On the local chaos in reaction-diffusion equations

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1 Chaotic Moeling an Simulation (CMSIM) 1: , 017 On the local chaos in reaction-iffusion equations Anatolij K. Prykarpatski 1, Kamal N. Soltanov, an Denis Blackmore 3 1 Department of Applie Mathematics, AGH University of Science an Technology, Krakow, Polan ( pryk.anat@ua.fm, pryk.anat@cybergal.com) Department of Mathematics, Faculty of Sciences, Hacettepe University, Beytepe, Ankara, TR-06800, Turkey ( soltanov@hacettepe.eu.tr, sultan kamal@hotmail.com) 3 Department of Mathematical Sciences an Center for Applie Mathematics an Statistics, New Jersey Institute of Technology, Newark, NJ 0710, USA ( eblac@m.njit.eu) Abstract. There is shown that a non-issipative reaction-iffusion partial ifferential equation with local nonlinearity can have an infinite number of ifferent nonstable solutions traveling along the space axis with varying spees, traveling impulses as well as an infinite number of ifferent states of spatio-temporal (iffusion) chaos. These solutions are generate by cascaes of bifurcations governe by the corresponing steay states. The behavior of these solutions is analyze in etail an, as an example, it is explaine how space-time chaos can arise. Questions of same type are also stuie in the case of a nonlocal nonlinearity. Keywors: Semilinear PDE, reaction-iffusion equation, behavior of solutions, chaos. Mathematics Subject Classification: Primary 35J10, 35K05, 35R30; Seconary 37G30, 37B5, 34C3 1 Introuction We are intereste in styying the chaotic behavior of solutions to reactioniffusion type equations with nonlocal an local nonlinearities; namely, we will eal with the following problems u u + g (t, x, u) = 0, (t, x) (0, T ) Ω, (1) t u (0, x) = u 0 (x) W 1, (Ω), x Ω, T > 0 () u [0,T ) Ω = 0, Ω R n, n 1, Ω Lip (3) Receive: 6 September 016 / Accepte: 1 December 016 c 017 CMSIM ISSN

2 10 Prykarpatski et al. where Ω R n is an open omain, the bounary Ω satisfies the Lipschitz conition, g : L p1 ((0, T ) Ω) L p ((0, T ) Ω) is a nonlinear operator an p 1, p > 1 are fixe numbers. We assume that g (t, x, u) is represente in one of the following forms: (α) : g (t, x, u) := a u ρ u+h (t, x), or (β) : g (t, x, u) := a (t, x) u ρ u+h (t, x), (4) where enotes the norm in L (Ω), h (t, x) an u 0 (x) are given functions an ρ > 0, a > 0 are given constants. The problems pose above are investigate for the both cases separately: in the case of a nonlocal nonlinearity (i.e. (4(α))), an in the case of a local nonlinearity (i.e. (4(β))). We shall emonstrate that the partial ifferential equation (1) in the case of the nonlocal nonlinearity (i.e. the case (4(α))) can possess an infinite number of ifferent non-stable solutions. In the local case (4(β)), which iffers markely from the nonlocal case (4(α)), the problem allows an infinite number of ifferent both non-stable solutions, traveling along the space axis with arbitrary spees, an traveling impulses, as well as an infinite number of ifferent spatiotemporal (iffusion) chaotic states. These solutions are generate by cascaes of static bifurcations of the evolution equation, which were stuie, in particular, in [0]. As Ya. Sinai asserts in [19]... the future of the chaos theory will be connecte with new phenomena in nonlinear PDEs an other infiniteimensional ynamical systems, where we can encounter absolutely unexpecte phenomena. The ynamics becomes much more complicate in the case of ynamical systems generate by partial ifferential equations (PDEs) largely ue to the formation of spatially chaotic patterns. More generally, such systems may isplay interactions between spatially an temporally chaotic moes. One of the most challenging problems in this fiel is that of turbulence which isplays statistical behavior in temporal an spatial irections, whose correlations ecay with istance in space an time, see e.g. [9,14,5]. It shoul be pointe out that there have been many investigations on this an relate topics (see, for example, [1,,6,7,5,11,15 17,19,5,6] an the references therein). In what follows we stuy the Cauchy problem for an equation of the nonissipative reaction-iffusion equation type with an infinite-imensional solution space; in particular, the corresponing steay-state problem has also infinite number of the ifferent solutions. We show that the trajectories of solutions in the phase space epen on choosing the starting point from a sphere of the initial values. To be more precise, this choice etermines how the solution behave beginning at this initial value an epening on the relate Lyapunov exponent. The choice of starting point allows to etermine the point at which the solution of the problem will en up. If the limiting set is not one-imensional, more complications can arise, incluing even the existence of absorbing manifols. Moreover, if such absorbing manifols exist, their associate ynamics tens to be chaotic. We stuy this type of ynamic behavior an explain how space-time chaos can arise.

3 Chaotic Moeling an Simulation (CMSIM) 1: , Existence in the autonomous case.1 The nonhomogeneous case We begin by stuying the problem in the case (4(α)) when g (t, x, u) := a u ρ u+ h (x), i.e. we consier the problem u t u a u ρ H u = h (x), (t, x) (0, T ) Ω, (5) u (0, x) = u 0 (x) W 1, 0 (Ω) := H 1 0 (Ω), u [0,T ) Ω = 0. (6) From (5) we compute that 1 t u (t) + u (t) which entails the inequalities a u (t) ρ+ = h, u, u (0) = u 0 (7) t u (t) u (t) + a u (t) ρ+ + h H 1 λ 1 u ) + a u (t) ρ+ + h H, 1 where u (t) H 1 0 := u (t) an λ 1 > 0 is the first eigenvalue of the Laplace operator : H 1 0 (Ω) H 1 (Ω). Now we consier the solvability of this problem, which will be analyze making use of the general results from [1]. We take u 0 B H1 0 r 0 (0), where r 0 < λ 1, an stuy the operator A generate by the problem: it acts, by efinition, from X := W ( 1, 0, T ; H 1 (Ω) ) L ( 0, T ; H0 1 (Ω) ) {u (t, x) u (0, x) = u 0 } to L ( 0, T ; H 1 (Ω) ). Next, we stuy the image of this operator A on the ball Br X (0) for r (0, r 0 ); more precisely, we efine a subset M of the space L ( 0, T ; H 1 (Ω) ) an a number r (0, r 0 ), such that A ( Br X (0) ) M L ( 0, T ; H 1 (Ω) ). In other wors, we shall show that the problem is solvable in B X r 0 (0) for any (h, u 0 ) M B H1 0 r 0 (0). A etaile investigation requires some preliminary estimates. So, let u 0 B H1 0 r 0 (0) for some number r 0 < λ 1 ; then we obtain t u (t) λ 1 u (t) + a u (t) ρ+ + h H. 1 Consequently, it is enough to stuy the following initial problem (y + c 1 ) /t + λ 1 (y + c 1 ) a (y + c 1 ) ρ1+1, y (0) = u 0 H, (8) 0 1 where y (t) := u (t), ρ 1 = ρ. Assume that the constant c 1 is chosen in such a way that the inequality hols. a (y + c 1 ) ρ1+1 λ 1 c 1 h H 1 + ayρ1+1

