The Global Dynamics of Isothermal Chemical Systems with Critical Nonlinearity

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1 Wright State University COE Scholar Mathematics and Statistics Faculty Publications Mathematics and Statistics 5-23 The Global Dynamics of Isothermal Chemical Systems with Critical Nonlinearity Yi Li Wright State University - Main Campus, yi.li@csun.edu Yuanwei Qi Follow this and additional works at: Part of the Applied Mathematics Commons, Applied Statistics Commons, and the Mathematics Commons epository Citation Li, Y., & Qi, Y. (23). The Global Dynamics of Isothermal Chemical Systems with Critical Nonlinearity. Nonlinearity, 16 (3), This Article is brought to you for free and open access by the Mathematics and Statistics department at COE Scholar. It has been accepted for inclusion in Mathematics and Statistics Faculty Publications by an authorized administrator of COE Scholar. For more information, please contact corescholar@ library-corescholar@wright.edu.

2 The Global Dynamics of Isothermal Chemical Systems with Critical Nonlinearity Y. Li Y. W. Qi Abstract In this paper, we study the Cauchy problem of a cubic autocatalytic chemical reaction system u 1,t = u 1,xx u α 1 u β 2, u 2,t = du 2,xx + u α 1 u β 2 with non-negative initial data, where the exponents α, β satisfy 1 < α, β < 2, α + β = 3 and the constant d > is the Lewis number. Our purpose is to study the global dynamics of solutions under mild decay of initial data as x. We show the exact large time behaviour of solutions which is universal. 1 Introduction In this paper, we study the Cauchy problem of the chemical reaction system with non-negative initial data u 1,t = u 1,xx u α 1 u β 2 (1) u 2,t = du 2,xx + u α 1 u β 2 (2) u 1 (x, ) = a 1 (x), u 2 (x, ) = a 2 (x), a 1 (x), a 2 (x) L 1 () L () (3) of arbitrary size, where the exponents α, β satisfy 1 < α, β < 2, α + β = 3 and the constant d > is the Lewis number. Our purpose is to study the global dynamics of solutions under mild decay of initial data as x. The system (1)-(2) arises as a model for cubic autocatalytic chemical reaction of the type αa + βb 3B, Department of Mathematics, University of Iowa, Iowa City, IA 52242, this author wishes to thank the financial support of the Chinese National Science Foundation Grant, USA. Department of Mathematics, Hong Kong University of Science & Technology, Hong Kong, maqi@ust.hk, this author was partially supported by HK GC grant HKUST6129/p. 1

3 with isothermal reaction rate proportional to u α 1 u β 2. Here u 1 is the concentration of reactant A, u 2 is the concentration of auto-catalyst B. The simple power rate is obtained by the usual assumption of the exponential dependence on the temperature rise in flame systems. Such system is also used to model thermal-diffusive combustion problems, see [7]. For recent works on similar systems with application in mathematical biology, see [8]. The system (1)-(2) on a bounded domain is well-studied in the literature, in particular for the case of α = 1, β = 2, see Alikakos [1], Hollis, Martin and Pierre [1], Martin and Pierre [12] and Masuda [13]. Among other things, the authors established boundedness, global existence and large time behavior of solutions. For homogeneous Dirichlet or Neumann boundary conditions, the large time behaviour is that (u 1, u 2 ) converges to a constant vector (c 1, c 2 ) such that c 1 c 2 =, see Masuda [13]. More recently, Billingham and Needham [4], [5] showed the existence of traveling front, again for the case of α = 1, β = 2 using shooting argument and phase plane methods. In addition, the large time behaviour of solutions are also studied by formal methods and numerical computation by the authors. Most recently, motivated by thermal-diffusion models with Arrhenius reactions [2], [14], Berlyand and Xin [3] and Bricmont, Kupiainen and Xin [7] considered system (1)-(2) with initial data (3) that have sufficient polynomial decay at infinity for the case of α = 1 and β = 2. The authors of [7], in particular Bricmont and Kupiainen [6], along with Goldenfeld et al [9], poineered in applying the enormalization Group to nonlinear parabolic equations. In [3] and [7], upper and lower bounds by self-similar profiles were established and more importantly, the exact large time self-similarity of solutions with initial data decaying sufficiently fast as x was proved. We give a detailed description of the main result of [7] in what follows. Suppose (a 1, a 2 ) B B, where B is the Banach space of continuous functions on with the norm Let φ d be the Gaussian f = sup x f(x) (1 + x ) q, q > 1. (4) φ d = 1 4πd e x2 /4d. For A >, let ψ A be the normalized ( ψ 2 A(x) dµ(x) = 1, see below) first eigenfunction of differential operator L A = d2 dx x d dx A2 φ 2 d(x) on L 2 (, dµ), with dµ(x) = e x2 /4 dx and E A, which is positive for A >, be the corresponding eigenvalue, then the main result of [7] is as follows. 2

