Global solution of reaction diffusion system with full matrix

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1 Global Journal of Mathematical Analysis, 3 (3) (2015) c Science Publishing Cororation doi: /gjma.v3i Research Paer Global solution of reaction diffusion system with full matrix Maroua Mebarki, Abdelkader Moumeni Deartement of Mathematics,Badji Mokhtar University,Annaba-Algeria *Corresonding author marwa.mebarki@hotmail.fr Coyright c 2015 Maroua Mebarki, Abdelkader Moumeni. 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. Abstract The urose of this aer is to rove the global existence in time of solutions for the strongly couled reactiondiffusion system: u t d 1 u d 2 v = f (u, v) in R + u t d 3 u d 4 v = g (u, v) in R + u η = v η = 0 in R+ u(., 0) = u 0 (.), v(., 0) = v 0 (.) in with full matrix of diffusion coefficients. Our techniques of roof are based on Lyaunov functional methods and some L estimates. we show that global solutions exist. Our investigation alied for a wide class of the nonlinear termsf and g. Keywords: Global Existence, Reaction Diffusion Systems, Lyaunov Functional. 1. Introduction In this aer we study the following semilinear arabolic system u t d 1 u d 2 v = f(u, v) v t d 3 u d 4 v = g(u, v) in R + in R + (1.1) Where is a regular and bounded domain of R n, (n 1), u = u(t, x) v = v(t, x), x, t > 0 are real valued functions, denotes the Lalacian oerator, and the constants of diffusion d 1, d 2, d 3, d 4 are assumed to be nonnegative. System (1.1) is subjected to the following boundary conditions u η = v η = 0 in R+ (1.2)

2 110 Global Journal of Mathematical Analysis and the initial data u(., 0) = u 0 (.), v(., 0) = v 0 (.) in (1.3) which are assumed to be nonnegative. The above system (1.1) (1.3) arises in hysics, chemistry and various biological rocesses including oulation dynamics. ( See [6, [23 and references therein). condition (1.2) means that there is no secies of immegration. Concerning the functions f and g, we assume the following hyothesis: (H1) f(r, s) and g(r, s) are continuously differentiable on R + R +, such that f(0, s) 0, g(r, 0) 0 r, s 0 (1.4) (H2) Assume further that there exists an integer 1 such that K 2i 1 f (r, s) + g(r, s) C(r + s + 1) i = 1,..., (1.5) For all r, s 0 and a real m 1 sach that: su( f(r, s), g(r, s) ) C(r + s + 1) m, r, s 0 (1.6) The main question we want to address is the existence of global solutions for system (1.1) (1.3). In fact the subject of the global existence of reaction diffusion systems has received a lot of attention in the last decades and several outstanding results have been roved by some of the major exerts in the field. See [3, 5, 14. This question has been investigated by many authors by considering secial forms of the nonlinear terms f and g. In the trivial case d 2 = d 3 = d 1 d 4 = 0; nonnegative solutions exist globally in time. In diagonal case d 2 = d 3 = 0 Note that, Alikakos[1, treated the following system { ut d 1 u = f(u, v) in R + v t d 4 v = g(u, v) in R + (1.7) with the same boundary conditions (1.2) and initial condition (1.3), f(u, v) = g(u, v) = uv σ and gave a ositive answer to the roblem of the global existence of system (1.7), (1.2), (1.3) under the assumtion 1 < σ < σ 0 (1.8) σ 0 = n (1.9) The method used in [1 is based on some Sobolev embedding theorems. Note that the exonent σ 0 given in (1.9) is exactly the critical exonents given by Fujita [7 for the arabolic roblem { ut = u + u σ u (x, 0) = u 0 (x) (1.10) u 0 in (1.10) is a nonnegative. Fujita roved that if 1 < σ < σ 0 then(1.10) ossesses no global nonnegative solutions while if σ > σ 0, both global and nonglobal nonnegative solutions exist, deending on the nature of the initial energy. Hollis, Martin, and Pierre [10 established global existence of ositive solutions for system (1.1)-(1.2) with the boundary conditions λ 1 u + (1 λ 1 ) u η = β 1, λ 2 v + (1 λ 2 ) v η = β 2on R +