4 1 Prykarpatski et al. Whence, one fins that which gives z λ 1 ρ 1 z aρ, z = (y + c 1 ) ρ1, z (0) = (y + c 1 ) ρ1 ( u 0 + c 1) ρ1, ( ) u 0 + c ρ1 1 e λ 1ρ 1t + a a e λ1ρ1t = λ 1 λ 1 ( ) [ u (t) + c 1 e λ1t u 0 + c 1 1 a ( ) u 0 λ + c ρ1 ( 1 1 e λ )] ρ 1 1 1ρ 1t. 1 (9) Therefore, we see from (9) that the functions u 0 an h shoul be selecte from balls of respective spaces so as to satisfy the inequality 1 a ( ) u 0 λ + c ρ1 1 > 0 = u0 + c 1 < 1 λ 1 ρ a. (10) Moreover, it follows reaily from (9) that c 1 < 0. So, we obtaine that for the solvability of the problem pose the initial ata an the exterior source shoul be [0,1] small enough. Consequently, one can formulate the following proposition. Proposition 1. Let the initial ata u 0 an h satisfy the inequality (10) with parameters λ 1, a, ρ an c 1 efine above hol. Then the problem (5) - (6) is solvable in the ball Br X 0 (0) for any h M H 1 where M := { h H 1 h, u A (u), u, u Sr X 0 (0) } { } an ( r 0 < min λ1 ) ρ a ; λ 1 ρ 1 some number. Proof. We will make use of the formal solution to the problem (5) - (6): u (t) = exp {t } u 0 + a0t exp {(t τ) } u (τ) ρ H u (τ) τ+ 0+t exp {(t τ) } τ h. For the evolution operator exp {t } one has the estimate: exp {t } exp { λ 1 t} (see, for example, [13] ). Thus, we obtain: u (t) H exp { λ 1 t} u 0 H + a0t exp { λ 1 (t τ)} u (τ) ρ+1 H τ+ 0+t exp { λ 1 (t τ)} τ h H = e λ1t u 0 H + 1 e λ1t λ 1 h H + +a0t exp { λ 1 (t τ)} u (τ) ρ+1 τ,

5 Chaotic Moeling an Simulation (CMSIM) 1: , where λ 1 > 0 is, as before, the first Laplace operator eigenvalue. Whence, one observes that there exists a number r 1 = r 1 (λ 1, h, a, ρ) < λ 1 such that u 0 ρ+1 < r 1. Then u (t) ecreases as t (at least, for t (0, t 1 ) where t 1 > 0 is some positive value), an consequently for any t > 0. Now, taking account the above property of the solution in the equation t u (t) + u (t) a u (t) ρ+ + h H, 1 it follows from (7) an (8) we obtain a priori estimates, which suffice to complete the proof of this proposition. Moreover, we have the following estimates u, u a u (t) ρ u, u = u (t) a u (t) ρ+ u (t) ( a u (t) ρ λ u (t) = u (t) 1 a ) u (t) ρ δ u (t) 1 λ 1 as in this case u 0 ρ < λ1 a, an consequently u (t) ρ < λ1 a for t > 0. Hence, it is enough to use the existence theorem from [1] on the ball Br X 0 (0), where r 0 ( ) λ 1 ρ a an u Sr X 0 (0). Inee, this implies following inequalities on W ( ) 1, 0, T ; H0 1 u t u a u ρ u, u t + u = u t + 1 t u a ρ + t u ρ+ + 1 t u + u a u ρ+ = u t + u a u ρ+ + [ 1 t u + 1 u a ] ρ + u ρ+ u t + u (1 a ) u ρ + [ 1 λ1 t u + 1 u a ] ρ + u ρ+, (11) which when integrate with respect to t yiel [ ] u 0T t + δ 1 u t + 1 u (T ) + 1 u (T ) a ρ + u ρ+ (T ) 1 u 0 1 u 0 + a [ ρ + u 0 ρ+ u 0T t + δ 1 u ] t+ 1 u (T ) + [ ] u 1 (T ) a λ 1 (ρ + ) u ρ (T ) 1 u 0 1 u 0 + a ρ + u 0 ρ+ [ ] u 0T t + δ 1 u t + 1 u (T ) + δ u (T ) C ( u 0 H 1, a, ρ).

6 14 Prykarpatski et al. Now we consier the following inequality (u v = w) u t u a u ρ u v t + v + a v ρ v, u v = w t w a ( u ρ u v ρ v), w = 1 w (t) + w (t) a u ρ u v ρ v, w 1 w (t) + w (t) a (ρ + 1) û ρ w (t) (1) where û = l (u, v) is a bilinear mapping. Thus, we conclue that all conitions of the general theorem from [1] are fulfille on the ball B X r 0 (0), thereby completing th proof of the proposition by virtue of the inequalities (11) an (1).. The homogeneous case Next, we consier the homogeneous case of (5) - (6) an stuy the behavior of its solutions. First, we nee note that in this case (i.e. h (t, x) = 0, a = 1) that (8) an (10) yiel the following inequalities: u (t) ρ a λ 1 + [ u 0 ρ a λ 1 ] e λ1 ρ t, or, at a = 1, [ ( u (t) + u 0 ρ 1 ) ] ρ e λ1 ρ t. λ 1 λ 1 ( ) Therefore, λ 1 + u 0 ρ 1 λ 1 e λ1 ρ t > 0 for any t > 0, for which the the inequality u 0 ρ λ 1 is sufficient. We now set h (t, x) = 0, a = 1 an investigate the problem (5) - (6) for the initial ata satisfying the conition u 0 B H1 0 r (Ω) 0 (0) H0 1 (Ω) if r ρ 0 < λ 1. In this case we have t u (t) + u (t) rρ (t) u (t) = 0, which gives rise to the ifferential inequality equivalent to t u (t) (λ 1 r ρ (t)) u (t), u (t) exp { ( t0 )} (λ 1 r ρ (τ)) τ u (0). Uner the conitions impose above there exists, owing to the continuity of the function u (t), an interval (0, t ), (t > 0) such that λ 1 r ρ (t) > 0 for t (0, t ). Thus, one easily computes that for any t > 0. u (t) < exp { (λ 1 r ρ 0 ) t} u 0 < u 0 < r 0 (13) As in the previous case, it is easy to prove the following result.

