S. Rionero. 1. Introduction. Let us consider the dynamical systems generated by the dimensionless nonlinear binary system of PDE s:

Transcription

1 J. Math. Anal. Appl. 319 (006) A rigorous reduction of the L -stability of the solutions to a nonlinear binary reaction diffusion system of PDE s to the stability of the solutions to a linear binary system of ODE s S. Rionero University of Naples Federico II, Department of Mathematics and Applications R. Caccioppoli, Complesso Universitario Monte S. Angelo, via Cinzia, 8016 Naples, Italy Received 3 December 004 Available online 5 July 005 Submitted by L. Debnath Abstract A basic peculiar Lyapunov functional V is introduced for the dynamical systems generated by a pair of nonlinear reaction diffusion PDE s, nonconstant coefficients. The sign of V and of its derivative along the solutions is linked through an immediate simple relation to the eigenvalues. By using V and the L -norm, the non-linear L -stability (instability) is rigorously reduced to the stability (instability) of the solutions to a linear binary system of ODE s. 005 Elsevier Inc. All rights reserved. Keywords: Reaction diffusion systems; Nonlinear stability; Lyapunov functional; Direct method 1. Introduction Let us consider the dynamical systems generated by the dimensionless nonlinear binary system of PDE s: address: rionero@unina.it X/$ see front matter 005 Elsevier Inc. All rights reserved. doi: /j.jmaa

2 378 S. Rionero / J. Math. Anal. Appl. 319 (006) ut = a 1 (x)u + a (x)v + γ 1 u + f(u,v, u, v), v t = a 3 (x)u + a 4 (x)v + γ v + g(u,v, u, v) f and g nonlinear and (1) a i : x a i (x) R, a i C(), i 1,, 3, 4}, γ i = const > 0, i = 1,, (u = v = 0) f = g = 0, u : (x,t) R + u(x,t) R, v: (x,t) R + v(x,t) R, () being a bounded domain in R 3 smooth boundary. To (1) we append the Dirichlet boundary conditions u = v = 0 on R + (3) or the Neumann boundary conditions (n being the unit outward normal to ) du dn = dv dn = 0 on R+ (4) the additional conditions ud = vd= 0, t R +, (5) in case (4). We denote by, the scalar product in L ();, the scalar product in L ( ), ; the L ()-norm; the L ( )-norm, ; H0 1 () the Sobolev space such that ϕ H0 1 () ϕ + ( ϕ) L (), ϕ = 0on } ; H 1 () the Sobolev space such that ϕ H 1 () ϕ + ( ϕ) L (), dϕ dn = 0on, } ϕd= 0 ; and study the stability of (u = v = 0) in the L ()-norm respect to the perturbations (u, v) belonging, t R +,to[h 1 0 ()] in case (3) and to [H 1 ()] in case (4) (5) [1 3,5,7,11,13]. We assume also that (cf. (ii) of Section 7) u, f + v,g = o ( u + v ). (6) To (1) we associate the binary linear system of ODE s dξ = b 1 (x)ξ + b (x)η, dη = b 3 (x)ξ + b 4 (x)η, (7)

3 S. Rionero / J. Math. Anal. Appl. 319 (006) x = (x 1,x,x 3 ) vectorial parameter varying on and b 1 = a 1 (x) γ 1 ᾱ, b = a (x), b 3 = a 3 (x), b 4 = a 4 (x) γ ᾱ, (8) ᾱ being the positive constant appearing in the Poincaré Wirtinger inequality 1 φ ᾱ φ (9) holding both in the spaces H0 1(), H 1 (). As it is well known, ᾱ =ᾱ() > 0isthe lowest eigenvalues λ of φ + λφ = 0, (10) respectively in H0 1() and H 1 () (i.e. the principal eigenvalue of ). Our aim is to show that the stability (instability) of the critical point (ξ = η = 0) of (7) implies the stability (instability) of the critical point (u = v = 0) of (1). The plan of the paper is as follows. In Section we introduce a suitable transformation for u and v and two basic Liapunov functionals. Section 3 is dedicated to the stability, while the instability is studied in Sections 4, 5. In the last section (Section 6) the instability condition for a double diffusive convection in a rotating porous medium, uniformly heated and salted from below, is obtained. The paper ends two final remarks (Section 7).. Preliminaries: The basic Liapunov functionals Setting f = γ 1 ( u +ᾱu), g = γ ( v +ᾱv) (11) (1) becomes ut = b 1 u + a v + f + f, v t = a 3 u + b 4 v + g + g (1) b i (i = 1, 4) given by (8). Denoting by α and β α: x α(x) R, (13) β: x β(x) R, two functions suitably defined on and setting u = αū, v = β v, (14) in view of (1) we obtain ūt = b 1 ū + ˆb v + f + f, v t = ˆb 3 ū + b 4 v +ḡ (15) +ḡ 1 When is a cell of periodicity in three dimensions like : x = (x,y,z) 0 x a, 0 y b, z 1 u and v periodic in x and y directions of period a and b, respectively, then (3) and (4) are required only on z = 1 ([4, p. 37], [14, pp ]).