4 Theorem (Bricmont, Kupiainen and Xin). Suppose α = 1, β = 2, (a 1, a 2 ) B B and a i, a i, i = 1, 2. Let A = a 1(x) + a 2 (x)dx, which is conserved in time. Then the system (1)-(2) has a unique global classical solution (u 1 (x, t), u 2 (x, t)) B B t. Moreover, there exists a q(a), which is an increasing function of A and tending to as A, such that if q q(a) in (4), there is a positive number B depending continuously on (a 1, a 2 ) such that /2+E A u 1 ( t, t) Bψ A ( ) /2 u 2 ( t, t) Aφ d ( ) as t. emark. As was pointed out in [7] that the extra decay power E A in time is due to the critical cubic nonlinearity of the system (1)-(2). In others word, the scaling law which works for non-cubic nonlinearity no longer works for cubic nonlinearity, and thus the appearance of the anomalous exponent E A. In this paper, following the work of [7] we continue to investigate the critical cubic nonlinearity (i.e. α + β = 3) of the system (1)-(2). The main result of the present work is: Theorem 1 Suppose 1 < α, β < 2 and α+β = 3. Consider initial data (a 1, a 2 ) B B and a i, a i, i = 1, 2, where q > 1 is fixed. Let A = a 1(x) + a 2 (x)dx > be the total mass, which is conserved in time. Then the system (1)-(2) has a unique global classical solution (u 1 (x, t), u 2 (x, t)) B B for t. Furthermore, there exists B = B(A, d, α, β) > such that as t. /2 (log t) 1/(α 1) u 1 ( t, t) Bφ 1 ( ) (5) /2 u 2 ( t, t) Aφ d ( ) (6) emark. Unlike the case of α = 1 and β = 2, where the value of B is not explicitly, we know the precise value of B. The exact expression of the constant B is as follows: ( 4πd β/2 ) 1/(α 1) B = (α 1)A β (α + β/d) 1/2. emark. The asymptotic of the solution is in certain sense universal since the large time spatial-temporal profile is independent of the decay rate at x = and the size of the initial data except its apparent dependence on the total mass. In addition, the constant B depends explicitly on the known parameters A, d, α and β. emark. The log t term in (5) is not out of a simple scaling argument, rather it is the by-product of the critical cubic nonlinearity and competition between diffusion and mass transfer. The power of log t is an anomalous exponent. (7) 3

5 emark. In order to understand intuitively why the asymptotic stated in Theorem 1 holds, let us consider a more general case of α, β 1. If α + β > 3, it can be shown as in [3] that both u 1 and u 2 go to zero like the solution of pure diffusion as t. For, α + β < 3, one can apply the maximum principle and a simple a priori estimate to bound u 1 by u, which is a solution of u t = u xx O ( t (α+β 1)/2 φ β d ( x t ) ) u. Then, by applying the Feynmann-Kac formula, one obtains u exp ( O ( t 3 α β)/2)) (8) for x O( t). For x O( t), one gets a diffusive behaviour, depending on the rate of decay, as x of the initial data. Next, by inserting the fast decay (8) for u 1 in (2), one shows the nonlinear term is insignificant and that u 2 diffuses to zero. Clearly, the critical case of α + β = 3 is the most delicate and interesting one. In particular, instead of (8), one gets exp( O(log log t)), which, after detailed analysis, gives rise to (5). emark. In comparison of our result with that of Bricmont, Kupiainen and Xin [7], it shows that (i) For our case, only minimum requirement on the decay of initial data as x is needed, the decay rate is independent of the total mass A, unlike the case in [7] where the decay rate is related to A. We believe our result is optimal. But our method fails for their case. (ii) The appearance of log t indicates the analysis is more involved and subtle. In particular, it is well known in the scientific computation field that a scaling of log t is hardly detectable in computation. (iii) The nonlinear dependence of u 1, in addition to the cubic nonlinearity, is solely responsible for the arise of log t term, the power of which tends to infinity as α 1. This may explain the appearance of extra decay power of t in the limiting case of α = 1. emark. To model chemical fronts (flames) propagating down a tube, one has to feed the system at one end of the tube or in an idealized situation at spatial infinity by specifying nonzero value of u 1 or u 2. But, when the feeding is turned off at a later stage of reaction process and leaves the system to relax freely by itself, the decaying initial condition emerges. Therefore, the dynamics of the present case is vastly different from the case where traveling front solutions develop. In that case, the front solutions in general undergo transitions to chaos as Lewis number d is sufficiently large or the order of autocatalytic reactions is high enough, see Sivashinsky [17] and Metcalf et al [15]. For a more detailed description, see [7]. But, for the critical nonlinearity case studied 4