3 Global Journal of Mathematical Analysis λ 1, λ 2 1 or λ 1 = λ 2 = 1 and β 1 0, β 2 0 or λ 1 = λ 2 = β 1 = β 2 = 0 under the conditions f(r, s) + g(r, s) C(r, s)(r + s + 1); r, s 0, i = 1,..., In [20 Masuda obtained a global existence result for a large class of the arameter σ. In fact, by using some L estimates, he showed that the solution of roblem (1.1) (1.3) exists globally in time if σ > 1. The same result in [20 was obtained by Hollis et al [19 by exloiting the duality arguments on L techniques, allowing to derive the uniform boundeness of the solution. Following Masuda s aroach, Haraux and Youkana [9 established a global existence result of system (1.1) (1.3) for a large class of the function f and g. More recisely they showed that for f(u, v) = g(u, v) = uφ(v) (1.11) the roblem (1.1) (1.3) admits a global solution rovided that the following condition holds: log (1 + Φ (v)) lim = 0 (v + ) v In the general case, that is to say for f(u, v) = g(u, v) (1.12) the ositivity of the function g(u, v) together with the maximum rincile of the heat oerator give the following uniform estimate of the solution in L () u(t) u 0 (t) t [0, T max [ Where T max is the maximal time of existence. See Pazy [24 for more details.based on the Lyaunov functional method and for f and g satisfying (1.12), Kouachi [12 roved that the solution of roblem (1.1) (1.3) exists globally in time if log(1 + f(u, v)) lim (v + ) v 8αβ n(α β) 2 u 0 Moumeni and Salah Derradji [21 have established the existence of global solution using an aroach that involves the Lyaunov s functional for the system (1.1) (1.3) the functions f and g are assumed to satisfy the condition f (r, s) + g(r, s) C(r + s + 1). If d 1 d 4, an imortant articular case is that when f 0, which means that the first substance is absorbed by the reaction, in this case, the roblem of the global existence of system (1.7) reduces to obtaining a uniform estimate for v, since by the maximal rincile we have u(x, t) u 0. Here the global existence when d 1 d 4 has been treated by Kanel and Kirane [12 for a bounded domain and by Martin and Pierre [14 for whole sace R n. Still in the case d 1 d 4, but without assuming d 1 d 4, the answer is again ositive to the roblem of the global existence of system (1.7) under condition (1.13) and a olynomial growth assumtion on g: g(u, v) C(u + v + 1) γ,for allu, v 0 and some γ 1,see [10 for more details. If the diffusion coefficients are the same, that is, if d 1 = d 4, then system (1.7) has a global solution under the condition f(u, v) + g(u, v) 0 (1.13),which is known as the mass dissiative structure condition. Indeed if d 1 = d 4,then the solution (u, v) of (1.7) satisfies (by summing u the two equations in (1.7))

4 112 Global Journal of Mathematical Analysis (u + v) t d 1 (u + v) = f + g 0 Then the maximal rincile imlies 0 u + v u 0 + v 0 Therefore, the global existence follows. In tridiagonal case d 3 > 0 and d 2 = 0, Moumeni and Salah Derradji [22 have established the existence of global solution of the roblem (1.1) (1.3) using the Lyaunov method combined with some L estimates. For d 3 > 0 and d 2 0 In [12 J. I. Kanel and M. Kirane roved the global existence of solutions for a strongly couled reaction-diffusion system with homogeneous Neumann boundary conditions and f(u, v) = g(u, v) = uv m, m 0 m is an odd integer, Later they imroved their results in [13 they obtained the global existence with f(u, v) = g(u, v) = uf (v) On the same direction, S. Kouachi [17 has roved the global existence of solutions for two-comonent reactiondiffusion systems with a general full matrix of diffusion coefficients, nonhomogeneous boundary conditions and olynomial growth conditions on the nonlinear terms and he obtained in [18 the global existence of solutions for the same system with homogeneous Neumann boundary conditions and g(u, v) = ρf (u, v), f(u, v) = σf (u, v)ρ 0, σ 0 B. Rebiai and S. Benachour[25treat the case of a general full matrix of diffusion coefficients with the homogeneous boundary conditions with nonlinearities of exonentiel growth. finally in[4 K. Boukerrioua generalize a result obtained in [22. Our techniques are based on invariant regions and Lyaunov functional methods. In the resent work we consider roblem (1.1) (1.3) with d 2 > 0 and d 3 > 0, the function f and g are assumed to satisfy the condition (1.6), and by adoting the Lyaunov method combined with some L estimates we establish a global existence result of the solution. The content of this aer is as follows. In section 2, we introduce some notations and give a local existence result. Our main result is stated in section Local existence Throughout this work, we denote by., [1; + )the norm in L and. the norm in C() or L, resectively, defined by u = u dx 1 and u = esssu u(x) x The study of local existence and uniqueness of solutions (u; v) of (1.1)-(1.3) follows from the basic existence theory for arabolic semi linear equations (see, e.g., [2, [10, [24 and [27). As a consequence, for any initial data in L there exists a T max (0; + such that (1.1)-(1.3) has a unique classical solution on (0, T max [. Furthermore, if T max then lim t Tmax { u(t,.), v(t,.) } = + Therefore, if there exists a ositive constant C such that u(t,.) + v(t,.) C t [0, T max ) then T max = + Remark2.1 Under condition (H1), it follows from the invariant region method that system (1.1) (1.3) reserves ositivity. In other words, if the initial data u 0 and v 0 in (1.3) are nonnegative, then the functions u and v of the corresonding solution of (1.1) (1.3) are also nonnegative on 0, T max [. See [ Statement of the main results 3.1. Existence of global solutions In this section, we state and rove our global existence result of system (1.1) (1.3). Our main theorem reads as follows.