4 380 S. Rionero / J. Math. Anal. Appl. 319 (006) f = α 1 f (u=αū), ḡ = β 1 g (v=β v), ˆb = β α b, f = α 1 f (u=αū,v=β v), ḡ = β 1 g (u=αū,v=β v), ˆb 3 = β α b 3. (16) In the sequel we will use two basic Liapunov functionals. The first one is given by [ V(ū, v) = 1 A(ū + v )d + b 1 v ˆb 3 ū + ˆb v b 4 ū ], (17) A = b 1 b 4 ˆb ˆb 3 = b 1 b 4 a a 3, I = b 1 + b 4. (18) By virtue of dv = Aū, ū t + A v, v t + ( b1 + ˆb ) ( v, vt + ˆb 3 + )ū, b 4 ūt ( b 1 ˆb 3 + ˆb b 4 ) v,ūt ( b1 ˆb 3 + ˆb b 4 )ū, vt, (19) taking into account that along the solutions of (15) one immediately obtains ū, ū t = b 1 ū, ū + ˆb ū, v + ū, f + f, v, v t = ˆb 3 ū, v + b 4 v, v + v,ḡ +ḡ, v,ū t = b 1 ū, v + ˆb v, v + v, f + f, ū, v t = ˆb 3 ū, ū + b 4 ū, v + ū, ḡ +ḡ, by straightforward calculations it turns out that along the solution of (1) dv = AI (ū + v )d + Ψ + Ψ (1) Ψ = α 1 ū α 3 v, f + α v α 3 ū, ḡ, Ψ = α 1 ū α 3 v, f + α v α 3 ū, ḡ, α 1 = A + ˆb 3 + b 4, α = A + b1 + ˆb, α 3 = b 1 ˆb 3 + ˆb b 4. () The second basic Liapunov functional is simply given by E(ū, v) = 1 [ ū + v ]. (3) Along the solution of (15) it turns out that ( de a β = b 1 ū, ū + α + a ) 3α ū, v b 4 v, v + ū, f + f + v,ḡ +ḡ. (4) β (0)

5 S. Rionero / J. Math. Anal. Appl. 319 (006) Stability Lemma 1. Let and a.e. on b i (x)<0, i = 1, 4, b a a 3 > 0, 1 b 4 a a 3 >ε > 1, m a a 3 M, (5) m< M positive constants such that M m < ε + ε 1 ε ε 1. (6) Then the quadratic form ( α P(ū, v) = b 1 ū + β a 3 + β ) α a ū v + b 4 v (7) α = M m, β = [ 1 ( M + m)ε ( M m) ε 1 ] (8) is negative definite a.e. on. Proof. By virtue of (5) 1 (5), the negativeness is guaranteed by ( β α a + α ) β a 3 < 4b 1 b 4, a.e on (9) and hence by Z + 1 Z < ε, Z = β a α a 3, a.e. on, (30) i.e. by ε ε 1 < β α m<z= β a < β α a 3 α M <ε+ ε 1. (31) In view of (8) 1, it is requested that M ( ε ε 1 ) <β< m ( ε + ε 1 ) (3) which is obviously verified by (8). Lemma. Let and a.e. on b i (x)<0, i = 1, 4, b a a 3 < 0, 1 b 4 a a 3 >ε = const > 0, m < a a 3 <M, (33)