7 Chaotic Moeling an Simulation (CMSIM) 1: , Proposition. Let h (t, x) = 0 an a = 1. Then for any u 0 B H1 0 r (Ω) 0 (0) H0 1 (Ω) the problem (5) - (6) is solvable for any t > 0 if r ρ 0 < λ 1. Moreover ( the mapping ) (semi-flow) F (t) : u 0 u (t) is such that L strongly F (t) B H1 0 r (Ω) 0 (0) 0 as t. 3 Longtime behavior of solutions Let h L ( (0, ) ; H 1 (Ω) ) an u 0 H0 1 (Ω) with the norm H0 1 (Ω) :=. We assume that the Laplace operator : H1 0 (Ω) H 1 (Ω) has only a point spectrum, i.e. σ ( ) := σ P ( ) (0, ). Denote the eigenvalues of the Laplace operator by λ j, j = 1,,... (σ P ( ) := {λ j j N }). This, of course, requires that the omain Ω is sufficiently regular in a geometric sense. Now we consier the problem (5) - (7) in the case g (t, x, u) := u ρ u + h (t, x), an investigate the inverse mapping of the operator f (t) : B H1 0 r (Ω) 0 (0) H0 1 (Ω) H 1 (Ω), where f (t) u := u/ t u u ρ u an r 0 > 0 is some positive number. For simplicity, we assume that the eigenfunctions an ajoint eigenfunctions are total in the space H0 1 (Ω) ; moreover, we may assume without loss of generality that they generate an orthogonal basis in this space. Let inf {λ j σ P ( ) : λ j > r ρ 0, j = 1,,...} = λ k 0. Then, we can represent [1,13,18,0,4] the space H0 1 (Ω) in the form H0 1 (Ω) := H k0 H k0, where the subspace H k0 H0 1 (Ω) is relate to {λ j } k0 1 j=1 an has imension im H k0 = k 0 1 an H k0 is a subspace of coimh k0 = k 0 1. We can now introuce the projections Q k0 an P k0 ; Q k0 : H0 1 (Ω) H k0 H0 1 (Ω) an P k0 : H0 1 (Ω) H k0 H0 1 (Ω), giving rise to the splitting u := Q k0 u + P k0 u. (It is well known that such a ecomposition allows to introuce either a spectral measure or a family of spectral projections (see, for example, [10,?], etc.) Thus, it is easy to see that : H k0 H k 0 an : H k0 H k 0, where the subspaces H k 0, H k 0 possess bi-orthogonal bases (see, for instance, [10,?,?] ), an owing to the evient commutativity of operators P k0 an Q k0 with the Laplacian in H0 1 (Ω), one can rewrite the problem as t P k 0 u P k0 u u ρ P k 0 u = Pk 0 h (t, x), (14) t Q k 0 u Q k0 u u ρ Q k 0 u = Q k 0 h (t, x), (15) P k0 u (0, x) = P k0 u 0 (x) H k0 H0 1 (Ω), (16) Q k0 u (0, x) = Q k0 u 0 (x) H k0 H 1 0 (Ω), (17) with impose above the conition (3), where Pk 0 an Q k 0 are the ajoint operators to P k0 an Q k0, respectively. For the investigation of the problem above we shall consier the following ifferential-functional expression E (u) := 1 t u (t) + u (t) u ρ (t) u (t),

8 16 Prykarpatski et al. for u H 1 ((0, T ) Ω). As our aim is the investigation of the behavior of solutions of the problem uner the conition u 0 B H1 0 r (Ω) 0 (0), 0 < r ρ 0 < λ k 0 it is enough to stuy the homogeneous equation case. Therefore, we assume h (t, x) = 0. Then for the problem (14) - (15) we obtain E (P k0 u) = 1 ) t P k 0 u (t) + P k 0 u ( u (t) ρ P k 0 u (t) = 0, (18) P k0 u, P k0 u t=0 = P k0 u (0) = P k 0 u 0. (19) Whence, it follows that for some t 0 > 0 for t [0, t 0 ) we have u ρ (t) r ρ 0 + θ < λ k 0, for some θ > 0. Inee if u 0 = r 0, then we have from (18) - (19) that t P k 0 u (t) + (λ k 0 1 r ρ (t)) P k0 u (t) t P k 0 u (t) + (λ k 0 1 r ρ (0)) P k0 u (t) 0 an consequently we obtain the inequality P k0 u (t) exp { (λ k 0 1 r ρ 0 )t} P k 0 u 0. (0) Thus, we see that if Q k0 u (t) δ < θ < r 0 for some sufficiently small δ > 0 an t [0, t 0 ), then the solution of problem (18) - (19) exists an is an exponentially increasing function. Now consier the problem (16) - (17) for which one easily obtains E (Q k0 u) = 1 ) t Q k 0 u (t) + Q k 0 u ( u (t) ρ Q k 0 u (t) = 0, (1) Q k0 u, Q k0 u t=0 = Q k0 u (0) = Q k 0 u 0. Therefore, the solution of problem (1) exists an is an exponentially ecreasing function. Consequently for Q k0 u 0 + P k0 u 0 = r 0 if Q k0 u 0 < P k0 u 0 an Q k0 u 0 is sufficiently small, then the solution u (t) exists an is an increasing function up to some t 1 > 0. To stuy in etail the behavior of solutions to the problem we will make use of the following assumption: the system of eigenfunctions {w k } k=1 H1 0 (Ω) comprises an orthonormal basis of this space. Then each function u (t, x) L ( (0, T ) ; H 1 0 (Ω) ) has the representation u (t, x) = k = 1 u k (t) w k (x). Consequently, owing to (14) - (15), the problem is equivalent to stuying the system of equations 1 t u k (t) + λ k u k (t) with the Cauchy ata ( i = 1 u i (t) ) ρ u k (t) = 0 () u k (0) = u 0k, k = 1,,..., k 0.