5 Global Journal of Mathematical Analysis 113 Theorem3.1 Let mn 2. Assume that condition (H2) are satisfied. Then the solution (u(t,.), v(t,.)) of (1.1) (1.3) with initial ositive condition in L () exists globally in time. We note that to rove Theorem 3.1 it is sufficient to derive a uniform estimate of su( f(u, v) q, g(u, v) q ) for some q > n/2. (See [10 for more details). The following lemma is a useful tool in the roof of the Theorem 3.1. Lemma3.1 Let (u(t,.), v(t,.)) be the solution of (1.1) (1.3) and let L(t) = Ci K i2 u i v i dx wih a ositive integer and K is a serie of ositive numbers such that K max( d1+d4 2 1, d2+d3 d 1d ) d 3d 2 then the functional L is uniformly bounded on the interval [0, T T T max Proof Differentiating L with resect to t yields L (t) = = [ (ick i i2 u i 1 v i )u t + (( i)ck i i2 u i v i 1 )v t dx (ick i i2 u i 1 v i )(d 1 u + d 2 v + f(u, v))dx + (( i)ck i i2 u i v i 1 )(d 3 u + d 4 v + g(u, v))dx A simle comutation leads L (t) = (ick i i2 u i 1 v i )(d 1 u + d 2 v + f(u, v))dx + (( i + 1)C i 1 From the above equality, it follows that K (i 1)2 u i 1 v i )(d 3 u + d 4 v + g(u, v))dx L (t) = + + I + J + H d 1 ick i i2 u i 1 v i udx + d 2 ic i K i2 u i 1 v i vdx + ick i i2 u i 1 v i f(u, v)dx + d 4 ( i + 1)C i 1 K (i 1)2 u i 1 v i vdx d 3 ( i + 1)C i 1 K (i 1)2 u i 1 v i udx ( i + 1)C i 1 K (i 1)2 u i 1 v i g(u, v)dx By a simle use of Green s formula we have: ( I = A u 2 + B u v + C v 2) dx (3.1) A = : d 1 i (i 1) CK i i2 u i 2 v i

6 114 Global Journal of Mathematical Analysis B = d 1 i ( i) CK i i2 u i 1 v i 1 + d 4 (i 1) ( i + 1)C i 1 K (i 1)2 u i 2 v i C = d 4 ( i) ( i + 1)C i 1 K (i 1)2 u i 1 v i 1 Using the fact that : ic i = ( i + 1)C i 1 and also since i(i 1)C i+1 A = we get = C i 1 1 i = 1,..., (3.2) = i( i)c i = ( i) ( i + 1)C i 1 d 1 ( 1) C i 2 2 Ki2 u i 2 v i = ( 1)C i 2 2 (3.3) B = d 1 ( 1) C 2 i 2 u i 1 v i 1 + Ki2 = B 1 + B 2 and C = d 4 ( 1)C 2 i 1 u i 1 v i 1 K(i 1)2 Putting :j = i 2,we have : d 4 ( 1)C i 2 2 K(i 1)2 u i 2 v i 2 A = d 1 ( 1) C j 2 u j v j 2 K(j+2)2 2 B 2 = d 4 ( 1)C j 2 K(j+1)2 u j v j 2 and Putting :j = i 1,we get : 2 B 1 = d 1 ( 1) C j 2 K(j+1)2 u j v j 2 2 C = d 4 ( 1)C j 2 u j v j 2 Kj2 Then : 2 I = ( 1) C j 2 u j v j 2 Ψ( u, v)dx (3.4)

