Abdelmalek Salem. Received 4 June 2007; Accepted 28 September 2007 Recommended by Bernard Geurts

Transcription

1 Journal of Alied Mathematics Volume 2007 Article ID ages doi: /2007/12375 Research Article Invariant Regions and Global Existence of Solutions for Reaction-Diffusion Systems with a Tridiagonal Matrix of Diffusion Coefficients and Nonhomogeneous Boundary Conditions Abdelmalek Salem Received 4 June 2007; Acceted 28 Setember 2007 Recommended by Bernard Geurts The urose of this aer is the construction of invariant regions in which we establish the global existence of solutions for reaction-diffusion systems (three equations) with a tridiagonal matrix of diffusion coefficients and with nonhomogeneous boundary conditions after the work of Kouachi (2004) on the system of reaction diffusion with a full 2-square matrix. Our techniques are based on invariant regions and Lyaunov functional methods. The nonlinear reaction term has been suosed to be of olynomial growth. Coyright 2007 Abdelmalek Salem. This is an oen access article distributed under the Creative Commons Attribution License which ermits unrestricted use distribution and reroduction in any medium rovided the original work is roerly cited. 1. Introduction We consider the reaction-diffusion system u t aδu bδv = f (uvw) in R+ Ω (1.1) v t cδu aδv bδw = g(uvw) in R+ Ω (1.2) w t cδv aδw = h(uvw) in R+ Ω (1.3) with the boundary conditions λu +(1 λ) η u = β 1 λv +(1 λ) η v = β 2 λw +(1 λ) η w = β 3 in R + Ω in R + Ω in R + Ω (1.4)

2 2 Journal of Alied Mathematics and the initial data u(0x) = u 0 (x) v(0x) = v 0 (x) w(0x) = w 0 (x) in Ω. (1.5) (i) For nonhomogeneous Robin boundary conditions we use 0 <λ<1 β i R i = (1.6) (ii) For homogeneous Neumann boundary conditions we use λ = β i = 0 i = (1.7) (iii) For homogeneous Dirichlet boundary conditions we use 1 λ = β i = 0 i = (1.8) Here Ω is an oen bounded domain of class C 1 in R N with boundary Ω and / η denotes the outward normal derivative on Ω. The a bandcare ositive constants satisfying the condition 2a (b + c) which reflects the arabolicity of the system and imlies at the same time that the matrix of diffusion a b 0 A = c a b (1.9) 0 c a is ositive definite; that is the eigenvalues λ 1 λ 2 andλ 3 (λ 1 <λ 2 <λ 3 ) of its transosed are ositive. The initial data are assumed to be in the following region: 2μ v0 u 0 + μw 0 2μ β2 β 1 + μβ 3 (u 0 v 0 w 0 ) R 3 such that if u 0 μw 0 β 1 μβ 3 u 0 + μw 0 2μv 0 β 1 + μβ 3 2μβ 2 (u 0 v 0 w 0 ) R 3 such that if u 0 μw 0 β Σ = 1 μβ 3 2μ v0 u 0 + μw 0 2μ β2 β 1 + μβ 3 (u 0 v 0 w 0 ) R 3 such that if μw 0 u 0 μβ 3 β 1 u 0 + μw 0 2μv 0 β 1 + μβ 3 2μβ 2 (u 0 v 0 w 0 ) R 3 such that if μw 0 u 0 μβ 3 β 1 (1.10) μ = b c. (1.11) One will treat the first case the others will be discussed in the last section.