6 in the present work, we know the exact global dynamics of the problem, a rare treat in nonlinear problems. Our result is in spirit similar to the decay of turbulence results in fluid dynamics for the incompressible Navies-Stokes equations ([11], [17]). However, for such problems, only decay rates of solutions in proper Sobolev norms are known. The plan of the paper is as follows. In section 2, we derive a priori estimates on the solutions of the system (1)-(2) which is similar to the case of α = 1, β = 2 as in [7] and that of [13] for bounded domain. But, the extra freedom of choice for α, β means the estimates are a bit more involved. These estimates directly imply the solution is global, classical solution. In section 3, we derive key decay estimates using the maximum principle and a careful construction of super-solution. In section 4, we use the renormalization group method to find the exact large time dynamics of solution. In particular, we give a rigorous proof that the G map converges as is claimed. Throughout the paper, p stands for the L p ()-norm of a function and the norm defined in (4). Also, for simplicity of notation, we shall not distinguish generic constant C from line to line. 2 A Priori Estimates In this section, we show that there is a uniform bound in time of the L p ()-norm, 1 p, for both u 1 and u 2. Then, it follows from the classical theory that the solutions (u 1, u 2 ) are smooth and exist global in time. First we have the simple Lemma 1 The solutions (u 1, u 2 ) satisfy the following L 1 estimates: ( ) ( ) (u 1 + u 2 )dx (t) = (a 1 + a 2 )dx, u 1 dx (t) a 1 dx for t, ( ) u 2 dx (t) a 2 dx for t, Proof: By integrating (1)-(2) over [, t], we get u 1 (x, t)dx = u 2 (x, t)dx = a 1 (x)dx a 2 (x)dx + t t Together, (1) and (11) give the result of Lemma 1. u α 1 u β 2dxdt < +. (9) u α 1 u β 2dxdτ, (1) u α 1 u β 2dxdτ. (11) Next, using directly the maximum principle, we have the following result. Lemma 2 < u 1 (x, t) a 1, t > ; 5 Q.E.D.

7 < u 2 (x, t) u 2 (x, t) t >, where u 2 is a solution of u 2,t = du 2,xx, u 2 t= = a 2 (x); u 1 (x, t) u 1 (x, t), t >, where u 1 is a solution of u 1,t = u 1,xx u α 1 u β 2, u 1 t= = a 1 (x). The key estimates of this section are stated in the following lemma. Lemma 3 The solutions (u 1, u 2 ) of (1)-(2) are uniformly bounded in time in L p () norm: u 1 (, t) L p + u 2 (, t) L p C(a 1, a 2, p) <, 1 p <, (12) where c(a 1, a 2, p) is a constant depending on initial data and p is a positive integer. Proof: It is clear by Lemmas 1 and 2 that we only need to prove the bound for u 2. By standard local existence theory, all L p ()-norm of (u 1, u 2 ) and the L 2 ()-norm of (u 1,x, u 2,x ) are finite and continuous in time. Therefore, we can freely integrate by parts in what follows. By multiplying (2) by pu p 1 2 with p 2, integrating over +, we get u p 2dx = a p 2dx d t p(p 1)u 2 2,xu p 2 2 dxdτ + t In addition, with the help of integration by parts, we have the identity pu α 1 u β+p 1 2 dxdτ. (13) d (u u 1 )u p dt 2dx (14) = (1 + 2u 1 )(u 1,xx u α 1 u β 2)u p 2dx + (u u 1 )pu p 1 2 (du 2,xx + u α 1 u β 2)dx = 2 u 2 1,xu p 2dx (1 + 2u 1 )u 1,x u 2,x pu p 1 2 (1 + 2u 1 )u α 1 u β 2u p 2dx d (1 + 2u 1 )u 1,x u 2,x pu p 1 2 d (u u 1 )p(p 1)u p 2 2 u 2 2,xdx + (u u 1 )pu p 1 2 u α 1 u β 2dx = I + II + III + IV + V + V I. 6