9 Chaotic Moeling an Simulation (CMSIM) 1: , Let u (0, x) B H1 0 (Ω) r 0 (0) an u 0 < r 0, then we have u (t) := (i = 1 u i (t) ) r 0 + ε for sufficiently small t = t (ε, r 0 ) > 0. But u k (t) will increase in this case for k = 1,,..., k 0 k 0 epening on the relationship between u 0 ρ an λ k (therefore, between r 0 an λ k ). Consier the behavior of u k (t) for all k = 1,,... Define u 0 := r 0 for some r 0 > 0. Let us list all of possible cases: 1) r ρ 0 < λ 1, ) λ k0 : λ k0 1 < r ρ 0 < λ k 0 an 3) λ k0 : r ρ 0 = λ k 0. The case 1) was alreay investigate, therefore we will consier here only cases ) an 3). Consier either the case ) or 3), i.e. λ k0 : λ k0 1 < r ρ 0 < λ k 0 an λ k0 : r ρ 0 = λ k 0. We have the following system of equations 0 = 1 t u k (t) + λ k u k (t) r (t) ρ u k (t) = = 1 t u k (t) + (λ k r (t) ρ ) u k (t), u k (0) = u 0k, k = 1,,..., (3) where u (t, x) := k = 1 u k (t) w k (x). It is easy to see that this system of equations is of interest only for the cases k k 0, k k 0 1 an k = k 0 separately. In case ) if k k 0 this part of the system has a solution that is unique as t t (k, r 0 ) for some t (k, r 0 ) > 0. Formally, we can etermine the solution of each equation from (1) to be { ( )} u k (t) = exp λ k t t0 r (τ) ρ τ u 0k. (4) Thus, if we consier the expression (1) for k k 0 1 in the case ), it follows from () that { ( )} u k (t) = exp λ k t t0 r (τ) ρ τ u 0 exp { (r (0) ρ λ k ) t} u 0, as r (t) ρ > λ k for 1 k k 0 1 an some t > 0. Consequently, the sequence u k (t) increases for each k : 1 k k 0 1 leaing to the the increase of r (t) as long as P k0 u 0 is sufficiently greater than Q k0 u 0. For case 3) for some k = k 0 one has r ρ 0 = λ k 0 for which { } u k0 (t) = exp t0 (λ k0 r (τ) ρ ) τ u 0k0 by virtue of (). Here the function r 1 (t) := λ k0 r (t) ρ equals zero at t = 0, but in general its variation is not known. Thus, it is impossible to obtain a monotonicity result for a solution to this equation, since the behavior of r (t) is not known. As we shall explain further in the sequel, the behavior of r (t) epens on the geometrical properties of the initial ata u 0 of sphere S H1 0 r (0), 0 < r r 0. From the previously mentione relationships it is clear that in orer to investigate the behavior of the parameter r (t) one shoul stuy both P k0 u

10 18 Prykarpatski et al. an Q k0 u. It is easy to see that if the conition ) is assume, then P k0 u increases, an Q k0 u ecreases as t > 0 at least nearby zero in virtue of (18) an (). Using the orthogonal splitting u = P k0 u + Q k0 u, we also fin that u = P k 0 u + Q k 0 u. (5) Thus, the behavior of the functional u (t) epens on the relationship between the values P k0 u 0 an Q k0 u 0. Let u 0 := r 0, λ k0 1 < r ρ 0 < λ k 0 an consier (5), i.e. u (t) = P k 0 u (t) + Q k 0 u (t) = k k 0 uk (t) + k > k 0 uk (t). (6) It is now necessary to investigate the following three cases: a) u 0 := k k 0 1 ( u 0k w k P k0 H 1 0 (Ω) ) ( := H k0 ; b) u 0 := k k 0 u0k w k Q k0 H 1 0 (Ω) ) := H k0 an c) u 0 := k 1 u 0k w k, when c 1 ) Q k0 u 0 < P k0 u 0 an c ) Q k0 u 0 P k0 u 0, separately. Consier a): in this case we have u k (t) = exp { ( λ k t t0 r ρ (τ) τ )} u 0k for any k = 1,..., k 0 1, thus, u k (t) = 0 for k k 0 since r0 = r (0) := k k 0 1 (u 0k ) an r (t) := k k 0 1 u k (t). On the other han, in this case ( λ k t t0 r ρ (τ) τ ) < 0 as r ρ 0 > λ k for each k = 1,..., k 0 1 an r ρ (t) increases as t. Now let us consier case b), i.e. r0 = r (0) := k k 0 (u0k ). Then u k (t) = exp { ( λ k t t0 r ρ (τ) τ )} u 0k for any k k 0, giving rise to u k (t) = 0 for k = 1,,..., k 0 1, since ( λ k t t0 r (τ) ρ τ ) > 0 as r ρ 0 < λ k for each k k 0 an r ρ (t) ecreases as t. It follows from continuity that u (t) = r (t) < r 0 for all t > 0 an k k 0 for any solution to the problem 1 t u k (t) + (λ k r (τ) ρ ) u k (t) = 0, u k (0) = u 0k. The above results show that for the cases λ k0 > r ρ (0) > λ k0 1 an c) we nee to consier the space ecomposition H := H k0 H k0 = H0 1 (Ω) (P k0 (H) := H k0, Q k0 (H) := H k0 ), by means of which one analyze the intrinsic behavior of the corresponing solutions. Let the space H := H0 1 (Ω) be representable, if r0 := u 0k 0 + H u + k0 0k 0, in the vector form H k0 H := { u = ( u + k 0, u k 0 ) : u + k 0 H k0, u k 0 H k0 }. Then the following proposition follows irectly from the above iscussion. Proposition 3. Uner the above conitions if u 0k 0 H k0 > u + 0k 0 if the rate of ecrease of the norm u k 0 (t) H k0 H k0 an is greater than the rate of

