Invariant Sets for non Classical Reaction-Diffusion Systems

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1 Global Journal of Pure and Applied Mathematics. ISSN Volume 1, Number 6 016, pp Research India Publications Invariant Sets for non Classical Reaction-Diffusion Systems Safia Meftah 1 Echahid Hamma Lakhdar University, P.O. Box 789, El-Oued, Algeria, Laboratory of Applied Mathematics, Badji Mokhtar University - Annaba, Algeria. Lamine Nisse Laboratory of Applied Mathematics, Badji Mokhtar University - Annaba, Algeria. Abstract The question discussed in this study concerns the invariant sets for a system of coupled equations for a binary mixtures of isotropic and homogeneous heat conductors. Conditions are given for the existence of an invariant set in the first quadrant of the phase plan. Furthermore, we show that the positively invariant set is not only an attracting set to R + but it is also an absorbing set in R +. AMS subject classification: 35K45, 35B41, 74F0. Keywords: Parabolic systems, positive solutions, invariant sets, mixture effects. 1. Introduction The study of invariant sets or invariant regions for partial differential equations systems was extensively developed in the literature, see for example in [3, 4, 5, 7, 16, 18]. The question on the existence of invariant sets for reaction-diffusion systems is the subject of considerable interest in the study of evolution problems see [17]. The existence of such regions is a very powerful tool, which is used to obtain some interesting properties of the solutions, like their global existence [8, 13, 1], their positivity [], and some other qualitative properties. Our aim in this study is to treat the same subject but 1 Corresponding author: safia-meftah@univ-eloued.dz

2 5106 Safia Meftah and Lamine Nisse for non classical type of dynamical systems which come from the area of thermodynamics in mixtures, and especially we care for coupled equations for a binary mixture of isotropic and homogeneous heat conductors. For the background literature and recent developments we refer for example to [1, 9, 10, 1, 15] and the references cited therein. In [6] Gurtin and De la Penha introduced the following system of partial differential equations, governing the heat flow through a mixture S { a11 t ut, x + a 1 t vt, x = k 1 ut, x + f ut, x, vt, x a 1 t ut, x + a t vt, x = k vt, x + g ut, x, vt, x for t>0 and x in a bounded domain in R n n = 1, or 3, with a smooth boundary. Here u and v are the temperature variations of the constituents, is the laplacian operator the diffusion part, the entries of the matrix a ij and k1,k are the specific heats and the thermal conductivities of the constituents the reaction part, f and g are the functions of heat transfer between constituents. The system S is considered with the Neumann boundary conditions ut,x ν = v t,x ν = 0 for t>0 and x, and the initial conditions u0,x= u 0 x and v0,x= v 0 x and x. The results on existence and uniqueness of the solutions for S are obtained in [15]. For reliability of the model proposed through S we have to care about the non negativity of the solutions, since these constraints are imposed by certain physical considerations. For this purpose we present in our study, a mathematical analysis based on the well-known characterization of flow invariance by means of tangential conditions. Conditions are given for the existence of invariant set noted in the first quadrant of the u, v-plan. Furthermore we show that is not only a positively invariant but it is also an attracting set in R +. We deduct that any neighborhood of is an absorbing set, that is any trajectory of system S starting within R + but outside will enter into a neighborhood of in a finite time. Moreover we show that under certain conditions, is an absorbing set in R + Ṫhe rest of the paper is organized as follows. In the next section, we reformulate the previous system and we give assumptions and transformations for the existence of invariant sets in the positive quadrant of the phase plan. In section 3 we determine an invariant set with the linear reaction term F for which the study of the existence of positive solution is made in [1]. In section 4 we prove that is not only an attractive set in R + but also it is an absorbing set in R +. Finally, in section 5 we give an example which illustrates our results.