8 It is clear that I + II + IV (1 + 2 a 1 )(1 + d) u 1,x u 2,x pu p u 2 1,xu p 2 dx u 2 1,xu p 2 dx + C( a 1, p) u 2 2,xu p 2 2 dx 2 u 2 1,xu p 2 dx C( a 1, p) u 2 2,xu p 2 2 dx u 2 1,xu p 2 dx. Moreover, and Integrating (14) from to t yields III u α 1 u β 2u p 2 dx, V V I ( a 1 + a 1 2 ) pu α 1 u β 2u p 1 2 dx. ( (u u 1 )u p 2 dx)(t) (a a 1 )a p 2 dx + C(p, a 1 ) The combination of (13) and (15) then gives u p 2 + t C(a 1, a 2, p)(1 + t t +( a 1 + a 1 2 ) t u α 1 u β 2u p 2 dxdτ For p = 1, proceed as in Lemma 2.3 of [7], we derive t t u 2 2,xu p 2 2 dxdτ (15) u α 1 u β 2pu p 1 2 dxdτ u 2 1,xu p 2 dxdτ. (u 2 2,xu p u 2 1,xu p 2 + u α 1 u β+p 2 ) dxdτ (16) t u α 1 u β+p 1 2 dxdτ). (u α 1 u β u 2 1,xu 2 ) dxdτ C(a 1, a 2 )(1 + t u α 1 u β 2 dxdτ). (17) Apparently, a simple induction on p with the help of (13), (16), (17) and Lemma 1 give the desired uniform bound for u 2. This completes the proof of the lemma. Q.E.D. Lemma 4 The following estimates for the derivative of (u 1, u 2 ) of (1)-(2) hold if t t > : where t > is arbitrary. u 1,x (, t) 2 + u 2,x (, t) 2 C(a 1, a 2 ) <, (18) 7

9 Proof: First, multiplying (1) by u 1 and integrating on imply d u 2 1 dx + u 2 1,x dx. dt Integrating the above inequality over [, t] gives t u 2 1 dx + u 2 1,x dx dτ Similarly, we have u 2 2 dx + t u 2 2,x dx dτ t a 2 1 dx. u α 1 u β+1 2 dx dτ + a 2 2 dx C(a 1, a 2 ) by Lemma 3. By Fubini s theorem, there exists t such that ( (u 2 1,x + u 2 2,x) dx)( ) C(a 1, a 2 )/t. (19) Next, multiplying (1) by u 1,xx and integrating over yield 1 d 2 dt u 1,x 2 2 = u 2 1,xx dx u α 1 u β 2u 1,xx dx (2) = u 2 1,xx dx + α u α 1 1 u β 2u 2 1,x dx + β u α 1 u β 1 2 u 1,x u 2,x dx. Similarly, we have 1 d 2 dt u 2,x 2 2 = d u 2 2,xx dx β Add (2) and (21), and integrate from to t then gives α +β t t u α 1 u β 1 2 u 2 2,x dx α u α 1 1 u β 2u 1,x u 2,x dx. (21) ( u 1,x u 2,x 2 2)(t) ( u 1,x u 2,x 2 2)( ) + (22) t u1 α 1 u β 2u 2 1,x dxdτ + β u α 1 u β 1 2 u 1,x u 2,x dxdτ u α 1 u β 1 2 u 2 2,x dxdτ + α t u α 1 1 u β 2 u 1,x u 2,x dxdτ. We now derive bounds for each of the last four terms on the right-hand side of (22). t t u1 α 1 u β 2u 2 1,x dxdτ a 1 α 1 u β 2u 2 1,x dxdτ (23) a 1 α 1 ( t ) β 1 ( t u 2 2u 2 2,x dxdτ u 2 u 2 2,x dxdτ ) 2 β by Hölder s inequality. Similarly, t u α 1 u β 1 2 u 1,x u 2,x dxdτ (24) a 1 α ( t u 2(β 1) 2 u 2 1,x dxdτ 8 t u 2 2,x dxdτ ) 1/2.