3 Abdelmalek Salem 3 We suose that the reaction terms f g andh are continuously differentiable olynomially bounded on Σ satisfying ( 2μv ) ( 2μv ) ( 2μv ) f μwvw 2μg μwvw + μh μwvw 0 (1.12) f (μwvw)+μh(μwvw) 0 (1.13) ( f ) 2μv μwvw + ( 2μg ) ( 2μv μwvw + μh ) 2μv μwvw 0 (1.14) for all vw 0. And for ositive constants C 1 1/μ C 1 2/μ α 1 1/μ and β 1 2/μ C 1 f (uvw)+c 1 g(uvw)+h(uvw) C 1 ( α1 u + β 1 v + w +1 ) ; (1.15) for all u v w in ΣC 1 is a ositive constant. In the trivial case b = c = 0 nonnegative solutions exist globally in time. This class of systems motivated us to construct this tye of functionals considered in this aer in the aim to rove global existence of solutions. 2. Existence In this section we rove that if ( f gh) oints into Σ on ΣthenΣ is an invariant region for roblem (1.1) (1.5) that is the solution remains in Σ for any intial data in Σ. Once the invariant regions are constructed both roblems of the local and global existensce become easier to be established Local existence. The usual norms in saces L (Ω) L (Ω) and C(Ω) are denoted resectively by u = 1 u(x) dx Ω Ω u = max u(x) (2.1). x Ω For any initial data in C(Ω) orl (Ω) (1+ ) local existence and uniqueness of solutions to the initial value roblem (1.1) (1.5) follow from the basic existence theory for abstract semilinear differential equations (see Friedman [1] Henry [2] and Pazy [3]). The solutions are classical on [0;T max [; T max denotes the eventual blowing-u time in L (Ω) Invariant regions. The main result of this subsection is the following. Proosition 2.1. Suose that the functions f g andh oint into the region Σ on Σ then for any (u 0 ;v 0 w 0 ) in Σ the solution (u(t; );v(t; )w(t; )) of the roblem (1.1) (1.5) remains in Σ for any time [0;T [.

4 4 Journal of Alied Mathematics Proof. The roof follows the same way to that of Kouachi (see [4]). Then we multily (1.1) byc (1.2) by 2bc and(1.3) byb. Adding the first result to the third one and subtracting the second result we get (2.2). Subtracting the first result from the third one we get (2.3). Addingthe first thesecond andthethird resultsto eachother we get (2.4): U t λ 1ΔU = F(UVW) in ]0T [ Ω (2.2) V t λ 2ΔV = G(UVW) in ]0T [ Ω (2.3) W λ 3 ΔW = H(UVW) t in ]0T [ Ω (2.4) with the boundary conditions λu +(1 λ) η U = ρ 1 λv +(1 λ) η V = ρ 2 λw +(1 λ) η W = ρ 3 in ]0T [ Ω in ]0T [ Ω in ]0T [ Ω (2.5) and the initial data U(0x) = U 0 (x) V(0x) = V 0 (x) W(0x) = W 0 (x) inω (2.6) U(tx) = cu(tx) 2bcv(tx)+bw(tx) V(tx) = cu(tx)+bw(tx) W(tx) = cu(tx)+ 2bcv(tx)+bw(tx) (2.7) for all (tx)in]0t [ Ω F(UVW) = ( cf 2bcg + bh ) (uvw) G(UVW) = ( cf + bh)(uvw) H(UVW) = ( cf + 2bcg + bh ) (uvw) (2.8) for all (uvw)in Σ λ 1 = a 2bc λ 2 = a λ 3 = a + 2bc ρ 1 = cβ 1 2bcβ 2 + bβ 3 ρ 2 = cβ 1 + bβ 3 ρ 3 = cβ 1 + 2bcβ 2 + bβ 3. (2.9) First let us notice that the condition of the arabolicity of the system (1.1) (1.3) imlies the one of the (2.2) (2.4) system; since 2a (b + c) a> 2bc.