3 Invariant Sets for non Classical Reaction-Diffusion Systems Preliminaries By use of the notation see [14] a11 a 1 M = a 1 a ut, x, Ut, x = vt, x and k1 u t,x fut,x,vt,x, FU t,x = gu t,x,vt,x ku t,x = k v t,x with t,x [0, T[, where T > 0 and is a bounded domain of R n n = or3 with a smooth boundary, the system S can be written as M t U t,x = ku t,x + F U t,x. If we assume that the entries of matrix M satisfies the following conditions Cg 0 <a a 11 a 1 a 1 < 1 4 a 11 + a, a a 11 > 0, and a 1 a 1 < 0, then M is a defined positive matrix. So it has two real non-negative eigenvalues λ 1 and λ, and we are allowed to rewrite the system S as S 1 t U t,x = M 1 ku t,x + M 1 F U t,x, where M 1 is the inverse matrix of M. By the following transformations we obtain a simpler formulation of system S 1. Let d = a a 11 a 1 a 1, and k 1, k be positive real numbers. For t,x [0, T[, we define W t,x = w 1 t,x,w t,x by w 1 t,x = u, k 1 x, and w t,x = v, k x where U t,x = u t,x,vt,x is a solution of S, and therefore the system S 1 is transformed to { t w 1 = a w 1 a 1 w + a f w 1,w a 1 g w 1,w t w = a 1 w 1 + a 11 w a 1 f w 1,w + a 11 g w 1,w which can be written more compactly as S t W t,x = E W t,x + E F W t,x, where E is the matrix defined by dm 1.,

4 5108 Safia Meftah and Lamine Nisse Remark.1. 1 Under conditions C g the matrix E has two left eigenvectors v1 = A 1, 1 and v = A, 1 corresponding respectively to the eigenvalues λ 1 and λ with where A 1 = 1 a 11 a δ and A = 1 a 11 + a δ a 1 a 1 δ = a 11 a + 4 a 1 a 1 It is important to note here, that because of conditions C g, λ1 and λ are both positive while A 1 < 1 and 0 <A < Invariant sets and positivity of the solutions in the linear case Some sufficient conditions which assure the existence of positive solutions for system S when F is a linear map are given in [8]. By the use of the tangential conditions see [11], we determine a positively invariant set in the positive quadrant of w 1,w -plan. Definition 3.1. A subset w R is called a positively invariant set for the system S if any solution W t,x with initial and boundary values in w satisfies W t,x w for all t, x [0,T[ where T is the maximal time of existence of W t,x. In the case of the reaction term given in [6], for which the existence has been studied in [15], we obtain the invariant set given by the following proposition Proposition 3.. If the conditions C g are satisfied and such that c>0, the set defined by F w 1,w = c w 1 w,cw 1 w, w ={w 1,w R, such that w 1 0 and A w 1 w A 1 w 1 }, is an invariant set for S. Proof. Let G 1 and G be two smooth real functions defined on R as follows W = w 1,w R, G 1 W = A 1 w 1 + w and G W = A w 1 w. It is easy to see that G 1 and G satisfy the conditions of theorem 4.3 in []. Thus, the set which is equal exactly to w, and given by { W = w1,w R, such that w 1 0 and G i W 0, i = 1, }, is an invariant set for S.

5 Invariant Sets for non Classical Reaction-Diffusion Systems 5109 Definition 3.3. It is said that F w 1,w is quasi-positive if it satisfies for all w 1,w R + : F 1 0,w 0 or F w 1, 0 0, where F 1 and F are the components of F. At first, it is important to point out here, that in spite of the quasi-positivity of the reaction term F w 1,w, the positively invariant set w that we obtained, is strictly included in the first quadrant of the phase plan, which is not the case for the classical systems like in [11]. Furthermore we show that it is not only a positively invariant set, but it is also an attracting set in R +. Indeed by the help of the maximum principle via comparison theorem, we prove that all trajectories W t = w 1 t,x,w t,x which are in the positive quadrant of the w 1,w -plan are attracted by the invariant set w. Definition 3.4. We say that the set w R + is an attractor if it satisfies 1 w is an invariant set. w admits an open neighborhood O such that, for any solution W t of S with W 0 O; dist W t, w 0 when t +. Here dist u, L means the distance between the point u and the set L. Proposition 3.5. Let us assume that assumptions of Proposition 3. are satisfied. If W t is a positive global solution of S with initial condition then it is attracted by the set w. W 0 = w 1 0,x,w 0,x R +, x, Proof. Under the assumptions of Proposition 3., if we take the dot product of S by the left eigenvector v 1 we have v 1, W = λ 1 v 1,W + λ 1 v 1,F W. t If we take G 1 = A 1 w 1 + w > 0, we obtain G 1 t = λ 1 G 1xx + λ 1 ca 1 w 1 w + c w 1 w = λ 1 G 1xx + λ 1 c G 1 + A 1 w + w 1 λ 1 G 1xx + λ 1 c w A 1 w 1 = λ 1 G 1xx + λ 1 c G 1 thus, G 1 t λ 1 G 1xx + λ 1 cg 1 0.