11 Chaotic Moeling an Simulation (CMSIM) 1: , ecrease of the norm u + k 0 (t) H k0, then there exists t > 0 such that u k (t) ecreases for t t. On the other han, if u 0k 0 < H u + k0 0k 0 an the rate H k0 of increase rate of the norm u + k 0 (t) is greater than the rate of ecrease of H k0 u k 0 (t), then there exists t > 0 such that u H k (t) increases for t t. k0 Now we will procee to investigation of the behavior of solutions that start at a fixe function with the norm fixe by the number r 0 : λ k0 1 < r ρ 0 < λ k 0. As H = H k0 H k0 an for each u H there hols the ecomposition u (t) = u + k 0 (t) + u k 0 (t) for any t > 0, then it is enough to stuy the case when u 0k 0 u + H k0 0k 0. Inee if we set u H 0 = r 0 an r 0 satisfies the k0 inequality λ k0 1 < r ρ 0 < λ k 0, then each of solutions to the problem (14) - (15) satisfies one of the following statements: 1. If u 0 lies in H k0 or in a small neighborhoo of the subspace H k0, then u k (t) as t for k = 1, k 0 1, moreover since in this case r (t) increases graually an so in time is greater than each λ k for k k 0, i.e. in this case u k (t) graually increases for all k;. If u 0 lies in H k0 or in a small neighborhoo of the subspace H k0, then u k (t) 0 as t for k k 0, moreover since in this case r (t) ecreases graually so that in time it is less than each λ k, i.e. in this case u k (t) graually ecreases for all k; 3. If u 0 H such that u 0k 0 H k0 u + 0k 0 H k0, then there is a relation between of u 0k 0 (t) an u + 0k 0 such that the behavior of the u k (t) is chaotic for all k for which u k (0) If u 0 H such that u 0k 0 an u + H k0 0k 0 are ifferent, then the H k0 solutions are still connecte by some relations. As the Claims 1. an. were prove above, we nee only stuy Claim 3. Consier the representation of the formal solutions () to the problem (1): u k (t) = exp { ( λ k t t0 r (τ) ρ τ )} u 0k, k = 1,,... It is known that u k (t) increases for k : 1 k k 0 1 an ecreases for k k 0 by virtue of Proposition 3, epening on the ifference λ k r (0) ρ. But as r (t) := u k 0 (t) + H u + k0 k 0 (t) hols in a vicinity t = 0, r (t) can H k0 change epening on the behavior of u k (t) as k k 0 an k k 0 1 vary; consequently, the corresponing subspaces of H change, i.e. r (t) ρ can become greater than λ k0 or less than λ k0 1. This variation of r (t) is very complicate, as the variation epens on relations among the behaviors of u k (t) in the case when k k 0 an k k 0 1, which may give rise to chaos. We now investigate Claim 4 with an eye towar the question of whether or not there is an attractor for the operator resolving the problem (14) - (15). Towar this en, we will consier the following system of ifferential equations E k (u) := t u (t), w k + u (t), w k (t) u ρ (t) u (t), w k = h, w k,

12 130 Prykarpatski et al. where {w k (x)} k=1 are eigenfunctions of the Laplacian in H1 0 (Ω) corresponing to the eigenvalues {λ k } k=1, respectively, by virtue of the impose conitions. Whence, it follows that E k (u) := t u k (t) + λ k u k (t) u ρ (t) u k (t) = h k (t), k = 1,,... As our aim is the investigation of the behavior of solutions of the problem uner the conition u 0 S H1 0 r (Ω) 0 (0), λ k0 1 < r ρ 0 < λ k 0 it is enough to stuy the homogeneous equation case. So we assume h (t, x) = 0. Then for the problem (14) - (15) we obtain the following problem Cauchy E k (u) = t u k (t) + λ k u k (t) u ρ (t) u k (t) = 0, (7) u (t), w k t=0 = u k (t) t=0 = u 0k, k = 1,,..., k 0 1 (8) Whence, for some t 0 > 0 for t [0, t 0 ) we have u ρ (t) rρ 0 + ε < λ k 0, for some ε > 0. Inee, we have from (18) - (19) that t u k (t) + (λ k u ρ (t))u k (t) = 0, u k (0) = u 0k, which leas to the formal solution of the Cauchy problem { } { u k (t) = exp 0t (λ k r ρ (τ))τ u 0k = exp λ k t + 0t } r ρ (τ) τ u 0k (9) Hence, if λ k0 1 < r ρ 0, then u k (t, x) increases in the vicinity of zero if u 0k (x) > 0 for k = 1,,..., k 0 1 an the part P k0 u 0 is sufficiently greater than Q k0 u 0. Let u 0 H 1 0 (Ω) an u 0 = r 0, then the above expression implies that the behavior of the solution u k (t) epens on the relationship between r 0 an λ k an also between P k0 u 0 an Q k0 u 0. Consier the behavior of u k (t) for all k = 1,,..., in the case when λ k0 : λ k0 1 < r ρ 0 < λ k 0. We have the following system of equations 0 = t u k (t) + λ k u k (t) r (t) ρ u k (t) = = t u k (t) + (λ k r (t) ρ ) u k (t), u k (0) = u 0k, k = 1,,..., (30) where u (t, x) := k = 1 u k (t) w k (x). It is easy to see that this system of equations is of interest only for the cases k k 0, k k 0 1 an k = k 0 separately. If k k 0 this part of the system has a solution that is unique as t t (k, r 0 ) for some t (k, r 0 ) > 0 if Q k0 u 0 is sufficiently greater than P k0 u 0. Formally we can etermine the solution of each equation from (1) in the following form: u k (t) = exp { ( )} λ k t t0 r (τ) ρ τ u 0k. (31)