10 If < 2(β 1) < 1, then by Hölder s inequality we find t u 2(β 1) 2 u 2 1,x dxdτ ( t ) 2(β 1) ( t u 2 u 2 1,x dxdτ u 2 1,x dxdτ ) 3 2β. (25) If 1 = 2(β 1), the right-hand side of (24) is already bounded by C(a 1, a 2 ) by Lemma 3. If 1 < 2(β 1) < 2, again by a simple application of Hölder s inequality, we get t u 2(β 1) 2 u 2 1,x dxdτ ( t ) 1/p ( t u 2 2u 2 1,x dxdτ where p = 1/(2(β 1) 1). Continuing the procedure, t a 1 α t a 1 α 1 a 1 α 1 ( t u α 1 u β 1 2 u 2 2,x dxdτ a 1 α ( t u 2 u 2 2,x dxdτ t ) β 1 ( t u 2 u 2 1,x dxdτ ) (p 1)/p, (26) u (β 1) 2 u 2 2,x dxdτ (27) u 2 2,x dxdτ ) 2 β. u α 1 1 u β 2 u 1,x u 2,x dxdτ (28) ( t ( t u β 2u 2 1,x dxdτ u 2 2u 2 2,x dxdτ t ) 1/2σ ( t u 2 2u 2 1,x dxdτ ) 1/2 u β 2u 2 2,x dxdτ ) (β 1)/2 ( t u 2 u 2 1,x dxdτ ) (2 β)/2 u 2 u 2 2,x dxdτ ) (σ 1)/2σ where σ = 1/(β 1). Combining (22)-(28) and Lemma 3, we obtain Hence, by (19), ( u 1,x u 2,x 2 2)(t) ( u 1,x u 2,x 2 2)( ) + C(a 1, a 2 ) ( u 1,x u 2,x 2 2)(t) C(a 1, a 2 ), for all t t. Q.E.D. Proposition 1 The system (1)-(2) has a unique classical solution satisfying, for all t >, u 1 L p + u 2 L p C(a 1, a 2 ), 1 p, (29) where the constant depends only on the initial data (a 1, a 2 ) (L 1 () L ()) 2. 9

11 Proof: The bound for u 1 follows directly from Lemmas 1 and 2. For u 2, the classical theory of local existence yields the bound for small t, say t t, where t >. For t > t, we can use Sobolev embedding, (16) for p = 2 and Lemma 4 to get u 2 C(a 1, a 2 ). This completes the proof of the proposition. Q.E.D. 3 Sharp Decay Estimates The purpose of this section is to prove sharp decay estimates for (u 1, u 2 ) in time. We assume from now on that the initial data (a 1, a 2 ) B B with q > 1 being a fixed constant. To put the solutions in the right scale we let ũ 1 (x, t) = tu 1 ( tx, t), ũ 2 (x, t) = tu 2 ( tx, t). Proposition 2 The solutions (u 1, u 2 ) of (1)-(2) satisfies the bounds ũ 1 (t) C(a 1, a 2 )[log(1 + t)] 1/(α 1), (3) ũ 2 (t) C(a 1, a 2 ), (31) where stands for the norm in (4). emark. We note that as a direct consequence of (3) and (31) we have u 1 C(a 1, a 2 )(1 + t) 1/2 [log(1 + t)] 1/(α 1), (32) u 2 C(a 1, a 2 )(1 + t) 1/2. Lemma 5 The estimate (3) holds. Proof: Let s = log(t + T ), y = x t + T, u 1 (x, t) = v 1(y, s) e s/2 s 1/(α 1), u 2(x, t) = v 2(y, s) e s/2. where T > 1 is fixed. Then, it is easy to verify that v 1, v 2 satisfy, for s s = log T >, Set v 1,s = v 1,yy v yv 1,y + v 1 (α 1)s vα 1 v β 2 s, v 2,s = dv 2,yy v yv 2,y + vα 1 v β 2 s α α 1, L (v) = v s v yy 1 2 v 1 2 yv y 1 v (α 1)s.

12 It is easy to see that if there exists a function v which has the property that L(v) on (s, ) and v(y, s ) v 1 (y, s ) on, then v(y, s) v 1 (y, s) on (s, ). In another word, such v is a super-solution of v 1. To construct such a super-solution, we let v = Ae y2 /4 ( 1 + f(y) s ) + B (1 + y2 ) k s 2, where k > 1/2 is arbitray and A, B > are constants to be fixed later and Note that f has a bound f(y) = f(y) α y e y2 1 /4 dy 1 e y2/4 2 dy 2. α 1 y 1 α (πe + log yh(y 1)), α 1 where H is the Heaviside function. Detailed calculations show that if k > 1/2, ( /4 L(v) = Ae y2 1 αf(y) ) + B(1 + y 2 ) k s (α 1)s ( (2k 1)y 4 2(4k 2 + k + 1)y 2 + 4k 1 2α 1 ). 2(1 + y 2 ) 2 s 2 (α 1)s 3 Since (2k 1)y 4 2(4k 2 + k + 1)y 2 + 4k 1 > m(1 + y 2 ) 2 for any m < 2k 1 with y suitably large, we have, if k > 1/2, that ( /4 L(v) Ae y2 1 αf(y) ) + B s (α 1)s ( 2k 1 2α 1 ) (1 + y 2 ) k. 4s 2 (α 1)s 3 Clearly, there exist y 1 (k) > and s 1 (k) > such that the right hand side is positive if y y 1 (k) and s s 1 (k) > if A = B >. For y < y 1 (k), we can choose s 2 (k) 1 such that L(v) > for all s s 2 (k). Hence, v is a super-solution for s max(s 1 (k), s 2 (k)) >. It follows from our assumption on initial data and the above construction that the estimate (3) holds by noting that A, B can be arbitarily large independent of k. Q.E.D. To prove Proposition 2, we only need to show Lemma 6 There exists C = C(a 1, a 2 ) such that u 2 C(a 1, a 2 )(1 + t) 1/2. 11