6 38 S. Rionero / J. Math. Anal. Appl. 319 (006) m <M positive constants such that M ε < ε. (34) m ε + 1 ε Then (7) α = M m, β = 1 (M + m ) ε [ + 1 (M ] m )ε (35) is negative definite a.e. on. Proof. The definite negativeness is guaranteed by (a.e. on ) ( β α a α ) β a 3 < 4b 1 b 4 (36) and hence by ( ) Z 1 Z < 4ε, Z = β (37) a α a 3, i.e. by Z 1 Z < ε, Z Z 1 > ε (38) and consequently by ε + 1 ε < β α m <Z= β a α < β α M < ε ε. (39) In view of (35) 1, it is requested that M ( ε + 1 ε ) ( <β<m ε ε ) which are obviously verified by (35). a 3 (40) Lemma 3. Let and a.e. on b i (x)<0, i = 1, 4, a = 0, b 1 b 4 >ε = const > 0, (41) a 3 <M 3 = const > 0. Then (7) α = ε, β = M 3 (4) is negative definite a.e. on. Proof. The proof is immediately obtained by substitution of (4) in (7).

7 S. Rionero / J. Math. Anal. Appl. 319 (006) Lemma 4. Let and a.e. on b i (x)<0, i = 1, 4, a 3 = 0, b 1 b 4 >ε = const > 0, (43) a <M = const > 0. Then (7) α = M, β = ε (44) is negative definite a.e. on. Proof. Cf. the proof of Lemma 3. Lemmas 1 4 allow to obtain conditions guaranteeing the stability of the critical point (u = v = 0) of (1). To this end we begin by recalling that the eigenvalues of (7) are given by λ = I ± I 4A. Therefore if x, (45) b i (x)<0, i = 1, 4, A(x)>0, (46) the asymptotic stability of the critical point (ξ = η = 0) of (7) is guaranteed. Theorem 1. Let (6) and (46) a.e. on, hold. Then (u = v = 0) is nonlinearly asymptotically exponentially stable respect to the L ()-norm. Proof. We denote by i, i = 1,, 3, 4, the largest subdomains of such that x 1 a a 3 > 0, a.e. on 1, x a a 3 < 0, a.e. on, x 3 a = 0, a.e. on 3, x 4 a 3 = 0, a.e. on 4, and by ε 1,ε two positive constants such that b1 b 4 a a 3 >ε1, a.e. on 1, b 1 b 4 a a 3 >ε, a.e. on / (48) 1. We notice that 4 ε 1 > 1, = i. (49) i=1 Let us consider now a partition of 1, n 1 = i, n N, (50) i=1 (47)

8 384 S. Rionero / J. Math. Anal. Appl. 319 (006) such that for i = 1,,...,n M i < ε 1 + ε 1 1 m i ε 1 (51) ε1 1. a a M i = ess.sup, m i = ess.inf. (5) a i 3 i a 3 and a partition of, = n+m i=n+1 i, m N, (53) such that for i = n + 1,...,n+ m M i ε < ε (54) m i ε + 1 ε M i = ess.sup i a a 3, m i = ess.inf i a a 3, i = n + 1,...,n+ m. (55) Choosing α i = M i m i, x i,i= 1,...,n,...,n+ m, α = α n+m+1 = ε, x 3, α n+m+ = M, x 4, β i = 1 [ ( M i + m i )ε 1 ( M i m i ) ε1 1 ], x i,i= 1,...,n, β i = 1 [ ( M i + m i ) 1 + ε ( ] M i m i )ε, x i, β = i = n + 1,...,n+ m, β n+m+1 = M 3, x 3, β n+m+ = ε, x 4, M = ess.sup a, M 3 = ess.sup a 3, (58) 4 3 it turns out that ū, f = n+m+ 1αi i=1 ū, f(α i ū) = n+m+ i i=1 ū, f (ū) i = ᾱ ū ), (59) v,ḡ = ( v ᾱ v ), n+m+1 = 3, n+m+ = 4. (60) (56) (57)

