and σp+1 σ p are sufficiently large if we choose θ and σ sufficiently large, using the condition (1.7) and the relation (.6a) successively we get, for an appropriate constant C 3, [ ] J C 3 (z 1 + z + z 3 + 1) C p Cq pz q 1 zp q z () p 3 dx. To prove that the functional L is uniformly bounded on the interval [0, T ], we first write (z 1 + z + z 3 + 1) C p Cq pz q 1 zp q z () p 3 = = R n (z 1, z, z 3 ) + S (z 1, z, z 3 ), where R n (z 1, z, z 3 ) and S (z 1, z, z 3 ) are two homogeneous polynomials of degrees n and n 1, respectively. First, since the polynomials H n and R n are of degree n, there exists a positive constant C 4 such that R n (z 1, z, z 3 ) dx C 4 H n (z 1, z, z 3 ) dx.

14 106 Said Kouachi and Belgacem Rebiai Applying Hölder s inequality to the integral S (z 1, z, z 3 ) dx, one gets n S (z 1, z, z 3 ) dx (meas ) 1 n (S (z 1, z, z 3 )) n dx. Since for all z 1 0 and z, z 3 > 0 where ξ 1 = z 1 z, ξ = z z 3 (S (z 1, z, z 3 )) n H n (z 1, z, z 3 ) and lim ξ 1 + ξ + = (S (ξ 1, ξ, 1)) n H n (ξ 1, ξ, 1) (S (ξ 1, ξ, 1)) n H n (ξ 1, ξ, 1) < +, one asserts that there exists a positive constant C 5 such that (S (z 1, z, z 3 )) n H n (z 1, z, z 3 ) C 5 for all z 1, z, z 3 0. Hence the functional L satisfies the differential inequality L (t) C 6 L (t) + C 7 L n (t),, which for Z = L 1 n can be written as nz C 6 Z + C 7. A simple integration gives a uniform bound of the functional L on the interval [0, T ]. This completes the proof of Theorem 1. Corollary 1. Suppose that the functions f (r 1, r, r 3 ), g (r 1, r, r 3 ) and h (r 1, r, r 3 ) are continuously differentiable on Σ, point into Σ on Σ and satisfy the condition (1.7). Then all uniformly bounded on solutions of (1.1) (1.5) with the initial data in Σ are in L (0, T ; L p ()) for all p 1. Proof. The proof of this Corollary is an immediate consequence of Theorem 1, the trivial inequality (z 1 + z + z 3 ) p dx L (t) on [0, T [, and (.6a). Proposition. Under the hypothesis of Corollary 1, if f (r 1, r, r 3 ), g (r 1, r, r 3 ) and h (r 1, r, r 3 ) are polynomially bounded, then all uniformly bounded on solutions of (1.1) (1.4) with the initial data in Σ are global in time.

15 Reaction-Diffusion Systems 107 Proof. As has been mentioned above, it suffices to derive a uniform estimate of F 1 (z 1, z, z 3 ) p, F (z 1, z, z 3 ) p and F 3 (z 1, z, z 3 ) p on [0, T ], T < T for some p > N. Since the reactions f (u, v, w), g (u, v, w) and h (u, v, w) are polynomially bounded on Σ, by using relations (.6a) and (.7a) we get that so are F 1 (z 1, z, z 3 ), F (z 1, z, z 3 ) and F 3 (z 1, z, z 3 ), and the proof becomes an immediate consequence of Corollary 1. Acknowledgments The authors would like to thank the anonymous referee for his/her valuable comments and suggestions that helped to improve the presentation of this paper. References 1. E. L. Cussler, Multicomponent diffusion. Chemical Engineering Monographs, Vol. 3, Elsevier Publishing Scientific Company, Amesterdam, E. L. Cussler, Diffusion, mass transfer in fluid system. Second Edition, Cambridge University Press, Cambridge, A. Friedman, Partial differential equations of parabolic type. Prentice-Hall, Inc., Englewood Cliffs, N.J., A. Haraux and A. Youkana, On a result of K. Masuda concerning reaction-diffusion equations. Tôhoku Math. J. () 40 (1988), No. 1, D. Henry, Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin New York, J. I. Kanel and M. Kirane, Pointwise a priori bounds for a strongly coupled system of reaction-diffusion equations with a balance law. Math. Methods Appl. Sci. 1 (1998), No. 13, J. I. Kanel, M. Kirane, and N. E. Tatar, Pointwise a priori bounds for a strongly coupled system of reaction-diffusion equations. Int. J. Differ. Equ. Appl. 1 (000), No. 1, M. Kirane and S. Kouachi, Global solutions to a system of strongly coupled reaction-diffusion equations. Nonlinear Anal. 6 (1996), No. 8, S. Kouachi, Existence of global solutions to reaction-diffusion systems via a Lyapunov functional. Electron. J. Differential Equations 001, No. 68, 10 pp. (electronic). 10. S. Kouachi, Existence of global solutions to reaction-diffusion systems with nonhomogeneous boundary conditions via a Lyapunov functional. Electron. J. Differential Equations 00, No. 88, 13 pp. (electronic). 11. S. Kouachi, Global existence of solutions for reaction-diffusion systems with a full matrix of diffusion coefficients and nonhomogeneous boundary conditions, Electron. J. Qual. Theory Differ. Equ. 00, No., 10 pp. (electronic). 1. S. Kouachi, Invariant regions and global existence of solutions for reaction-diffusion systems with a full matrix of diffusion coefficients and nonhomogeneous boundary conditions. Georgian Math. J. 11 (004), No., S. Kouachi and A. Youkana, Global existence for a class of reaction-diffusion systems. Bull. Polish Acad. Sci. Math. 49 (001), No. 3, K. Masuda, On the global existence and asymptotic behavior of solutions of reactiondiffusion equations. Hokkaido Math. J. 1 (1983), No. 3, A. Pazy, Semigroups of operators in Banach spaces. In: Equadiff 8 (Wurzburg, 198), , Lecture Notes in Math., 1017, Springer, Berlin, 1983.

16 108 Said Kouachi and Belgacem Rebiai 16. M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass. SIAM J. Math. Anal. 8 (1997), No., J. Smoller, Shock Waves and Reaction-Diffusion Equations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], 58. Springer-Verlag, New York-Berlin, V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations, J. Soviet Math. 8 (1977), Authors addresses: (Received ; revised ) Said Kouachi Department of Mathematics College of Science Qassim University P.O.Box 6644, Al-Gassim, Buraydah 5145 Saudi Arabia kouachi.said@caramil.com Belgacem Rebiai Department of Mathematics University of Tebessa, 100 Algeria brebiai@gmail.com