13 Chaotic Moeling an Simulation (CMSIM) 1: , Thus, consiering the expression (1) for k k 0 1, it follows from () that { ( )} u k (t) = exp λ k t t0 r (τ) ρ τ u 0k exp {(r (0) ρ λ k ) t} u 0k, as r (t) ρ > λ k for 1 k k 0 1 an some t > 0. Consequently, the sequence u k (t) increases for each k : 1 k k 0 1, if the part P k0 u 0 is sufficiently greater than Q k0 u 0, that reners the increase of r (t). Consier the representation of the formal solutions () to the problem (1): { ( )} u k (t) = exp λ k t t0 r (τ) ρ τ u 0k, k = 1,,... (3) It is known that u k (t) increases for k : 1 k k 0 1 an ecreases for k k 0 by virtue of Proposition 3, epening on the ifference λ k r (0) ρ an the relation between P k0 u 0 an Q k0 u 0. Thus, concerning the system () for k we nee to stuy the expression λ k r (0) ρ which is negative owing to the conitions impose. But the behavior of functions u k (t) cannot exactly explain the behavior of functions u k (t), an also the behavior of the solution u(t, x). Consequently, we nee to stuy the behavior of functions u k (t) in greater etail. So, from the expression (3) uner the corresponing relation between P k0 u 0 an Q k0 u 0 is clear that if u 0k 0 (u 0k 0) then u k (t) 0 (u k (t) 0) an in aition if λ k r (0) ρ > 0 then in the case u 0k 0 we see that u k (t) ecreases, in the case u 0k < 0 we see that u k (t) increases, but u k (t) will ecreases at least in some vicinity of zero. An next let λ k r (0) ρ < 0, then when u 0k 0 we see that u k (t) increases in the case u 0k < 0 we see that u k (t) ecreases, but u k (t) increases at least in a vicinity of zero. If λ k r (0) ρ = 0 then u k (t) oes not vary at least in some vicinity of zero. Thus, we obviously nee to investigate the behavior of r (t) for various initial function u 0 (x) in the case when u 0 = r 0. So as λ k r ρ 0 > 0, the functions u k (t) ecreases an converge to zero when t for k k 0, where k 0 1 is such that λ k0 r ρ 0 > 0 an λ k 0 1 r ρ 0 0. Therefore, the behavior of r (t) essentially epens on the selections of u k (0) for 1 k k 0 1. It is clear from the above analysis that nee to consier the expression u (t, x) = k 1 u k (t) w k (x) for the solution an the expression u 0 (x) = k 1 u 0k w k (x) for initial ata. Let r 0 > 0, then there exists k 0 such that λ k0 > r ρ 0 r (0)ρ > λ k0 1 an u 0 = r 0. Using the orthogonal splitting u (t) = P k0 u (t) + Q k0 u (t), we obtain the following expressions: u 0 (x) = k 1 u 0k w k (x) = k 0 1 k 1 u 0k w k (x)+k k 0 u0k w k (x) an u (t, x) = k 1 u k (t) w k (x) = = k 0 1 k 1 u k (t) w k (x) + k k 0 uk (t) w k (x). (33) There exist t 1 > 0 an t > 0 such that P k0 u (t) of (33) can increases in (0, t 1 ) an Q k0 u (t) of (33) ecreases in (0, t ) if any of their terms are

14 13 Prykarpatski et al. nonnegative functions. Let min {t 1, t } = t 1, then when t > t 1 these summans can behave quite ifferently. Here are the possibilities: 1) the velocity u (t) becomes greater than r 0 for t t 1, moreover r (t) ρ > λ k0 for t > t 1, so the orthogonal splitting u = P k0 u + Q k0 u changes an becomes, at least, u = P k0+1u + Q k0 1u; ) Q k0 u ecreases of to a point where u (t) is smaller than r 0 for t t 1, moreover r (t) ρ < λ k0 1 for t > t 1, so the orthogonal splitting u = P k0 u+q k0 u changes an becomes, at least, u = P k0 1u+Q k0+1u; 3) there exist a t 0 t 1 an an R 0 P k0 u R 1 > 0 such that beginning at t 0 the changes of P k0 u an Q k0 u become such that r (t) = u (t) = k 0 1 k 1 u k (t) + k k 0 uk (t) satisfies R 1 r (t) R 0 for t t 0. Consier the case 1). In this case we have the following possibilities: a) P k0 u increases with such velocity that u (t), which can takes place when u 0 (x) is chosen in the vicinity of the subspace H k0 (this scenario is stuie in Proposition 3); b) rate of growth of P k0 u iminishes beginning at time t an the function u (t, x) behaves as in case 3, which we will explain in what follows. The case ) has have variants: a ) Q k0 u ecreases with such velocity that u (t) 0 leaing to the inequality r (t) ρ < λ 1, which can take place when u 0 (x) is chosen in the vicinity of the subspace H k0 (this variant is also stuie in Proposition 3); b ) rate of ecrease of Q k0 u iminishes beginning at some time t an leaing to case 1)b). Consequently, it remains only to investigate case 3). It is clear that this case can occur when P k0 u 0 assumes both positive an negative values. Therefore, we consier special initial atum an try to explain case 3 for such functions. So, let P k0 u 0 = u 0k0 1w k0 1, i.e. u 0 (x) = u 0k0 1w k0 1 (x) + Q k0 u 0 (x) an u 0 = r 0. Then we obtain the following: u k0 1 (t) changes with such way that u k0 1 (t) increases with t an u k0 1 (t) ρ λ k0 1 when t ; moreover, Q k0 u (t) ecreases with increasing t an therefore Q k0 u (t) 0 when t. Hence, u ρ (t) λ k0 1 as t. In other wors, the increase of P k0 u an ecrease of Q k0 u (t) compensate for each other in such a way that this process leas to the case escribe above. Thus, it not is ifficult to see that in orer to obtain the above result, we nee to select u 0k0 1 in the vicinity of the subspace H k0, which epens on the given r 0 : r ρ 0 (λ k 0 1, λ k0 ). Accoringly it follows in the case when P k0 u 0 increases, the corresponing u 0k, 1 k k 0 1, must be chosen as one previously. Moreover, in this case there is a λ j0 such that P k0 u (t) λ j0 = inf {λ k 1 k k 0 1, u 0k 0} when t. Therefore, there exists a ouble cone with the vertex at zero that contains the subspace H k0 an all elements are containe in some neighborhoo of H k0. In aition, the maximal istance between of the elements of this subset an the subspace H k0 epens on the given r 0. Now, we enote this subset { by H H0 1. It follows from } this efinition that any subset of H B H1 0 r 1 (0) B H1 0 r (0), r 1 > r > 0 converges to a set, which we can efine as H k0 B H1 0 λ j0 (0), where r 1, r are some numbers with λ k1 1 r ρ < λ k 1

15 Chaotic Moeling an Simulation (CMSIM) 1: , an there is a λ j1 = inf {λ k 1 k k 1 1, u 0k 0} an k 1 = k 1 (r 1 ). This shows that the obtaine set is subset of a finite-imension space an it is local attractor in some sense. Thus, we have prove the following result. Theorem 1. Let all the impose above conitions hol an u 0 H0 1 (Ω) whose norm u 0 = r 0 satisfies the inequality λ k0 1 < r ρ 0 < λ k 0. Then each solution to the problem (14) - (15) satisfies one of the following properties: 1. If u 0 lies in H k0 or in a sufficiently small neighborhoo of the subspace H k0, then u k (t) as t for k = 1, k 0 1, moreover since in this case r (t) increases an graually it will be greater than each λ k for k k 0, i.e. in this case u k (t) will graually increase for all k;. If u 0 lies H k0 or in a small neighborhoo of the subspace H k0, then u k (t) 0 as t for k k 0, moreover since in this case r (t) ecreases an graually it will be less than each λ k, i.e. in this case u k (t) graually ecreases for all k; 3. If u 0 H, P k0 u 0 Q k0 u 0 an if there are small numbers δ (λ k0 ) > ɛ (λ k0 ) > 0 such that for the Hausorff istance ɛ ( H k0 ; { u + 0k k = 1, k 0 1 }) δ (34) hols, then the behavior of the u (t, x) is chaotic for sufficient large t. An also, if u 0k 0 H u + k0 0k 0, then there is a relationship between u H k0 0k 0 (t) an u + 0k 0 for which the behavior of the u k (t) is chaotic for all k satisfying u k (0) 0 Remark 1. If 3. of the above theorem obtains, then the following claim is reasonable: for any λ k0 there is a subset B λk0 H0 1 (Ω) for which (34) hols an for any u 0 B λk0 the corresponing solution u (t) satisfies the conition an λ j0 u ρ (t) < λ k 0 foranyt > 0, lim t u (t) ρ = λ j 0 = inf {λ k 1 k k 0 1, u 0k 0} then there is an absorbing chaotic set in L (Ω). 4 Problem (1) - (): the case (β) Let the mapping g have the local nonlinearity as in the case (β), i.e. now we consier the following problem: u t u µ u ρ u = h (x), (t, x) (0, T ) Ω, (35) u (0, x) = u 0 (x) H 1 0 (Ω), u [0,T ) Ω = 0. (36) Here µ, ρ > 0 are some numbers, Ω R n is a omain with sufficiently smooth bounary Ω or Ω := R n, h L (Ω).