9 S. Rionero / J. Math. Anal. Appl. 319 (006) From (3), by virtue of (59), we obtain dē n+m+ i=1 [ b 1 ū b i 4 v ( αi + a i 3 + β ) ] i a ū, v β i α i i + ū, f + v,ḡ. (61) In view of (49) (58) and Lemmas 1 4 there exist n + m + positive constants ε i < 1 such that a.e. on i a α i β i 3 + a β i α ε i b1 b 4, (6) i and hence i 1,...,n+ m + } ( αi a 3 + β ) ( i a ū, v ε i b1 ū + b β i α i i 4 v ). (63) i i Setting ε = infε 1,ε,...,ε n+m+ }, δ= (n + m + ) ε, (64) from (61) (64) it turns out that de δ( b1 ū + b 4 v ) + ū, f + v,ḡ. (65) By virtue of (6) there exist two positive constants ε and μ such that u, f μ( u + v ) 1+ε, v,g μ( u + v ) 1+ε. Therefore setting α = minα 1,...,α n+m+ }, β = minβ 1,...,β n+m+ }, k = maxα1,...,α n+m+,β 1,...,β n+m+ }, k 1 = μ ( ) (k ) 1+ε, k= δ inf(δ α β 1,δ ), δ i = ess.sup bi, by virtue of ū, f = 1 u, f 1 u, f α α 1 μ[ u + v ] 1+ε μ (k α α ) 1+ε (E) 1+ε, v,ḡ μ (k β ) 1+ε (E) 1+ε δ ( b 1 ū + b 4 v ) ke it turns out that de ( k k 1 E ε) E. (69) For any 0 <r 0 <k, in view of (69) it easily follows that k 1 E ε (0)<r 0 (70) (66) (67) (68)

10 386 S. Rionero / J. Math. Anal. Appl. 319 (006) implies k 1 E ε (t) < r 0, t >0, (71) and hence de (k r 0 )E, t >0, (7) i.e. E E 0 e (k r0)t, t >0. (73) 4. Instability By virtue of (45), the critical point (ξ = η = 0) of (7) is unstable iff x 0 : or I(x 0 )>0, A(x 0 )>0 (74) A(x 0 )<0. (75) Passing to the instability of (u = v = 0) of (1) in the L ()-norm we will distinguish two kinds of instability according to (74) or (75) are verified only on a subdomain of or a.e. on. In the first case we define the instability localized on ; in the second case we define the instability distributed on. We begin by considering the instability distributed overall on. Theorem. Let (74) hold a.e. on. Then (u = v = 0) is linearly unstable respect to the L ()-norm. Proof. Choosing α = β = 1, x, on any kinematically admissible perturbation u = p = ϕ1 (t)φ, (76) v = q = ϕ (t)φ ϕ i, i = 1,, arbitrary functions of t and φ =ᾱφ, (77) respectively, on H0 1() or H 1 () (according to the boundary conditions), misregarding the contributions Ψ of the nonlinear terms f and g, from (1) it follows that dv = AI (p + q )d. (78) By virtue of (17), there exist two positive constants δ i,i = 1,, such that δ 1 V u + v δ V. (79) Hence setting

11 S. Rionero / J. Math. Anal. Appl. 319 (006) m = δ 1 ess.inf (AI) (80) it follows that dv = AI (p + q )d > m ( p + q ) >mv, (81) δ 1 i.e., V V 0 e mt. (8) Theorem 3. Let (6) and (74) a.e. on hold. Then (u = v = 0) is nonlinearly unstable respect to the L ()-norm. Proof. Choosing α = β = 1, x }, in view of (6) and (79) it follows that ( Ψ k 1 u + v ) 1+ε kv 1+ε (83) k,k 1,ε positive constants. From (1), (81) and (83) it turns out that dv mv kv 1+ε, (84) i.e. 1 dv V 1+ε m k. (85) V ε Setting Z = V 1 ε, it follows that dz + εmz εk, i.e. V ε (86) me εmt. mz 0 + keεmt (87) Theorem 4. Let A = b1 b 4 a a 3 < 0 a.e. on, meas 4 = 0, 4 being the largest subdomain of such that (88) x 4 a = a 3 = 0. (89) Then (u = v = 0) is unstable. Proof. In view of the procedure of Theorem 3, it is enough to show that (u = v = 0) is linearly unstable. Let us consider the kinematically admissible perturbations (76) u = αū = αϕ1 (t)φ, v = β v = βϕ (t)φ, ϕ 1 = ϕ = ϕ(t), t 0,