7 Global Journal of Mathematical Analysis 115 Ψ( u, v) = d 1 K (j+2)2 u 2 + (d 1 + d 4 )K (j+1)2 u v + d 4 K j2 v 2 The quadratic forms are ositive since : ((d 1 + d 4 )K (j+1)2 ) 2 4d 1 d 4 K j2 K (j+2)2 0 j = 0,..., 2 (3.5) Using the relation K max( d1+d4 Then 2 1, d2+d3 d 1d ) d 3d 2 I 0 (3.6) By a simle use of Green s formula we have: ( J = D v 2 + E v u + F u 2) dx (3.7) : D = d 2 i ( i) CK i i2 u i 1 v i 1 E = d 2 i (i 1) CK i i2 u i 2 v i + d 3 ( i) ( i + 1)C i 1 K (i 1)2 u i 1 v i 1 F = d 3 (i 1) ( i + 1)C i 1 K (i 1)2 u i 2 v i Using the relation (3.2) we get D = d 2 ( 1) C 2 i 2 u i 1 v i 1 Ki2 E = d 2 ( 1) C i 2 E 1 + E 2 2 u i 2 v i + Ki2 d 3 ( 1)C i 1 2 K(i 1)2 u i 1 v i 1 and F = d 3 ( 1)C 2 i 2 K(i 1)2 u i 2 v i utting :j = i 1,we have : 2 D = d 2 ( 1) C j 2 K(j+1)2 u j v j 2 2 E 2 = d 3 ( 1)C j 2 u j v j 2 Kj2 and utting :j = i 2,we get :

8 116 Global Journal of Mathematical Analysis 2 E 1 = d 2 ( 1) C j 2 K(j+2)2 u j v j 2 2 F = d 3 ( 1)C j 2 K(j+1)2 u j v j 2 Then : 2 J = ( 1) C j 2 u j v j 2 Φ ( v, u) dx (3.8) Φ ( v, u) = d 2 K (j+1)2 v 2 + (d 2 K (j+2)2 + d 3 K j2 ) v u + d 3 K (j+1)2 u 2 The quadratic forms are ositive since : ((d 2 K (j+2)2 + d 3 K j2 )) 2 4d 2 d 3 K (j+1)2 K (j+1)2 0 j = 0,..., 2 (3.9) Using the relation K max( d1+d4 Then 2 1, d2+d3 d 1d ) d 3d 2 J 0 (3.10) H = Using the relation (3.2), in the third integral, yields : [ ( ) K i2 f(u, v) + K (i 1)2 g(u, v) C 1 i 1 ui 1 v i dx Using the relation(1.5) we deduce [ H c 3 (u + v + 1)C 1 i 1 ui 1 v i dx To rove that the functional L is uniformly bounded on the interval[0, T first we write L (t) c 3 L (t) c 3 L (t) c 3 [ [ [ C i 1 1 ui v i + C i 1 Using the fact that C 1u i i v i 1 = (u + v) 1 C i 1 1 ui 1 v i+1 + C i 1 1 ui 1 v i dx 1 ui v i + C 1u i i v i + C 1u i i v i 1 dx Cu i i v i + C 1u i i v i 1 dx

9 Global Journal of Mathematical Analysis 117 Therefore, the last inequality can be written as L (t) c 1 ()L(t) + c 3 (u + v) 1 Alying Hôlder s inequality to the second term in the right hand side of the above inequality, we obtain L (t) c 1 ()L(t) + c 3 (mes) 1 ( Since the following inequality holds, (u + v) dx) ( 1) ) (u + v) = Cu i i v i su 0 i C i min 0 i CK i i2 CK i i2 u i v i Then, we have L su (t) c 1 ()L(t) + c 3 (mes) 1 0 i C i ( ) ( 1) min 0 i CK i i2 (L(t)) ( 1) t T max Hence,L(t) the functional satisfies the following differential inequality: L (t) c 1 ()L(t) + c 2 ()(L(t)) ( 1) t T max su c 2 () = c 3 (mes) 1 0 i C i ( ) ( 1) min 0 i CK i i2 which gives us, by a simle integration [ (L(t)) 1 (L(0)) 1 c + 2() c 1 () ex(c 1()t) c 2() c 1 () (3.11) c 1() = c 1() c 2() = c 2() By using the inequality L(t) = ( CK i i2 u i v i )dx (CK 2 u + CK 0 02 v )dx it follows that L(t) min(ck 0 02, CK 2 ) su( Hence, (L(t)) 1 [min(c 0 K 02, CK 2 ) 1 su(( And therefore, u dx, v dx) u dx) 1, ( v dx) 1 ) (L(t)) 1 su( u(t,.), v(t,.) ) [min(ck 0 02, CK 2 ) 1 t T max (3.12)