13 Proof: Consider the equation for u 2. Write the equation in integral form: t u 2 (x, t) = 1 4πdt Take L -norm then yields Now, and e y2 /(4dt) a 2 (x y)dy + 1 4πds e y2 /(4ds) (u α 1 u β 2)(x y, t s)dyds. u 2 c a 2 t (2 + t) + C s 1/2 u α 1/2 1 u β 2 1 (t s)ds. (33) u α 1 u β 2 1 u 1 α u β 2 1 C(2 + t) α/2 u β 1 2 u 2 1 C(2 + t) α/2 (34) Thus, t s 1/2 (2 + t s) α/2 ds C(2 + t) min(1,(α 1))/2 = C(2 + t) (α 1)/2. We can use (35) to improve (34) into which, when inserted into (33) gives u 2 C(2 + t) (α 1)/2. (35) u α 1 u β 2 1 C(2 + t) (2α 1)/2, u 2 C(2 + t) (α 1), (36) and a finite number of iterations gives the desired estimate. This completes the proof of the lemma. Q.E.D. Proof of Proposition 2: By the bound of u 1 and Lemma 6, u 2 u 2, where u 2 is the solution of the linear heat equation { u2,t = du 2,xx + C(2 + t) 1 log α/(α 1) (2 + t)u 2, u 2 t= = a 2. The bound in (31) then follows from the classical theory. This ends the proof of Proposition 2. Q.E.D. 12

14 4 The G Method and Asymptotics In this section, we prove Theorem 1. The method we use is the renormalization group method. We refer the reader to the pioneering work of Bricmont, Kupiainen and Lin [6] and the reference therein for a thorough and stimulating discussion of its application to a wide range of PDE problems. Our approach here is close in spirit to the one used in [7]. Define for n N and for L > 1 large u n i = L n u i (L n x, L 2n t), t [1, L 2 ], i = 1, 2. (37) (u n 1, u n 2) satisfy the system (1)-(2) with initial data (when t = 1): The decay estimates we get in Section 3 imply a n i = L n u i (L n x, L 2n ), i = 1, 2. (38) a n 1 C(L)n 1/(α 1) and a n 1 α a n 2 β C(L)n α/(α 1). (39) We shall consider the G map (a 1, a 2 ) (a 1 1, a 1 2) defined in B B with a 1 i = Lu i (Lx, L 2 ), i = 1, 2 inductively, (4) where (u 1, u 2 ) solve (1)-(2) with initial data (a 1, a 2 ). That is, a 1 1 = L[e (L2 1) a 1 (Lx)] Ln 1 (Lx, L 2 ), (41) a 2 1 = L[e d(l2 1) a 2 (Lx)] + Ln 2 (Lx, L 2 ), (42) where represents the differential operator d 2 /dx 2 and n 1 (x, t) = n 2 (x, t) = t 1 t 1 G(x y, t s)u α 1 (y, s)u β 2(y, s)dyds, (43) G d (x y, t s)u α 1 (y, s)u β 2(y, s)dyds (44) with G and G d being corresponding Gaussians of u t = u xx and u t = du xx respectively. If we denote B + = {f B f }. The map S L from B + B + defined as (S L f)(x) = Lf(Lx) is a bounded operator. In fact, we have the following result. Lemma 7 Suppose L 1. Then, (a) S L L, (b) e µ(t s) e c(t s), for µ = 1, d, c <. 13