5 Abdelmalek Salem 5 Now it suffices to rove that the region {(U 0 V 0 W 0 )suchthatu 0 0V 0 0W 0 0}=R + R + R + (2.10) is invariant for system (2.2) (2.4). Since from (1.12) (1.14) we have that F(0VW) 0suffices to be f ( 2μv μwvw) 2μg( 2μv μwvw)+μh( 2μv μwvw) 0 for all VW 0andallvw 0 then U(tx) 0 for all (tx) in]0t [ Ω thanks to the invariant region method (see Smoller [5]) and Because G(U0W) 0suffices to be f (μwvw)+μh(μwvw) 0 for all UW 0andallvw 0 then V(tx) 0forall(tx)in]0T [ ΩandH(UV0) 0suffices to be f ( 2μv μwvw)+ 2μg( 2μv μwvw)+μh( 2μv μwvw) 0forallUV 0andallvw 0 then W(tx) 0forall(tx)in]0T [ ΩthenΣis an invariant region for the system (1.1) (1.3). Then system (1.1) (1.3) with boundary conditions (1.4) and initial data in Σis equivalenttosystem(2.2) (2.4) with boundary conditions (2.5) and ositive initial data (2.6). As it has been mentioned at the beginning of this section and since ρ 1 ρ 2 andρ 3 are ositive for any initial data in C(Ω) andl (Ω) (1+ ). (The local existence and uniquness of the solutions to the initial value roblem (2.2) (2.6) gives us directly those of (1.1) (1.5).) Once invariant regions are constructed one can aly Lyaunov technique and establish global existence of unique solutions for (1.1) (1.5) Global existence. As the determinant of the linear algebraic system (2.7) withre- gard to variables uv andw isdifferent from zero then to rove global existence of solutions of roblem (1.1) (1.5) one needs to rove it for roblem (2.2) (2.6). To this subject; it is well-known that (see Henry [2]) it sufficies to derive anuniform estimate of F(UVW) G(UVW) and H(UVW) on [0T [forsome>n/2. Let us ut A 12 = (a bc/2)/ a(a 2bc) A 13 = a/2 a 2 2bc A 23 = (a + bc/2)/ a(a + 2bc) θandσ are as two ositive constants sufficiently large such that θ>a 12 (2.11) ( θ 2 )( A 2 12 σ 2 A 2 ) ( ) > A13 A 12 A 23 (2.12) Let us definefor any ositive integern the two finite sequences θ q = θ ( q)2 q = 0... σ = σ (n )2 = 0...n. (2.13) Themainresultoftheaerisasfollows.

6 6 Journal of Alied Mathematics Theorem 2.2. Let (U(t )V(t )W(t )) be any ositive solution of (2.2) (2.6) and assume the functional ( ) L(t) = H n U(tx)V(tx)W(tx) dx (2.14) Ω H n (UVW) = n CnC q θ q σ U q V q W n. (2.15) Then the functional L is uniformly bounded on the interval [0T ]T <T max. Corollary 2.3. Suose that the functions f gand h are continuously differentiable on Σ oint into Σ and satisfy condition (1.15).Then all solutions of (1.1) (1.5)withinitialdata in Σ and uniformly bounded on Ω are in L (0T ;L (Ω)) for all 1. Proosition 2.4. Under the hyothesis of Corollary 2.3 if the reactions f gandh are olynomially bounded then all solutions of (1.1) (1.5) with the initial data in Σ and uniformly bounded on Ω are global time. Proofs. For the roof of Theorem2.2 we need some rearatory lemmas. Lemma 2.5. Let H n be the homogeneous olynomial defined by (2.15). Then n 1 U H n = n Cn 1C q θ (q+1) σ (+1) U q V q W (n 1) n 1 V H n = n Cn 1C q θ q σ (+1) U q V q W (n 1) n 1 W H n = n Cn 1C q θ q σ U q V q W (n 1). Proof of Lemma 2.5. See Kouachi [6]. Lemma 2.6. The second artial derivatives of H n are given by (2.16) n 2 U 2H n = n(n 1) Cn 2C q θ (q+2) σ (+2) U q V q W (n 2) n 2 UV H n = n(n 1) Cn 2C q θ (q+1) σ (+2) U q V q W (n 2) n 2 UW H n = n(n 1) Cn 2C q θ (q+1) σ (+1) U q V q W (n 2) n 2 V 2H n = n(n 1) Cn 2C q θ q σ (+2) U q V q W (n 2)

