Pullback exponential attractors for nonautonomous equations Part II: Applications to reaction-diffusion systems

Pullbac exponential attractors for nonautonomous equations Part II: Applications to reaction-diffusion systems Radoslaw Czaja a,1,, Messoud Efendiev b a CAMGSD, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal b Helmholtz Center Munich, Institute of Biomathematics and Biometry, Ingolstädter Landstraße 1, 85764 Neuherberg, Germany Abstract The existence of a pullbac exponential attractor being a family of compact and positively invariant sets with a uniform bound on their fractal dimension which at a uniform exponential rate pullbac attract bounded subsets of the phase space under the evolution process is proved for the nonautonomous logistic equation and a system of reaction-diffusion equations with time-dependent external forces including the case of the FitzHugh-Nagumo system. Keywords: exponential attractor, pullbac attractor, fractal dimension, nonautonomous dynamical systems, reaction-diffusion equations. 000 MSC: Primary 35B41; Secondary 37B55, 35K57, 37C45. 1. Pullbac exponential and global attractors for semilinear parabolic problems In Part I of this wor see [5] we have constructed a pullbac exponential attractor for an evolution process. By this we mean a family of compact and positively invariant sets with uniformly bounded fractal dimension which under the evolution process at a uniform exponential rate pullbac attract bounded subsets of the phase space. We have also compared this object with a better nown notion of a pullbac global attractor see for example [], [3] being a minimal family of compact invariant sets under the process and pullbac attracting each bounded subset of the phase space. Moreover, we have formulated conditions under To appear in J. Math. Anal. Appl. 011, doi:10.1016/j.jmaa.011.03.05. Corresponding author. Email addresses: czaja@math.ist.utl.pt Radoslaw Czaja, messoud.efendiyev@helmholtz-muenchen.de Messoud Efendiev 1 The author was partially ported by FCT/Portugal. Preprint submitted to Elsevier April, 011

which the mentioned abstract results apply to nonautonomous semilinear parabolic problems. For completeness we recall here the main result see [5, Theorem 3.6] and refer the reader for the proof and details to Part I of this wor. We consider a positive sectorial operator A: X DA X in a Banach space X having a compact resolvent see [7]. Denoting by X γ, γ 0, the associated fractional power spaces, we fix α [0, 1 and consider a function F : R X α X satisfying the following assumption G X α, bounded 0<θ=θG 1 T1,T R,T 1 <T L=LT T 1,G>0 τ1,τ [T 1,T ] u1,u G F τ 1, u 1 F τ, u X L τ 1 τ θ + u 1 u X α. F1 Note that L depends only on the difference T T 1 and on G. Under this assumption for any R and u 0 X α there exists a unique forward local X α solution to the problem { u τ + Au = F τ, u, τ >, 1.1 u = u 0, defined on the maximal interval of existence [, τ max, i.e. a function u C[, τ max, X α C, τ max, X 1 C 1, τ max, X satisfying 1.1 in X and such that either τ max = or τ max < and in the latter case Furthermore, we denote lim uτ X α =. τ τ max T = {τ R: τ τ 0 } with τ 0 fixed and assume that for some M > 0 F τ, 0 X M. τ T F In order to prove that the local solutions can be extended globally forward in time and obtain the existence of a bounded absorbing set in X α in specific examples we will verify an appropriate a priori estimate. Here we assume that each local solution can be extended globally forward in time, i.e. τ max =, F3a there exists a constant ω > 0 and a nondecreasing function Q: [0, [0, both independent of such that uτ X α Q u 0 X αe ωτ + R 0, τ, τ T, F3b

holds with a constant R 0 = R 0 τ 0 > 0 independent of, τ and u 0 and in case τ 0 < for any T > 0 there exists R T, > 0 and a nondecreasing function Q T, : [0, [0, such that uτ X α Q T, u 0 X α + R T,, τ [, + T ]. F3c Note that hypotheses F3a F3c can be replaced by a single stronger requirement that 1.1 admits the following dissipativity condition in X α uτ X α Q u 0 X αe ωτ + Rτ, τ [, τ max, F3 where ω > 0, Q: [0, [0, is a nondecreasing function and R: R [0, is a continuous function such that for some positive constant R 0 independent of u 0,, τ Rτ R 0. τ T Because of F3a we define the evolution process {Uτ, : τ } on X α by Uτ, u 0 := uτ, τ, u 0 X α, 1. where uτ is the value at time τ of the X α solution of 1.1 starting at time from u 0. Thus we have Uτ, U, ρ = Uτ, ρ, τ ρ, τ,, ρ R, Uτ, τ = I, τ R, 1.3 where I denotes an identity operator on X α. Theorem 1.1. Under the conditions stated above for any β α, 1 there exists a family {Mτ: τ R} of nonempty compact subsets of X β such that i {Mτ: τ R} is positively invariant under the process Uτ,, i.e. Uτ, M Mτ, τ, ii Mτ has a finite fractal dimension in X β uniformly with respect to τ R, i.e. there exists d < such that d Xα f Mτ d Xβ f Mτ d, τ R, iii {Mτ: τ R} has the property of pullbac exponential attraction, i.e. ϕ>0 B1 X β, bounded τ R lim t e ϕt dist X βuτ, τ tb 1, Mτ = 0 3