15 For each step, we write a 1 (x) = A 1 φ(x) + b 1, a 2 (x) = A 2 φ d (x) + b 2. where A i = a i(x)dx, i = 1, 2 and b i(x)dx =, i = 1, 2, φ(x) = φ 1 (x). In order to estimate the change of coefficients A i and b i, we make some preliminary analysis. First, we look at the equation for u 2, Consider the ball u 2 (x, t) = e d(t 1) a 2 + n 2 (x, t) = u 2 (x, t) + n 2 (x, t). (45) B = {u 2 : u 2 Suppose u 2 B. First we have and by Lemma 7. Next, by Lemma 2, and Lemma 7 (b), we find n 1 where ɛ = a 1 α a 2 β 1. Similarly, In consequence, sup u 2 (, t) a 2 }. t [1,L 2 ] u 1 (, s) e (t 1) a 1 C(L) a 1 u 2 (, s) e (t 1) a 2 C(L) a 2 u 1 (, s) a 1 C a 1 t sup ds G(x y, t s)(c a 1 ) α ( a 2 ) β dy t [1,L 2 ] 1 C(L) β a 1 α a 2 β C(L, )ɛ a 2, n 2 C(L, ))ɛ a 2. Ln i (L, L 2 ) = S L n i (, L 2 ) C(L, )ɛ a i, i = 1, 2. Therefore, the right-hand side of (45) defines a map from B to B if > is sufficiently large and ɛ sufficiently small. A more close look reveals it is a contraction map. Hence, there is a unique fixed point in B. 14

16 For the G map, we have the relations A 1 1 = A 1 Ln 1 (Lx, L 2 )dx, A 1 2 = A 2 + b 1 1 = e (L2 1) b 1 Ln 1 (Lx, L 2 ) + (A 1 A 1 1)φ, b 1 2 = e d(l2 1) b 2 + Ln 2 (Lx, L 2 ) + (A 2 A 1 2)φ d, where the major terms in n 1, n 2 are given by w(y, s) = u α 1 (y, s)u β 2(y, s) (A 1 φ(y)) α (A 2 φ d (y)) β = (u 1 n 1 ) α (u 2 + n 2 ) β (A 1 φ) α (A 2 φ d ) β Ln 2 (Lx, L 2 )dx, (46) = (u 1 n 1 ) α (u 2 + n 2 ) β (u 1 n 1 ) α (A 2 φ d ) β + (u 1 n 1 ) α (A 2 φ d ) β (A 1 φ) α (A 2 φ d ) β = (u 1 n 1 ) α [(A 2 φ d + e d(s 1) b 2 + n 2 ) β (A 2 φ d ) β ] + (A 2 φ d ) β [(A 1 φ + e (s 1) b 1 n 1 ) α (A 1 φ) α ] = (u 1 n 1 ) α β(a 2 φ d + θ 1 (e d(s 1) b 2 + n 2 )) β 1 (e d(s 1) b 2 + n 2 ) + (A 2 φ d ) β α(a 1 φ + θ 2 (e (s 1) b 1 n 1 )) α 1 (e (s 1) b 1 n 1 ), where θ i 1, i = 1, 2, such that w(., s) β u 1 α (A 2 φ d + θ 1 (e d(s 1) b 2 + n 2 )) β 1 (e d(s 1) b 2 + n 2 ) (47) +α (A 2 φ d ) β (A 1 φ + θ 2 (e (s 1) b 1 n 1 )) α 1 (e (s 1) b 1 n 1 ) C a 1 α (A 2 + b 2 + C(L, )ɛ a 2 ) β 1 ( b 2 + C(L, )ɛ a 2 ) + CA β 2(A 1 + b 1 + C(L, )ɛ a 2 ) α 1 ( b 1 + C(L, )ɛ a 2 ). Now for any initial data (a 1, a 2 ) B B, estimates in the previous section imply that if we consider the solution (u 1, u 2 ) after large enough time T, then a 1 = u 1 (, T ) will be very small while a 2 = u 2 (, T )) is preserved in B B and the iterated G map will be contractive for some = (a 1, a 2 ). Therefore for the rest of this section we will fix and take ɛ = a 1 α a 2 β 1 be as small as necessary. Lemma 8 Suppose L 1. There exists ɛ (L) > such that if a 1 α a 2 β 1 < ɛ ɛ (L), the following results hold: (a) A 1 i A i C(L)ɛ a 2, i =1, 2, (b) b 1 2 L 2δ b 2 + C(L)ɛ a 2, (c) A 1 1 A 1 + γa α 1 A β 2 η, where ( (α + β/d) 1/2 γ = and 4πd β/2 ) 1/(α 1) η = η( a 1, a 2, b 1, b 2, A 1, A 2, ɛ) = C(L) ( a 1 α (A 2 + b 2 + ɛ a 2 ) β 1 ( b 2 + ɛ a 2 ) ) +C(L) ( A β 2(A 1 + b 1 + ɛ a 2 ) α 1 ( b 1 + ɛ a 2 ) ), 15