16 134 Prykarpatski et al. Rewriting (35) in the form u t = u + µ u ρ u + h (x) := ( + µ u ρ ) u + h (x), we can easily express the formal solution to this problem in the form { } u (t) = exp t + µt0 u (τ) ρ τ (u 0 + h). (37) Whence, we obtain the estimate { ( ) } u (t) exp t0 λ 1 + µ u ρ ρ (τ) τ ( u 0 + h ) for u (t) H0 1 for a.e. t 0, where λ 1 is first eigenvalue of the operator : H0 1 H 1. Moreover, it is not ifficult to see that from (35) - (36) one can obtain the following problem 1 t u (t) = u (t) + µ u ρ+ ρ+ (t) + h, u (t), u (0) = u Existence of a Solution First, we consier the case h (x) = 0. Then from (4) we obtain the estimate ( ) } u {t0 (t) exp λ 1 + µc 0 u ρ ρ+ (τ) τ u 0, where c 0 > 0 is the constant of the embeing inequality for L ρ+ (Ω) L (Ω). This inequality shows that we can stuy problem (35) - (36) using the previous 1 approach from Section in the case of the ball Br X λ 0 (0) for r 0 < 1 ρ µc 0c 1, where c 1 is the constant of the embeing inequality for H0 1 L ρ+ (Ω). In the other wors, we can stuy the solvability only locally for u 0 B H1 0 r 0 (0) an 0 ρ 4 n. Now, we eal with problem (4) in the other way. Let ρ < 4 n, then the following interpolation inequality (G-N-S inequality) hols u ρ+ c u 1 θ u θ, θ = ρn (ρ + ), ρ < 4 n, for u (t) H 1 0 an n 3. Thus, we get u ρ+ ρ+ (t) = u ρ ρ+ (t) u ρ+ (t) c u ρ ρ+ (t) u (1 θ) u θ, where θ = ρn 4 ρ (n ) < an (1 θ) = < (ρ + ) (ρ + )

17 Chaotic Moeling an Simulation (CMSIM) 1: , an consequently u ρ+ ρ+ (ρ+) 4 ρ(n ) (t) C (ε, c) u ρ+ (t) u (t) + ε u (t). (38) Thus if ρ satisfies the inequality 0 < ρ 4 (ρ+) n, then ρ 4 ρ(n ) ρ +, where ε > 0 is the small parameter. Taking into account the inequality (38) an equation (4), we fin that 1 t u (t) = u (t) + µ u ρ+ ρ+ (t) u (t) + µc (ε, c) u ρ (ρ+) 4 ρ(n ) ρ+ (t) u (t) + ε u (t) = (1 ε) u (ρ+) 4 ρ(n ) (t) + µc (ε, c) u ρ+ (t) u (t). Thus, if we choose ε = ε 0 > 0 a fixe small number, then we fin, owing to the previous approach, that ] t u [ (t) (1 ε 0 ) λ 1 + µc (ε 0, c) u ρ (ρ+) 4 ρ(n ) ρ+ (t) u (t), or u { t0 [(1 (t) exp ε 0 ) λ 1 + µc (ε 0, c) u (τ) ρ (ρ+) 4 ρ(n ) ρ+ ] } τ u (0). (39) Consequently, in this case we nee to stuy the problem uner the variants (i) - the case when u 0 ρ+ ρ+ < 1 ε0 µc(ε λ 0,c) 1; an (ii) - the case when u 0 ρ+ ρ+ 1 ε 0 µc(ε λ 0,c) 1, separately. It follows from the above estimate that in the case (i) we can prove of the existence theorem on the ball Br X 0 (0) using the approach in Section (or using of the metho of compactness [,3], since it is not ifficult to see the operator generate by the problem (35) - (36) is weakly compact). Hence, (39) implies that u (t) ecreases if u 0 B H1 0 r 0 (0) in the case [ ] 1 (ρ+) when r 0 < (here C 0 > 0 is the constant of the embeing 1 ε 0 µc 0C(ε 0,c) λ 1 theorem). Therefore, u H (t) u 0 H for t 0, i.e. the following statement is verifie. Theorem. Let h (x) = 0 an the following conitions be satisfie: (1) 0 < [ ] ρ 4 n, u 0 ρ+ ρ+ C 0 u 0 ρ+ 1 < 1 ε0 H0 1 µc(ε λ 1 ε 0,c) 1, r 0 < 0 µc λ (ρ+) 0C(ε 0,c) 1 (here 1 C 0 > 0 is as above); or () 0 < ρ 4 n an u 0 B H1 0 λ (0) for r 0 < 1 ρ r 0 µc 0c 1 (here c 0 > 0 an c 1 > 0 are such as in the starting part of this section). Then problem (35) - (36) is solvable in Br X 0 (0) (or B X r (0)) for all u 0 0 B H1 0 r 0 (0) (or u 0 B H1 0 (0)), moreover, the solutions u (t) are containe in the r 0 close ball B H1 0 r 0 (0) (or B H1 0 (0)) for t 0. r 0