and if τ 0 =, the pullbac attraction is uniform with respect to τ ϕ>0 B1 X β, bounded lim e ϕt dist X βuτ, τ tb 1, Mτ = 0. t This property is equivalent to the uniform forwards exponential attraction τ R ϕ>0 B1 X β, bounded lim e ϕt dist X βut + τ, τb 1, Mt + τ = 0. t τ R Furthermore, the pullbac exponential attractor {Mτ: τ R} contains a finite dimensional pullbac global attractor {Aτ: τ R}, i.e. a family of nonempty compact subsets of X β, invariant under the process {Uτ, : τ } pullbac attracting all bounded subsets of X β Uτ, A = Aτ, τ, B1 X β, bounded τ R lim dist X βuτ, τ tb 1, Aτ = 0 t and minimal in the sense that if {Ãτ: τ R} is a family of closed sets in Xβ pullbac attracting all bounded subsets of X β, then Aτ Ãτ, τ R. In this paper we apply Theorem 1.1 to nonautonomous reaction-diffusion equations and systems. In Section we verify the above hypotheses in an introductory example of the nonautonomous logistic equation with Dirichlet boundary condition and in Section 3 we consider a system of reaction-diffusion equations perturbed by a time-dependent external forces. This system satisfies an anisotropic dissipativity condition that holds, for example, for the FitzHugh-Nagumo system or in some chemical reaction systems see Remar 3.1.. Nonautonomous logistic equation We consider Dirichlet boundary problem for the nonautonomous logistic equation cf. [8] in a sufficiently smooth bounded domain R N, N 3, of the form { τ u = D u + λu bτu 3, τ >, x,.1 u, x = u 0 x, x, uτ, x = 0, τ, x. Here u = uτ, x is an unnown function, λ R and b is Hölder continuous on R with exponent θ 0, 1] and satisfies 0 < bτ M, τ R,. 4

for some positive M. Moreover, we assume that there exist τ 0 and m > 0 such that m bτ, τ T,.3 where we denoted T = {τ R: τ τ 0 }. We rewrite the problem.1 as an abstract Cauchy problem 1.1, where A = D in X = L with the domain DA = H H 1 0 is a positive sectorial operator with compact resolvent. We also consider its fractional power spaces and have for α 1 4, 1 X α = H α 0 = {φ H α : φ = 0}. Observe that F : R X 1 X given as F τ, u = λu bτu 3 is well defined and by. we have for u 1, u from a bounded subset G of X 1 = H0 1 and τ 1, τ R F τ 1, u 1 F τ, u L c 1 τ τ 1 θ + c u 1 u H 1 0. This shows, in particular, that assumption F1 is satisfied with α = 1. Moreover, we have F τ, 0 L = 0 for τ R. Hence F is satisfied trivially. Finally, we verify that F3 also holds. Multiplying the first equation in.1 by u and integrating over we get 1 d dt u L = u L + λu btu 4 dx. Note that by the Cauchy inequality we have d dt u L + u L 1 1 λ bt..4 Observe that by the Poincaré inequality we obtain d dt u L + λ 1 u L 1 1 λ bt, where λ 1 > 0 is the principal eigenvalue of D. Integrating over the time interval from to τ we get uτ L u L e λ 1τ + 1 e λ 1τ t λ dt..5 bt Now we proceed to obtain the a priori estimate in H0. 1 We multiply the first equation in.1 by D u, integrate over and use integration by parts to get 1 d dt u L + Du L = λ 3btu u dx. 5