6 5110 Safia Meftah and Lamine Nisse So if J 1 t,x = G 10 e λ1ct with G 10 = max G 1 0,x we have x Then G 1 t J 1 t = λ 1 J 1xx λ 1 cg 10 e λ 1ct = λ 1 J 1xx λ 1 cj 1. λ 1 G 1xx + λ 1 cg 1 0 = J 1 λ 1 J 1xx + λ 1 cj 1 t G 1 ν = 0 = J 1 ν G 1 0,x G 10 = J 1 0,x Then according to the theorem.10.1 in [19] we can say that G 1 t,x J 1 t,x = G 10 e λ 1ct. In other words G 1 t,x converge to 0 when t tends to + ; G 1 t,x = A 1 w 1 t,x + w t,x t In the same way, taking the dot product of the system S by the left eigenvector v,if we assume that G = A w 1 w > 0, we obtain a similar result for G G t,x = A w 1 t,x w t,x t Finally, 3.1 and 3. imply that w is an attractive set to R +. According to the compactness of the trajectories, and taking the result of Proposition 3.5 into account, we show that the set w has some absorbing neighborhoods. 4. Absorbing set The positively invariant set w is not only an attracting set in R +, but we show that it is also an absorbing set in R +. Definition 4.1. Let w R + and O an open set containing w. We say that w is an absorbing set in O, if the trajectory of all solution of S enters in w after some finite time. A direct consequence of Proposition 3.5 is that any open neighborhood of the attracting set w is an absorbing set. Corollary 4.. Under assumptions of Proposition 3.5, and for all ɛ>0 small enough, the set defined by ɛ ={w 1,w R, such that w 1 0, w 0 and A w 1 ɛ<w < A 1 w 1 +ɛ}

7 Invariant Sets for non Classical Reaction-Diffusion Systems 5111 is an absorbing set in R +. Proof. Under assumptions of Proposition 3.5, w is an attracting set in R +,soifwe note W t,x = w 1 t,x,w t,x such that w 1 > 0 and w + A 1 w 1 0, we have Therefore dist W t,x, w t + 0 uniformly for x. ɛ >0, T > 0, such that t >T : Inf σ w w W t,x σ w < Then, there exists σ w = σ 1,σ = σ 1, A 1 σ 1, ɛ 1 A 1, x. such that W t,x σ w < ɛ 1 A 1. So if we choose. 1 as a norm in R all the norms are equivalent in R wehave W t,x σ 1 w = w1,w σ1, A 1σ1 1 = w1 σ1 + w + A 1 σ1 ɛ < 1 A 1 w1 σ ɛ 1 1 A 1 w + A 1 σ1 ɛ 1 A 1 ɛ w 1 σ1 1 A 1 w + A 1 σ1 ɛ 1 A 1 A 1 σ1 A ɛ 1w 1 A 1 1 A ɛ 1 w A 1 σ1 1 A 1 ɛ w A 1 w 1 A 1 + ɛ 1 A 1 1 A 1 = A 1 w 1 + ɛ 1 A 1 = A 1 w 1 + ɛ 1 A 1 Then ɛ >0, T > 0, such that t>t : W t,x ɛ. So, ɛ is an absorbing set in R +. In the next, we give our main result, proving that any trajectory W t = w 1 t,x,w t,x

8 511 Safia Meftah and Lamine Nisse in the positive quadrant of w 1,w -plan, enters in w in a finite time. To reach this goal, we start with the following two lemmas. Lemma 4.3. Let t = 1 t, t = w 1 t,x ϕ x dx, w t,x ϕ x dx, where ϕ x is the eigenfunction corresponding to the principal eigenvalue λ of the laplacian with Neumann boundary conditions. Then is a solution of the following system of ordinary differential equations λa + c 1 λa 1 + c 1 where A =, λa 1 + c λa 11 + c c 1 = c a + a 1 and c = c a 11 + a 1. Furthermore t 0 e µ At, t 0, where µ A is the matrix measure of A. Proof. Let ϕ be the solution of { ϕxx x = λϕ x, ϕx > 0, x, ϕ x = 0, x. ν According to S we have d 1 = λa 1 + λa 1 + a c 1 a 1 c 1 d = λa 1 1 λa 11 a 1 c 1 + a 11 c 1 = A t and 0 = 0. Thus, by the Gronwall and Langenhop inequalities [0] page 47 we obtain t 0 e µ At, t Lemma 4.4. If we denote by η := λ 1 λ + c and we assume that the entries of matrix M satisfies conditions C g and a1 + a 1 0 then η µ<0 µ = µ A. Proof. Since the norms are equivalent in R, if we choose. defined by t = i=1 i t 1,