10 118 Global Journal of Mathematical Analysis With (3.11) and (3.12) we obtain : su( u(t,.), v(t,.) ) c(t) t T max (3.13) 1 c(t) = [min(ck 0 02, CK 2 ) 1 The roof of Lemma 3.1 is comlete. Proof of theorem3.1 From (1.6)we have su( f(u, v), g(u, v) ) c 2 (u + v + 1) m Then, it follows that su( f(u, v) m dx, g(u, v) m dx c m 2 which imlies : su( f(u, v) m m [ { (L(0)) 1 c + 2() c 1 () ex(c 1()t) c 2() c 1 ()} (u + v + 1) dx, g(u, v) m ) c m m 2 (u + v + 1) dx (3.14) On the other hand, we have (u + v + 1) dx = k (u + v + 1) dx = C k (u + v) k dx k=0 [1 + (u + v) dx + k=1 An alication of Hôlder s inequality leads [ (u + v) k k=1 C k k=1 C k Hence (u + v + 1) dx mes() + [ using (3.13) we get: 1 + k=1 C k C k ) ( (1 ( k) ( k) dx (u + v) k (u + v) dx ) k (u + v) dx (1) (mes()) ( k) ( ) k (u + v) dx ( (u + v) dx) 1 = u(t,.) + v(t,.) u(t,.) + v(t,.) 2c(t) and the inequality (3.15) can be written as follows (u + v + 1) dx mes() + 2 (c(t)) + C k [(mes()) ( k) (2c(t)) k k=1 k=0 C k [(mes()) ( k) (2c(t)) k

11 Global Journal of Mathematical Analysis 119 Therefore su(( f(u, v) m, g(u, v) m ) c m m m which gives that k=0 C k [(mes()) ( k) (2c(t)) k (3.16) su( f(u, v), g(u, v) ) c,m (t) t T max (3.17) m m c,m (t) = c[ k=0 2 k C k [(mes()) ( k) (c(t)) k m Remark3.1 From both Lemma 3.1 and Theorem 3.1, we have obtained an uniform estimate of su( f(u, v) q, g(u, v) q ) with q = /m > n/2. By the reliminary remarks, we conclude that the solution of the given roblem exists globally in time. References [1 N. Alikakos, L bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations 4 (1979), [2 H. Amann, Dynamic theory of quasilinear arabolic equations - I. Abstract evolution equations, Nonlinear Anal. 12 (1988), [3 S. Bonaved, D. Schmitt, Triangular reaction-diffusion systems with integrable initial data, Nonlinear. Anal 33 (1998), [4 K. Boukerrioua,existence of global solutions for a system of reaction-diffusion equations having a full matrix Ser. Math. Inform. Vol. 29, No 1 (2014), [5 T. Diagana, Some remarks on some strongly couled reaction-diffusion equations, J.Reine. Angew., [6 R. Fisher, The advance of advantageous genes, Ann. Eugenics 7 (1937), [7 H. Fujita, On the blowing u of solutions to the Cauchy roblem for u/ t = u + u (σ+1) J. Fac. Sci. Univ. Tokyo Sect. A Math. 16 (1966), [8 A. Haraux, M. Kirane, Estimation C1 our des roblèmes araboliques semi-linéaires,ann. Fac. Sci. Toulouse Math. 5 (1983), [9 A. Haraux, A. Youkana, On a result of K. Masuda concerning reaction-diffusion equations, Tohoku. Math. J. 40 (1988), [10 D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., 840, Sringer Verlag, New York, [11 S. L. Hollis, R. H. Martin, M. Pierre, Global existence and boundedness in reaction diffusion systems, SIAM. J. Math. Anal. 18(3) (1987), [12 J. I. Kanel and M. Kirane, Pointwise a riori bounds for a strongly couled system of reaction-diffusion equations with a balance law, Math. Methods Al. Sci. 21 (1998), [13 I. Kanel, M. Kirane, Global existence and large time behavior of ositive solutions to a reaction diffusion system, Differ. Integral Equ. Al. 13(1 3) (2000),

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