7 Abdelmalek Salem 7 n 2 VW H n = n(n 1) Cn 2C q θ q σ (+1) U q V q W (n 2) n 2 W 2H n = n(n 1) Cn 2C q θ q σ q U q V q W (n 2). Proof of Lemma 2.6. See Kouachi [6]. Proof of Theorem 2.2. Differentiating L with resect to t yields ( ) L U (t) = U H n Ω t + V VH n t + W WH n dx t ( = λ1 U H n ΔU + λ 2 V H n ΔV + λ 3 W H n ΔW ) dx Ω ( ) + F U H n + G V H n + H W H n dx = I + J. Ω Using Green s formula and alying Lemma 2.5wegetI = I 1 + I 2 ( I 1 = λ1 U H n η u + λ 2 V H n η v + λ 3 W H n η w ) dx Ω I 2 = n(n 1) n 2 Ω λ λ 1 σ (+2) θ 1 + λ 2 (q+2) 2 λ B q = 1 + λ 2 2 λ 1 + λ 3 2 Cn 2C q ( Bq T ) Tdx σ (+2) θ (q+1) λ 1 + λ 3 2 σ (+1) θ (q+1) λ 3 σ θ q λ σ (+2) θ (q+1) λ 2 σ (+2) θ 2 + λ 3 q σ (+1) θ q 2 λ σ (+1) θ 2 + λ 3 (q+1) σ (+1) θ q 2 (2.17) (2.18) (2.19) (2.20) for q = 0 = 0n 2andT = ( U V W) t. We rove that there exists a ositive constant C 2 indeendent of t [0T max [suchthat and that for several boundary conditions. I 1 C 2 t [0T max [ (2.21) I 2 0 (2.22)

8 8 Journal of Alied Mathematics (i) If 0 <λ<1 using the boundary conditions (1.4) we get ( ( I 1 = λ1 u H n γ1 αu ) ( + λ 2 v H n γ2 αv ) ( + λ 3 w H n γ3 αz )) dx (2.23) Ω α = λ/(1 λ)andγ i = ρ i /(1 λ) i = 123. Since K(UVW) = λ 1 u H n (γ 1 αu)+λ 2 v H n (γ 2 αv)+λ 3 w H n (γ 3 αw) P n 1 and Q n are olynomials with ositive coefficients and resective degrees (n 1) and n. Since the solution is ositive then lim su K(UVW) = (2.24) ( U + V + W ) + which roves that K is uniformly bounded on R 3 + and consequently (2.21). (ii) If λ = 0 then I 1 = 0on[0T max [. (iii) The case of homogeneous Dirichlet conditions is trivial since in this case the ositivity of the solution on [0T max [ Ω imlies that η U 0 η V 0and η W 0on[0T max [ Ω. Consequently one gets again (2.21)withC 3 = 0. Now we rove (2.22). The quadratic forms (with resect to U Vand W) associated with the matrixes B q q = 0... and = 0...n 2 are ositive since their main determinants Δ 1 Δ 2 andδ 3 are ositive too. To see this we have (1) Δ 1 = λ 1 σ (+2) θ (q+2) > 0forq = 0... and = 0...n 2; (2) λ λ 1 σ (+2) θ 1 + λ 2 (q+2) σ (+2) θ (q+1) Δ 2 = 2 λ 1 + λ 2 σ (+2) θ (q+1) λ 2 σ (+2) θ q (2.25) 2 = λ 1 λ 2 σ 2 ( (+2) θ2 (q+1) θ 2 ) A 2 12 for q = 0... and = 0...n 2. Using (2.11) we get Δ 2 > 0. (3) Δ 3 = B q =λ 1 λ 2 λ 3 σ 2 (+2) σ2 (+1) θ2 (q+1)θ q [(θ 2 A 2 12)(σ 2 A 2 23) (A 13 A 12 A 23 ) 2 ] for q = 0... and = 0...n 2. Using (2.12) we get Δ 3 > 0 (see Kouachi [6]). Substituting the exressions of the artial derivatives given by Lemma 2.5 in the second integralyields n 1 J = n Cn 1C q [ σ(+1) θ (q+1) F + σ (+1) θ q G + σ θ q H ] U q V q W (n 1) dx Ω n 1 [ = n Cn 1C q U q V q W (n 1) F+ θ q G+ σ ] θ q H σ (+1) θ (q+1) dx. Ω θ (q+1) σ (+1) θ (q+1) (2.26)