9 Invariant Sets for non Classical Reaction-Diffusion Systems 5113 the matrix measure is given by µ A = β max A + A t, such that A t is the transpose of the matrix A, and β max A + A t is the maximum of the eigenvalues of the matrix A + A t. Let λa + c 1 λa 1 + c 1 B = A =. λa 1 + c λa 11 + c Firstly, we calculate the eigenvalues of B + B t which are the roots of the characteristic polynomial P β = β β [λ a 11 + a + c 1 + c ] + 4 λa + c 1 λa 11 + c [λ a 1 + a 1 + c 1 + c ]. According to the conditions C g therefore a 11 a + a 1 + a 1 > 0, c 1 c [ a1 a 1 a 11 a ] + a 11 a a 1 + a 1 c c 1 < 0, and hence the discriminant of P is positive. Then the matrix B + B t admits two real eigenvalues. It follows that µ A = 1 } {λ max a 11 + a + c 1 + c +,λa 11 + a + c 1 + c, where = λ [ a 11 a + a 1 + a 1 ] + λ [a 11 a c c 1 + a 1 + a 1 c + c 1 ] + c 1 + c. The assumption a 1 + a 1 0 leads to λ a 11 + a + c 1 + c + µ A = 1 Finally, η µ = λ 1 λ + c 1 [ ] λ a 11 + a + c 1 + c [ a 11 + a a 11 + a 4 a 11 a a 1 a 1 = 1 1 [ ] λ a 11 + a + c 1 + c = 1 [ c a 1 + a 1 + c + λ. ] λ + c ] a 11 + a 4 a 11 a a 1 a 1 + < 0.

10 5114 Safia Meftah and Lamine Nisse Remark 4.5. If we denote by η := λ λ + c and we assume that the entries of matrix M satisfies conditions C g and a1 +a 1 0 with a 1 > 0, then η µ <0 µ = µ A. Theorem 4.6. Let the assumptions of Proposition 3.5 be satisfied, and α 0 = cosθ where θ is the angle between the lines A 1 w 1 = w and w 1 = 0. If we assume that 1 < cλ 1α 0 µ η and a 1 + a 1 0, then w is an absorbing set in R +. Proof. Let w 1 t,x,w t,x be a solution of S in R + outside w for all t>0. Suppose that A 1 w 1 t,x + w t,x > 0,w 1 t,x > 0, x, t >0. If we set according to 4.3 we have V t = A 1 1 t + t, which can be rewritten as dv t dv t = λev t + cev t = λ 1 λ + c V t + λ 1 c 1,A 1, 1,. 4.4 Since A 1 w 1 + w > 0, there exists α>α 0 > 0 such that Therefore, according to lemma 4.3 we get Consequently, the equation 4.4 implies 1,A 1, 1, = α v 1. 1,A 1, 1, α v 1 0 e µt. dv If we put κ = cλ 1 α v 1 0, we obtain dv λ 1 λ + c V cλ 1 α v 1 0 e µt. + ηv κe µt d t e ηt V t κe η µt e ηt V t V 0 κ e η µt ds e ηt V t V 0 κ e η µt 1 0 η µ V t V 0 + κ e η µt 1. µ η

11 Invariant Sets for non Classical Reaction-Diffusion Systems 5115 The function h defined by h t := V 0 + κ e η µt 1 µ η is a positive continuous function such that h 0 = V 0 > 0. According to Lemma 4.4 we have µ η>0, so lim h t = V 0 κ t + µ η v 1 0 cλ 1α v 1 0 µ η = v cλ 1α < v 1 0 µ η 1 cλ 1α 0 µ η By assumptions of the Theorem we have 1 < cλ 1α 0, which implies that lim µ η h t < t + 0. Then, there exists t 0 > 0, such that h t 0 = 0, therefore V t 0 h t 0 = 0 which contradicts the fact that V t > 0, t > 0. So there exists t > 0 such that w1 t,x,w t,x w. This proves that w is an absorbing set in R +.. Remark 4.7. The complementary set of w in R + consists of two positive cones, one situated below w limited by the half-lines w = 0, and w = A w 1 and the other above w limited by the half-lines w 1 = 0, and w = A 1 w 1. In the proofs of Corollary 4. and Theorem 4.6, we considered only the second situation, where the starting point of a solution outside the set w, is a point of R +, located above w.in the other situation, taking into account the Remark 4.5 the reasoning can be applied in a similar way. 5. Example To illustrate our results, we take as an example the following system { S t v t,x = k 1 u xx t,x c u t,x v t,x t u t,x + 3 t v t,x = k v xx t,x + c u t,x v t,x where k 1 and k are two positive real numbers, c is a positive real constant, and t,x ]0, T[ ]0, 1[. Here the matrix M is defined by 0 1 M =. 1 3 It can easily be checked that the entries of the matrix M satisfy the conditions C g. So, M possesses two real non-negative eigenvalues λ 1 = 3 5 and λ = 3 + 5, and