12 388 S. Rionero / J. Math. Anal. Appl. 319 (006) ϕ(t) being a suitable function of t. Then (7), along (76) becomes P(ū, v) = ϕ φ ( b 1 + α β a 3 + β α a + b 4 ). (90) Let ( ε>max b1 + b 4 ) and observe that α β a 3 + β α a > ε implies P(ū, v) > εϕ φ. Setting = 3 i=1 i (91) (9) (93) (94) 1 : x 1 a > 0,a 3 0, a.e., : x a 0,a 3 > 0, a.e., 3 : x 3 a a 3 > 0, a.e., and choosing 1, x 1, α = ε+ ε max + a a 3 3 a 3, x 3, ε+ ε max + a a 3 1 a β =, x 1, 1, x, ±1, x 3 according to a > 0ora < 0; then (9) is fulfilled a.e. on. By virtue of (59) i = 1,, 3, 4, from (4) misregarding the contribution of f and ḡ, it turns out that along (76) de ϕ (t)φ (t) = εe, (98) i.e. lim t E =. Theorem 5. Let (88) 1 hold or a a 3 0, b 1 > 0, a.e. on (99) a a 3 0, b 4 > 0, a.e. on. (100) Then (u = v = 0) is unstable. (95) (96) (97)

13 S. Rionero / J. Math. Anal. Appl. 319 (006) Proof. We consider case (99). The proof in case (100) is completely analogous. Let us consider the kinematically admissible perturbations (76) ϕ 1 (t) = Zϕ(t), ϕ (t) = ϕ(t), (101) Z being a real suitable parameter to be determined. Then (7), α = β = 1, along becomes Setting u = Zϕ(t)φ(x), v = ϕ(t)φ(x) (10) P(u,v)= ϕ φ [ b 1 Z + (a + a 3 )Z + b 4 ]. (103) 0 <ε= 1 ess.inf b 4 (104) it follows that b 1 Z + (a 3 + a 3 )Z + b 4 b 1 Z + (a + a 3 )Z ε. (105) Choosing Z in such a way that i.e. b 1 Z + (a 3 + a 3 )Z ε 0 a.e.on, (106) (a + a 3 ) + (a + a 3 ) Z> Z = ess.sup + 4εb 1 (107) b 1 along (10) from (4) Z>max Z,1} misregarding the contributions of f and ḡ it follows that de εϕ (t) φ d > ε Z E and hence lim t E(t) =+. Remark 1. We observe that (88) is not requested by Theorem 5. Remark. Let a 1 + a 4 > 0, a 1 a 4 a a 3 > 0, a.e. on. (108) Then according to Theorem in absence of diffusivity (u = v = 0) is unstable. Therefore when (108) and (46) hold, (u = v = 0) in the presence of diffusivity becomes stable i.e., the diffusivity has a stabilizing effect.

14 390 S. Rionero / J. Math. Anal. Appl. 319 (006) Localized instability Let, } be a smooth subdomain of non-zero measure and let H 0 1( ) denote the subspace of H0 1() of the functions φ such that φ H 0 1( ) implies φ1 (x), x φ =, 0, x (109) φ 1 H0 1( ). Theorem 6. Let (3) and I(x)>0, A(x)>0 a.e. on (110) hold. Then (u = v = 0) is unstable. Proof. Let α = β = 1, x and (p, q) be a kinematically admissible perturbation like (76) such that (p, q) [ H 0 1( )], t 0. Then the instability is immediately implied by Theorems, 3. In fact, along (p, q) the functional (17) reduces to [ ] V(p,q)= 1 (111) A(p + q )d + b 1 q a 3 p + a q b 4 p and misregarding the contributions of the nonlinear terms f and g it follows that dv = AI (p + q )d. (11) Starting from (111), (11), the procedures of Theorems, 3 can be used. Theorem 7. Let (3) and the assumptions of Theorems 4 or 5 hold, replaced by. Then (u = v = 0) is unstable. Proof. Cf. the proof of Theorems 4, 5. Remark 3. Let be the cube D given by x0 d x x 0 + d, y 0 d y y 0 + d, z 0 d z z 0 + d (113) x 0 = (x 0,y 0,z 0 ) and d positive constant. Then it easily follows that (ū, v) ū = v = ᾱd 3π ϕ(t)sin π x x 0 sin π y y 0 d d and ϕ(t), smooth arbitrary functions of t, belongs to [H 1 0 (D)]. sin π z z 0, (114) d