9 Abdelmalek Salem 9 Using the exressions (2.8) we obtain n 1 J = n Cn 1C q U q V q W (n 1) Ω (( 1 θ q + σ ) ( θ q cf + 1+ σ θ (q+1) σ (+1) θ (q+1) σ (+1) ( + 1+ θ q + σ ) ) θ q bh σ (+1) θ (q+1) dx. θ (q+1) σ (+1) θ (q+1) θ q θ (q+1) ) 2bcg (2.27) And so we get the following inequality: J Ω n 1 n Cn 1C q U q V q W (n 1) ( 1 θq /θ (q+1) + ( )( ) σ /σ (+1) θq /θ (q+1) 1+θ q /θ (q+1) + ( )( ) cf σ /σ (+1) θq /θ (q+1) + 1+ ( )( ) σ /σ (+1) θq /θ (q+1) 1+θ q /θ (q+1) + ( )( ) ) 2bcg + bh σ (+1) θ (q+1) dx σ /σ (+1) θq /θ (q+1) (2.28) using condition (1.15) and relation(2.7) successively we get J C 4 n 1 Ω Cn 1C q U q V q W (n 1) (U + V + W +1)dx. (2.29) Following the same reasoning as in Kouachi [6] a straightforward calculation shows that which for Z = L 1/n can be written as L (t) C 5 L(t)+C 6 L (n 1)/n (t) on[0t ] (2.30) nz C 5 Z + C 6. (2.31) A simle integration gives the uniform bound of the functional L on the interval [0T ]; this ends the roof of the Theorem 2.2. Proof of corollary. The roof of this corollary is an immediate consequence of Theorem 2.2 and the inequality Ω (U + V + W) dx C 8 L(t) on [0T [ (2.32) for some 1 taking into consideration exressions (2.7).

10 10 Journal of Alied Mathematics Proof of roosition. As it has been mentioned above; it suffices to derive a uniform estimate of F(UVW) G(UVW) and H(UVW) on [0T [forsome> N/2. Since the functions f gandhare olynomially bounded on Σ then using relations (2.5) (2.7) and (2.8) we get that F G andh are olynomially bounded too and the roof becomes an immediate consequence of Corollary Final remarks The second the third and the fourth cases are to be studied in the same way as we have done with the first case. The second case:.if { β 1 + μβ 3 2μβ 2 and β 1 μβ 3 }thensystem(1.1) (1.3) can be rewritten as follows: u t aδu bδv = f (uvw) in R+ Ω w t aδw cδv = h(uvw) in R+ Ω v cδu bδw aδv = g(uvw) t in R+ Ω (3.1) with the same boundary conditions (1.4) and initial data (1.5). In this case the diffusion matrix of the system becomes Then all the revious results remain valid in the region a 0 b A = 0 a c. (3.2) c b a Σ = { (u0 v 0 w 0 ) R 3 such that: u 0 + μw 0 2μv0 and u 0 μw 0 } (3.3) μ = b c. (3.4) And system (2.2) (2.4)becomes U t aδu = F 1(UV;W) V t ( a + 2 bc ) ΔV = G 1 (UV;W) W ( a 2 bc ) ΔW = H 1 (UV;W) t (3.5)