12 5116 Safia Meftah and Lamine Nisse its inverse matrix E = M 1 has two left eigenvectors v 1 = A 1, 1 and v = A, 1 corresponding respectively to the eigenvalues λ 1 and λ with A 1 = 3 5 and A = 3 5. Thus, after carrying out the transformations introduced in Section, according to Proposition 3., w defined by { w = w 1,w R, such that w 1 0 and 3 5 w 1 w 3 + } 5 w 1 is an invariant set. Finally, the condition η µ = 1 [ ] 5 λ + c + 9λ + 30λc + 6c < 0 being satisfied, with an appropriate choice of the constant c, Theorem 4.6 allows us to assert that w is an absorbing set in R +. References [1] R. J. Atkin and R. E. Craine, Continuum theories of mixtures: Basic theory and historical development, Quart. J. Mech. Appl. Math., , [] K. N. Chueh, C. C. Conly, and J.A. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Indiana. Univ. Math. J., , [3] C. Conley, R. Easton, Isolated invariant sets and isolating blocks, American mathematical society, Vol 158, No 1, July [4] H. Frid, Invariant regions under lax-friedrichs scheme for mulimensional systems of conservation laws, Discrete and continuous dynamical systemes, Vol 1, No 4, 1995, [5] H. Frid, Maps of convex sets and invariant regions for finite-difference systems of conservation laws, Arch. Rational Mech. Anal., , [6] M. E. Gurtin and G. M. De la Penha,On the thermodynamics of mixtures, Arch. Rational mech anal., , [7] D. Hoff, Invariant regions for systems of conservation laws, Transations of the American mathematical society, vol 89, No, [8] S. L. Hollis, R. H. Martin, M. Pierre, Global existence and boundedness in reactiondiffusion systems, Siam J. Math. Anal., Vol 18, No 3, [9] D. Iesan and R. Quintanilla, On the problem of propagation of heat in mixtures, Applied mechanics and engineering., ,

13 Invariant Sets for non Classical Reaction-Diffusion Systems 5117 [10] M. C. Leseduarte, R Quintanilla, On the decay of solutions for the heat conduction with two temperatures, Acta Mech., 4 013, [11] R. H. Martin Jr, Global existence questions for reaction-diffusion systems, Pitman res. Notes., , [1] F. Martinez and R. Quintanilla, Some qualitative results for the linear theory of binary mixtures of thermoelastic solids. Collect. Math., , [13] K. Masuda, On the global existence and asymptotic behavior of solutions of reaction-diffusion equations, Hakkaido math journal, Vol 1, 1983, [14] S. Meftah and L. Nisse, Invariant sets for systems in the thermodynamics of Mixture. Comm. RAMA-5, M Sila, avril 006. Prépublication de L.M.A. Départ. Math. Univ. Annaba., 006. [15] R. Quintanilla, Study of the solutions of the propagation of heat in mixtures, Dynamics of cont., Disc. & Impul. Syst., 8 001, [16] R Redheffer, W Walter, Invariant sets for systems of partial differential equations 1.Parabolic equations, Math inst uni Karlsruhe and dep of math uni of California at Los Angeles, [17] R Redlinger, Invariant sets for strongly coupled reaction-diffusion systems under general boundary conditions, Math inst uni Karlsruhe D-7500 Karlsruhe 1 West Germany, [18] D. Serre, Domaines invariants pour les systèmes hyperboliques de lois de conservation, J. of diff equa., , [19] J. Smoller, Shock waves and reation-diffusion equations, A Series in Comprehensive Studies in Math., 58. Springer-Verlag [0] M. Vidyasagar, Nonlinear Systems Analysis, Classics in applied mathematics, 4, Philadelphia, Pa., SIAM, 00. [1] J Zhang, Boundedness and blow-up behavior for reaction-diffusion systems in a bounded domain, Nonlinear analysis ,