15 S. Rionero / J. Math. Anal. Appl. 319 (006) Remark 4. Theorems analogous to Theorems 6, 7, hold also in the case of boundary data (4) (5). In fact, denoting by H 1( ) the subspace of H 1 () of the functions φ such that φ H 1( ) implies φ1 (x), x φ =, 0, x (115) φ 1 H 1( ), and following the procedures previously used, one immediately obtains the proof of Theorems 6, 7 under the boundary data (4), (5). Further, in the case of the cube (113), it easily follows that (ū, v) ū = v = ᾱd 3π ϕ(t)cos π x x 0 cos π y y 0 d d and ϕ(t), smooth arbitrary functions of t, belongs to [H 1 (D)]. Remark 5. Let cos π z z 0, (116) d I(x 0 )>0, A(x 0 )>0 (117) x 0. Then a i C(), i 1,, 3, 4}, guarantee that exists d>0 such that the cube (113) is a subdomain of I(x) I(x 0) > 0, A(x) A(x 0) > 0, x D. (118) Therefore, in view of Theorem 6, (117) imply instability. Analogously, let A(x 0 )<0 (119) x 0. Then a i C() guarantee that exists a d>0such that the cube (113) is a subdomain of A(x)<0, a (x)a 3 (x) 0, x D. (10) Therefore, in view of Theorem 7, (118) imply the instability. Remark 6. We underline the relevance of (117) and (119) for the onset of the instability. In fact, assume that a i = a i (x,r,c) R and C positive dimensionless parameters peculiar of the phenomenon described by (1) and let m = inf R + x j,r j,c j } R +. Then A(x,R,C)<0, A(x j,r j,c j ) = m, j = 1,,...,n, (11) R (1) C = inf R j, C (1) C = inf C j, j = 1,,...,n, (1) denote the critical values of R and C for the onset of the instability, computable by means of (119). Analogously critical values R () C and C() C of R and C can be obtained from

16 39 S. Rionero / J. Math. Anal. Appl. 319 (006) (117). Then comparing (R (1) C,C(1) C ) to (R() C,C() C ), one can evaluate the effective critical values for R and C. In the following section, we apply this methodology for obtaining the instability conditions for a double diffusive convection in a rotating porous medium, uniformly heated and salted from below. 6. Instability conditions for a double diffusive convection in rotating porous media The Darcy Oberbeck Boussinesq equations governing the motion of a binary porous fluid mixture bounded by two horizontal planes uniformly rotating around the vertical axis z are [6 1]: p = μ k v + ρ f g ρ 0 ɛ ω v, v = 0, (13) AT t + v T = k T T, ɛc t + v C = k C C, ρ f = ρ 0 [1 γ T (T T 0 ) + γ C (C C 0 )]; p 1 is the pressure field; p = p 1 1 ρ 0[ω x] ; ω = ωk is the constant angular velocity; γ C,γ T are, respectively, the thermal and solute expansion coefficients; ɛ is the porosity of the medium; T 0 is a reference temperature; C 0 is a reference concentration; v is the seepage velocity field; C is the concentration field; μ is the viscosity; T is the temperature field; k T,k C are, respectively, the thermal and salt diffusivity; c is the specific heat of the solid; ρ 0 is the fluid density at reference temperature T 0 ; A = (ρ 0 c) m /(ρ 0 c p ) f ; c p is the specific heat of fluid at constant pressure; and the subscript m and f refer to the porous medium and to the fluid, respectively. To (13) we append the boundary conditions TL = T 0 + (T 1 T )/, C L = C 0 + (C 1 C )/ onz = 0, (14) T U = T 0 (T 1 T )/, C U = C 0 (C 1 C )/ onz = d T 1 >T and C 1 >C. Let us introduce the dimensionless quantities x = d x, t = Ad t, v = k T k T d ṽ, P = k(p + ρ 0gz), T = T T 0, C = C C 0. μk T T 1 T C 1 C Omitting all the tildes, the dimensionless equations are: P = v + (RT CC)k + T v k, v = 0, T t + v T = T, εlec t + Lev C = C, where ν = μ/ρ 0 is the kinematic viscosity, ε = ɛ/a is the normalized porosity, T = kω/ɛν is the Taylor Darcy number, Le = k T /k C is the Levis number, (15)