11 Abdelmalek Salem 11 U(tx) = cu(tx)+bw(tx) V(tx) = cu(tx)+ 2bcv(tx)+bw(tx) (3.6) for all (tx)in]0t [ Ω W(tx) = cu(tx)+ 2bcv(tx) bw(tx) F 1 (UV;W) = ( cf + bh)(uvw) G 1 (UV;W) = ( cf + 2 bcg + bh ) (uvw) H 1 (UV;W) = ( cf + 2 bcg bh ) (uvw) (3.7) for all (uvw)in Σ; with the boundary conditions λu +(1 λ) η U = ρ 1 λv +(1 λ) η V = ρ 2 λw +(1 λ) η W = ρ 3 in ]0T [ Ω in ]0T [ Ω in ]0T [ Ω (3.8) ρ 1 = cβ 1 + bβ 3 ρ 2 = cβ 1 + 2bcβ 2 + bβ 3 (3.9) ρ 3 = cβ 1 + 2bcβ 2 bβ 3 and initial data (1.5). The conditions (1.12) (1.15) become resectively f (μwvw)+μh(μwvw) 0 f ( 2μv μwvw ) + 2μg ( 2μv μwvw ) + μh ( 2μv μwvw ) 0 f ( 2μv μwvw ) + 2μg ( 2μv μwvw ) μh ( 2μv μwvw ) 0 (3.10) for all vw 0 and for ositive constants C 2 1/2μ C 1 μ/2 α2 1/2μandβ2 μ/2 C 2 f (uvw)+g(uvw)+c 2 h(uvw) C 1 ( α2 u + v + β 2 w +1 ) ; (3.11) for all (uvw) inσc 1 is a ositive constant.

12 12 Journal of Alied Mathematics The third case: If { 2μ β 2 β 1 + μβ 3 and μβ 3 β 1 }thensystem(1.1) (1.3) canbe rewritten as follows: v t aδv bδw cδu = g(uvw) in R+ Ω w t cδv aδw = h(uvw) in R+ Ω (3.12) u t bδv aδu = f (uvw) in R+ Ω with the same boundary conditions (1.4) and initial data (1.5). In this case the diffusion matrix of the system becomes a b c A = c a 0. (3.13) b 0 a Then all the revious results remain valid in the region Σ = { (u0 v 0 w 0 ) R 3 such that: 2μ v 0 u0 + μw 0 and μw 0 u 0 } (3.14) And systems (2.2) (2.4)become μ = b c. (3.15) U t aδu = F 2(UV;W) V t ( a + 2 bc ) ΔV = G 2 (UV;W) W ( a 2 bc ) ΔW = H 2 (UV;W) t U(tx) = cu(tx) bw(tx) (3.16) V(tx) = cu(tx)+ 2bcv(tx)+bw(tx) (3.17) W(tx) = cu(tx) 2bcv(tx)+bw(tx) for all (tx)in]0t [ Ω F 2 (UV;W) = (cf bh)(uvw) G 2 (UV;W) = ( cf + 2 bcg + bh ) (uvw) (3.18) H 2 (UV;W) = ( cf 2 bcg + bh ) (uvw)

13 Abdelmalek Salem 13 for all (uvw)in Σ; with the boundary conditions λu +(1 λ) η U = ρ 1 λv +(1 λ) η V = ρ 2 λw +(1 λ) η W = ρ 3 in ]0T [ Ω in ]0T [ Ω in ]0T [ Ω (3.19) ρ 1 = cβ 1 bβ 3 and the initial data (1.5). The conditions (1.12) (1.15) become resectively ρ 2 = cβ 1 + 2bcβ 2 + bβ 3 (3.20) ρ 3 = cβ 1 2bcβ2 + bβ 3 f (μwvw) μh(μwvw) 0 f ( 2μv μwvw ) + 2μg ( 2μv μwvw ) + μh ( 2μv μwvw ) 0 f ( ) 2μv μwvw 2μg ( ) ( ) 2μv μwvw + μh 2μv μwvw 0 (3.21) for all vw 0 and for ositive constants C 3 2μ C 1 μ α 3 2μandβ 3 μ f (uvw)+c 3g(uvw)+C 3 h(uvw) C 1 ( u + α3 v + β 3 w +1 ) ; (3.22) for all (uvw) inσc 1 is a ositive constant. The fourth case. If { β 1 + μβ 3 2μβ 2 and μβ 3 β 1 }thensystem(1.1) (1.3)canbe rewritten as follows: w t aδw cδv = h(uvw) in R+ Ω u t aδu bδv = f (uvw) in R+ Ω v bδw cδu aδv = g(uvw) t in R+ Ω (3.23) with the same boundary conditions (1.4) and initial data (1.5). In this case the diffusion matrix of the system becomes a 0 c A = 0 a b. (3.24) b c a