17 (d) b 1 1 L 2δ b 1 + C(L)ɛ a 2, where δ > is independent of L. Proof: (a) follows directly from our previous analysis since A 1 i A i = Ln i(lx, L 2 )dx. For part (b), by our previous analysis on Ln 2 (Lx, L 2 ), (a) on A 1 2 A 2, and the classical theory for heat equation u t = u xx of bounded initial data b 2 (x) with b 2(x) dx =, we get (b). (c) follows from our estimates in (47) and some elementary calculation. (d) follows from the same argument as for (b). Q.E.D. Applying the Lemma inductively, we have A n+1 2 A n 2 C(L)ɛ n a n 2, A n+1 1 A n 1 C(L) a n 1 α ( (A n 2) β 1 + b n 2 β 1 + ɛ β 1 n a n 2 β 1) ( b n 2 + ɛ n a n 2 ) + C(L)(A n 2) β ( (A n 1) α 1 + b n 1 α 1 + (ɛ n a 2 ) α 1) ( b n 1 + ɛ n a n 2 ), b n+1 2 L 2δ b n 2 + C(L)ɛ n a n 2, b n+1 1 L 2δ b n 1 + C(L)ɛ n a n 2, n = 1, 2,, where ɛ n = a n 1 α a n 2 β 1. We show A n i, b n i, i = 1, 2 converge as n. Let Then, A n = A n 1 + A n 2, a n = a n 1 + a n 2, b n = b n 1 + b n 2, n = 1, 2,. By using the fact that A n+1 A n + C(L)ɛ n a n, b n+1 L 2δ b n + C(L)ɛ n a n, n = 1, 2,. ɛ n = a n 1 α a n 2 β 1 C/(logL 2n ) α/(α 1) C(L)n α/(α 1) and a n is uniformly bounded, we deduce that A n converges as n. Since, {A n 2} is an increasing sequence, this means both A n 1 and A n 2 converges as n. As a matter of fact, we know from the evolution of L 1 -norm, as deduced from (39) and Lemma 1, of (u 1, u 2 ) that A n 1 and A n 2 A as n. As to b n, we have b n+1 L 2δ b n + C(L)n α/(α 1) L 2δ (L 2δ b n 1 + C(L)(n 1) α/(α 1) ) + C(L)n α/(α 1) n 1 L 2nδ b 1 + C(L) L 2lδ (n l) α/(α 1) l= C(L)n α/(α 1) as n. 16

18 To calculate the exact limit of A n 1, we first note a n 1 C(L)n 1/(α 1) and therefore, A n 1 C(L)n 1/(α 1). Then from (c) of Lemma 8 and the above bound for b n we get A n+1 1 A n 1 + γ(a n 1) α (A n 2) β C(L){n α/(α 1) n α/(α 1) + ((A n 1) α 1 + n α )n α/(α 1) } C(L)(n 2α/(α 1) + (A n 1) α 1 n α/(α 1) + n α2 /(α 1) ) C(L)n (2α 1)/(α 1). Here we use the fact tha < α < 2. Hence, the difference equation which A n 1 satisfies takes the form A n+1 1 = A n 1 γ(a n 1) α (A n 2) β + O(n (2α 1)/(α 1) ). Taking into account that A n 2 A as n, we can write it as A n+1 1 = A n 1 γ(a n 1) α (A) β + o((a n 1) α ) + O(n (2α 1)/(α 1) ). Using the fact that the difference equation of the form with Γ > y n+1 = y n Γ(y n ) α + O(n (2α 1)/(α 1) ) has the maximum principle if the positive sequence {y n } is sufficiently small, and y n = cn 1/(α 1) is a sub-solution if Γc α 1 < 1/(α 1) but a super-solution if Γc α 1 > 1/(α 1), we get y n n 1/(α 1) has a positive finite limit c satisfying Γc α 1 = 1/(α 1) as n. Therefore, it is also true for A n 1. A n 1n 1/(α 1) B as n, where ( ) 1/(α 1) 1 B =. (α 1)γA β Proof of Theorem 1: It is clear that Theorem 1 follows directly from the limits A n 1n 1/(α 1) B and A n 2 A as n. Q.E.D. eferences [1] N. Alikakos, L p bounds of solutions of the reaction-diffusion systems, Comm. Partial Differential Equations 4, (1979) pp [2] J. D. Avrin, Qualitative theory for a model of laminar flames with arbitrary nonnegative initial data, J. Differential Equations 84 (199), pp

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