17 S. Rionero / J. Math. Anal. Appl. 319 (006) R = γ T g(t 1 T )dk νk T C = γ Cg(C 1 C )dk νk T is the thermal Rayleigh number, is the solutal Rayleigh number. To (15) we append the boundary data TL = 1/, C L = 1/ onz = 0, (16) T U = 1/, C U = 1/ onz = 1. (15) (16) admit the steady state solution (motionless state) vs = 0; p s (z) = ( R + C) ( z 1 ) k; T(z)= ( z 1 ) ( ) (17) ; C(z) = z 1 On denoting by u = (u,v,w),θ,γ,π the dimensionless perturbations to the (seepage) velocity, temperature, concentration and pressure fields, respectively, the equations governing the perturbations u = (u,v,w),θ,leγ,π are: π = u + (Rθ Le CΓ)k + T u k, u = 0, (18) θ t + u θ = w + θ, εle Γ t + Le u Γ = w + Γ the boundary conditions: w = θ = Γ = 0 onz = 0, 1. (19) In the sequel we shall assume that the perturbation fields are periodic functions of x and y of periods π/a x, π/a y, respectively, and we shall denote by =[0, π/a x ] [0, π/a y ] [0, 1] the periodicity cell. Finally to ensure that the steady state (17) is unique, we assume that ud = vd= 0. By taking the third component of the double curl of (18) 1 and linearizing we obtain w + T w zz = 1 (Rθ Le CΓ), θ t = w + θ, (130) εle Γ t = w + Γ, where 1 = +. We notice that the set I of the kinematically admissible perturbations is characterized by (19), (130) 1, the periodicity and regularity conditions. x y It is easily verified that setting w = γ(r θ Le CΓ), θ = ˆθ(x,y,t)sin(πz), Γ = Γ ˆ (x,y,t)sin(πz) (131)

18 394 S. Rionero / J. Math. Anal. Appl. 319 (006) a γ = ξ + T π, a = ax + a y, ξ = a + π, (13) and ˆθ, Γˆ verifying the plan-form equation 1 = a, (133) it follows that ( w, θ, Γ) I. Along this perturbations (130) (130) 3 become θ t = γr θ γ Le CΓ + θ, (134) εle Γ t = γr θ γ Le CΓ + Γ. By virtue of θ = ξ θ, Γ = ξγ it follows that the constant ᾱ appearing in (7) is given by ξ, and (134) can be written (omitting the bar) θt = b 1 θ + b Γ, (135) Γ t = b 3 θ + b 4 Γ b 1 = γr ξ, b = γle C, b 3 = 1 ε Le γr, b 4 = γ ε C 1 εle ξ (136) and hence A = ξ (γ R Le γ C ξ), εle I = γr γ ( ε C ) ξ. (137) ε Le In view of Theorem 4, Remark 5 and (119), A<0, i.e. R> ξ γ + Le C (138) implies instability. Since ( R B = inf ξ a R + γ )a = ( ξ ) =ā γ a =ā, ā = π 1 + T, R B = π ( T ), (138) immediately gives the instability condition (139) R>R (1) C = R B + Le C (140) for any Le, C and ε. On the other hand, (117) for εle > 1, C C = R B Le(εLe 1) (141) The instability condition (140) coincides the which one obtained in [13] in a different and more involved manner.

19 S. Rionero / J. Math. Anal. Appl. 319 (006) gives the instability condition [1] R>R () C = C ( ε ) R B. (14) εle But, by virtue of (140), it turns out that R () C R(1) C = 1 [ (1 εle)c + R ] B 0 (143) ε Le hence the critical value R C of R for the onset of instability is given by R () C hold and by R (1) C in any other case. when (141) 7. Final remarks (i) Let I denote the identity operator. The scalar E(u, v) = u, Iu + v,iv, (144) is usually interpreted as energy of the perturbation (u, v) to the basic state. Generalizing this point of view, the scalar Q = u, F u + v,gv (145) F and G operators acting on u and v, respectively, can be interpreted as energy dissipated or generated by the operators F, G, according to Q 0 or Q 0, respectively. In the case of the operators F = γ 1, G = γ (146) appearing in (1), in view of f, f = f, f = f, f H0 1 (), f H 1 (), (147) Q is given by Q = γ 1 u γ v (148) and hence the energy is dissipated by (10). By virtue of (9), Q max =ᾱ ( γ 1 ū + γ v ) (149) ū +ᾱū = 0, (150) v +ᾱ v = 0 denotes the lowest energy dissipated by (145) at each instant. The guideline of the present paper has been to show that the conditions guaranteeing the stability (instability) respect to the perturbations dissipating the lowest energy, guarantee the stability (instability) respect to any other perturbation.