14 14 Journal of Alied Mathematics Then all the revious results remain valid in the region { (u0 ) Σ = v 0 w 0 R 3 such that: } u 0 + μw 0 2μv0 and μw 0 u 0 (3.25) μ = b c. (3.26) And system (2.2) (2.4)becomes U t aδu = F 3(UV;W) V t ( a + 2 bc ) ΔV = G 3 (UV;W) W ( a 2 bc ) ΔW = H 3 (UV;W) t U(tx) = cu(tx) bw(tx) V(tx) = cu(tx)+ 2bcv(tx)+bw(tx) W(tx) = cu(tx)+ 2bcv(tx) bw(tx) (3.27) (3.28) for all (tx)in]0t [ Ω F 3 (UV;W) = (cf bh)(uvw) G 3 (UV;W) = ( cf + 2 bcg + bh ) (uvw) (3.29) H 3 (UV;W) = ( cf + 2 bcg bh ) (uvw) for all (uvw)in Σ; with the boundary conditions λu +(1 λ) η U = ρ 1 λv +(1 λ) η V = ρ 2 λw +(1 λ) η W = ρ 3 in ]0T [ Ω in ]0T [ Ω in ]0T [ Ω (3.30) ρ 1 = cβ 1 bβ 3 ρ 2 = cβ 1 + 2bcβ 2 + bβ 3 (3.31) ρ 3 = cβ 1 + 2bcβ 2 bβ 3.

15 Abdelmalek Salem 15 The conditions (1.12) (1.15) become resectively: f (μwvw) μh(μwvw) 0 f ( 2μv μwvw ) + 2μg ( 2μv μwvw ) + μh ( 2μv μwvw ) 0 f ( 2μv μwvw ) + 2μg ( 2μv μwvw ) μh ( 2μv μwvw ) 0 (3.32) for all vw 0 and for ositive constants C 4 1/2μ C 4 μ/2 α 4 1/2μandβ 4 μ/2 C 4 f (uvw)+g(uvw)+c 4 h(uvw) C 1 ( α4 u + v + β 4 w +1 ) ; (3.33) for all (uvw) inσc 1 is a ositive constant. References [1] A. Friedman Partial Differential Equations of Parabolic Tye Prentice-Hall Englewood Cliffs NJ USA [2] D. Henry Geometric Theory of Semilinear Parabolic Equations vol. 840 of Lecture Notes in Mathematics Sringer Berlin Germany [3] A. Pazy Semigrous of Linear Oerators and Alications to Partial DifferentialEquations vol. 44 of Alied Mathematical Sciences Sringer New York NY USA [4] S. Kouachi Global existence of solutions for reaction-diffusion systems with a full matrix of diffusion coefficients and nonhomogeneous boundary conditions Electronic Journal of Qualitative Theory of Differential Equations no [5] J. Smoller Shock Waves and Reaction-Diffusion Equations vol. 258 of Fundamental Princiles of Mathematical Science Sringer New York NY USA [6] S. Kouachi Existence of global solutions to reaction-diffusion systems with nonhomogeneous boundary conditions via a Lyaunov functional Electronic Journal of Differential Equations no Abdelmalek Salem: Deartment of Mathematics and Informatiques University Center of Tebessa Tebessa Algeria address: a.salem@gawab.com

16 Advances in Oerations Research Advances in Decision Sciences Journal of Alied Mathematics Algebra Journal of Probability and Statistics The Scientific World Journal International Journal of Differential Equations Submit your manuscrits at International Journal of Advances in Combinatorics Mathematical Physics Journal of Comlex Analysis International Journal of Mathematics and Mathematical Sciences Mathematical Problems in Engineering Journal of Mathematics Discrete Mathematics Journal of Discrete Dynamics in Nature and Society Journal of Function Saces Abstract and Alied Analysis International Journal of Journal of Stochastic Analysis Otimization