18 136 Prykarpatski et al. Remark. It shoul note that if we consier the problem (35) - (36) for h (x) 0, for example, as in the case (1) then using the same approach we get u t u µ u ρ u, u = 1 t u (t)+ u (t) µ u ρ+ ρ+ (t) h Y u H 1 0 an it is not ifficult to see that here, as in the case (1), we nee to etermine the balls B X r 1 (0) in X an B Y R 1 (0) in Y (here Y = H 1 or L (Ω)), where r 1 an R 1 epen on (λ 1, ρ, µ, C 0 ) as well as each other. However, we shall not go into this case here. Consequently, we consier here only the case h (x) = 0, so that we nee to stuy the pose problem in the variants (i) - the case when u 0 ρ+ ρ+ < 1 ε 0 µc(ε λ 0,c) 1; an (ii) - the case when u 0 ρ+ ρ+ 1 ε0 µc(ε λ 0,c) 1, separately. 4. Blow-up Solutions Consier the case (ii). Denote the functional F (t) := 1 u (t) µ ρ + u ρ+ ρ+ (t) for which t F (t) = u, u t µ u ρ u, u t = Ω uu t µω u ρ uu t = Ω u t = u t H, F (t) t=0 = F (0), then we get F (t) = F (0) t0 u t H = 1 u 0 H µ ρ + u 0 ρ+ ρ+ t0 u t H. Now consier the erivative of the functional G (t) = u (t) 1 t G (t) = u t, u = u + µ u ρ u, u = u ρµ ρ + u ρ+ ρ+ (t) = u 0 + µ ρ + u 0 ρ+ ρ+ + t0 i.e. 1 t G (t) = 1 t u (t) = F (0) + t0 (t)+µ u ρ+ ρ+ (t) = F (t) + u t + ρµ ρ + u ρ+ ρ+ (t) u t + µρ ρ + u ρ+ ρ+ (t) F (0) + c 1µρ ρ + u ρ+ (t), G (0) = 1 u 0 H.

19 Chaotic Moeling an Simulation (CMSIM) 1: , So we obtain the problem t G (t) 4F (0) + c ρ+ 1µρ G (t), G (0) = u 0 ρ + which shows the finite-time blow-up of solutions of the pose problem. Inee, there are two variants or F (0) 0 or F (0) > 0. If F (0) 0 then for G (t) we obtain t G (t) c ρ+ 1µρ G (t), G (0) = u 0 ρ +. (40) Clearly, if ρ > 0, the solutions of the problem (40) have finite-time blow-up. Accoringly, we have prove the following result. Theorem 3. Let the initial function u 0 H 1 0 L ρ+ (Ω) satisfy the conition 1 u 0 µ ρ + u 0 ρ+ ρ+ 0. Then if the local solution of problem (14) - (15) is sufficiently smooth, this solution has a finite-time blow-up in H. 4.3 Solution Behavior Now consier (35) - (36) in the general case. For a etaile investigation of the behavior of the solutions, we will use the same approach as in Section 3. Let the eigenfunctions of the operator : H0 1 H 1 be as in Section 3, then any function v H0 1 has the representation v (t, x) := k = 1 v k (t) w k (x). Thus, we have u t u µ u ρ u = 0; u (0, x) = u 0, = ( uj (t) j=1 t ) + λ j u j (t) w j (x) µ u j (t) w j (x) ρ u j (t) w j (x) = 0. j=1 j=1 (41) Now, if ρ n, then u ρ u := v u (t, x, ρ) L (Ω) for all u H 1. Consequently, we have v u (t, x, ρ) = k = 1 vu k (t, ρ) u k (t) w k (x) H. It is clear that the variation of vu k (t, ρ) epens on t in R +, as at any point t the function vu k (t, ρ) = ζ k (t). Moreover, as we saw in the previous section the variation of u (t, x) epens on the relationship between u 0 an λ k. Consequently, we can rewrite the problem (41) in the form j = 1 ( ) u j (t) + λ j u j (t) µvu j (t, ρ) u j (t) w j (x) 0 t u j (0) = u 0j, j = 1,,... Then, if we take into account that the system {w j (x)} j=1 is an orthogonal basis in L (Ω), the problem (41) is equivalent to the system of the problems u j (t) t + λ j u j (t) µv j u (t, ρ) u j (t) = 0, (4)

20 138 Prykarpatski et al. u j (0) = u 0j, j = 1,,... So to stuy the behavior of solutions of the problem (35)- (36,) it is enough to stuy the system of problems (4) with solutions { (λj u j (t) = exp t0 µvu j (t, ρ) ) } τ u 0j, j = 1,,... Thus, we obtain the flow which etermines the solution via these formal expressions. Now, if S(t) is the flow etermine by problem (4), we can rewrite S(t) as S(t) := S(t + c u (t, ρ)), as far as S(t) epens on x via u. ( ) u j (t) = e t0 (λj µv j u (t,ρ))τ u 0. (43) Here we get that the behavior of the solutions epens not only on the time, yet epens strongly on the location of points x in Ω belonging to the solution support as far as the quantity of v j u (t, ρ) epens of the locally variations of quantity of u in Ω. Consequently, (43) emonstrates us that this case of the spatiotemporal states essentially iffers from the case (5) - (6). In other wors, for fixe j Z + there are possible two choices: either λ j < µv j u (t, ρ) or λ j µv j u (t, ρ), epening on the state of (t, x) R + Ω. Consequently, to explain the possible variants in this case we can argue as in Section 3 to obtain the following result. Proposition 4. Suppose ρ n, u 0 B H1 0 r 0 (0) an there are subsets with nonzero measure in Ω on which λ i0+1 > u 0 (x) λ i0, for some i 0 1. Then, the behavior of the solution epens sensitively on the variation of vu i (t, ρ), i.e. it has spatio-temporal chaotic ynamics. Thus, we can conclue, that as in the previous problem, chaotic ynamics occur, but unlike the previous case, the chaos is spatio-temporal. 5 Acknowlegements D.B. is grateful to the CAMS center at NJIT for its support. A.P. corially appreciates the Hacettepe University of Ankara, Turkey, for the nice hospitality an financial support via the visiting grant TUBITAK- 1- Fellowships. References 1. Afraimovich, V., Babin, A. V., Chow, S. N. Spatial chaotic structure of attractors of reaction-iffusion systems. Trans. Am. Math. Soc. (1996) 348, 1, pp Babin, A. V. Topological invariants an solutions with a high complexity for scalar semilinear PDE. J. Dynam. Differ. Equ. (000) 1, 3, pp Barrachina X. an Conejero J. A. Devaney chaos an istributional chaos in the solution of certain partial ifferential equations. Abstract an Applie Analysis, (01), , 11 pages, oi: /01/457019

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