20 396 S. Rionero / J. Math. Anal. Appl. 319 (006) (ii) Instead of (6), one can assume ( )( u, f + v,g C u + v u + v ) (151) C positive constant. This happens, for instance, in the case f = c11 u + c 1 uv + c 13 v, g = c 1 u + c uv + c 3 v (15) c ij = const,i = 1,,j = 1,, 3, under Dirichlet boundary conditions. In fact in this case, by use of the Cauchy Schwarz inequality and Sobolev embedding inequality [7] u 4 d C 1 u 4, C 1 = C 1 () = positive constant, (153) one easily obtains that u3 d C u u, v3 d C v v, u vd C v u, uv d C u v, C = C () = positive constant, (154) and (151) immediately follows. Then one has to go back to (4). We here for the sake of simplicity consider, in the case a i = const, i = 1,, 3, 4, the nonlinear instability problem under the assumptions of Theorem 4. Along the kinematically admissible perturbations considered in the proof of Theorem 4, in view of (4), (48) and (151) one obtains de εe ke 3/ (155) ε and k positive constants, and hence E 1/ εe1/ 0 e εt/ ke 1/ 0 e εt/ + ε. (156) We end by observing that (15) reflect the nonlinearity encountered in many generalized Lotka Volterra models [14]. The nonlinear stability instability of these models will be considered in a next paper. Acknowledgments This work has been performed under the auspices of the G.N.F.M. of I.N.D.A.M. and M.I.U.R. (P.R.I.N.) Nonlinear Mathematical Problems of Wave Propagation and Stability in Models of Continuous Media. References [1] H. Amann, Quasilinear parabolic systems under nonlinear boundary conditions, Arch. Rational Mech. Anal. 9 (1986)

21 S. Rionero / J. Math. Anal. Appl. 319 (006) [] H. Amann, Dynamic theory of quasilinear parabolic equations II. Reaction diffusion systems, Differential Integral Equations 3 (1990) [3] H. Amann, Dynamic theory of quasilinear parabolic systems III. Global existence, Math. Z. 0 (1989) 19 50; Erratum: Math. Z. 05 (1990) 331. [4] J.N. Flavin, S. Rionero, Qualitative Estimates for Partial Differential Equations, CRC Press, Boca Raton, FL, [5] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice Hall, London. [6] D.D. Joseph, Stability of Fluid Motions I, II, Springer-Verlag, New York, [7] O.A. Ladyzenskaja, V.A. Solonikov, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr., vol. 3, Amer. Math. Soc., Providence, RI, [8] S. Lombardo, G. Mulone, B. Straughan, Non-linear stability in the Bénard problem for a double-diffusive mixture in a porous medium, Math. Methods Appl. Sci. 4 (001) [9] G. Mulone, On the nonlinear stability of a fluid layer of a mixture heated and salted from below, Contin. Mech. Thermodyn. 6 (1994) [10] D.A. Nield, A. Bejan, Convection in Porous Media, Springer-Verlag, New York, 199. [11] S. Rionero, A rigorous link between the L -stability of the solutions to a binary reaction diffusion system of PDE s and the stability of the solutions to a binary system of ODE s, Rend. Accad. Sci. Fis. Mat. Napoli (4) 71 (004). [1] S. Rionero, On the instability sources in dynamical systems, in: STAMM 00, Springer-Verlag, Berlin, ISBN , 004. [13] J. Smoller, Shock Waves and Reaction Diffusion Equations, Ser. Comprehensive Stud. Math., vol. 58, Springer-Verlag, Berlin, [14] B. Straughan, The Energy Method, Stability, and Nonlinear Convection, second ed., Appl. Math. Sci. Ser., vol. 91, Springer-Verlag, Berlin, 004.