Because b is a positive function, we obtain d dt u L λ u L..6 We add to both sides λ 1 u L, multiply by eλ 1t and integrate from to τ to obtain uτ L u L e λ 1τ + λ + λ 1 ut L eλ 1t τ dt..7 We return now to.4 and use the Poincaré inequality to get d dt u L + λ 1 u L + u L 1 1 λ bt. Multiplying by e λ 1t and integrating from to τ we conclude that ut L eλ 1t τ dt u L e λ 1τ + 1 λ Combining.5,.7 and.8 and using.3 we get e λ 1τ t bt dt..8 where and uτ H 1 0 1 + λ + λ 1 u H 1 0 e λ 1 τ + Rτ,.9 Rτ = R 0 λ 1 m R 0 = e λ 1τ t bt 1 + λ + λ 1 λ. λ 1 m 1 dt, τ R,.10 Note that the function R is well defined and Rτ R 0 for τ T. This shows that assumption F3 holds with α = 1. Therefore we may apply Theorem 1.1 and obtain the following Corollary.1. If. and.3 hold, then the problem.1 generates an evolution process {Uτ, : τ } in H0 1 and for any β 1, 1 there exists a family {Mτ: τ R} of nonempty compact subsets of H β 0 with the following properties: i {Mτ: τ R} is positively invariant under the process Uτ,, i.e. Uτ, M Mτ, τ, 6

ii Mτ has a finite fractal dimension in H β 0 uniformly w.r.t. τ R, i.e. d H1 0 f Mτ d Hβ 0 f Mτ d <, τ R, iii {Mτ: τ R} has the property of pullbac exponential attraction, i.e. ϕ>0 B1 H β 0, bounded τ R lim t e ϕt dist H β 0 Uτ, τ tb 1, Mτ = 0 and if τ 0 =, the pullbac attraction is uniform w.r.t. τ R ϕ>0 B1 H β 0, bounded lim t e ϕt τ R dist H β 0 Uτ, τ tb 1, Mτ = 0. Furthermore, the pullbac exponential attractor {Mτ: τ R} contains a finite dimensional pullbac global attractor {Aτ: τ R}, i.e. a family of nonempty compact subsets of H β 0, invariant under the process {Uτ, : τ } Uτ, A = Aτ, τ, and pullbac attracting all bounded subsets of H β 0 B1 H β 0, bounded τ R lim t dist H β 0 Uτ, τ tb 1, Aτ = 0. 3. Anisotropic nonautonomous reaction-diffusion systems Following [6] we consider the nonautonomous reaction-diffusion system { τ u + Au = fu + gτ, τ >, x, u, x = u 0 x, x, uτ, x = 0, τ, x, 3.1 where R 3 is a bounded domain with C +η. Here uτ, x = u 1 τ, x,..., u τ, x is an unnown function and fu = f 1 u,..., f u and gτ, x = g 1 τ, x,..., g τ, x are given functions. We pose that A is a second order elliptic differential operator of the form Au = A 1 u 1,..., A u, where A l u l x = 3 xi a l ijx xj u l x, x, l = 1,...,, 3. i,j=1 with the functions a l ij = a l ji from C 1+η and satisfying uniformly strong ellipticity condition 3 ν>0,..., x ξ=ξ1,ξ,ξ 3 R 3 a l ijxξ i ξ j ν ξ. 3.3 7 i,j=1

We also assume that for the nonlinear term f CR, R there exist constants p 1,..., p 0 and q 1,..., q 0 such that f satisfies the growth assumption c>0 u=u1,...,u,v=v 1,...,v R fu fv c u l v l 1 + u l p l + v l p l 3.4 and the anisotropic dissipativity assumption C>0 u=u1,...,u R f l uu l u l q l C. 3.5 As refers to the time- The restrictions on the range of constants will be imposed later. dependent perturbation we assume that g : R [L ] is globally Hölder continuous with exponent θ 0, 1] 3.6 and there is τ 0 such that gτ [L ] <, 3.7 τ T where we denoted T = {τ R: τ τ 0 }. Below in Remar 3.1 we present two particular cases of the system 3.1 concerning time-perturbed systems of two coupled reaction-diffusion equations. Remar 3.1. If =, we consider the perturbed FitzHugh-Nagumo system modelling transmission of nerve impulses in axons, i.e. for α, β, γ, δ R and ε > 0 f 1 u 1, u = αu 1 + βu 1 u 3 1 γu, f u 1, u = δu 1 εu. 3.8 Note that the following inequality holds q 0 C>0 u1,u R f l u 1, u u l u l q C. 3.9 Indeed, by the Young inequality it follows that for some positive c 1 αu 1 + βu 1 u 3 1 γu u 1 u 1 q + δu 1 εu u u q c 1 u 1 +q + β u 1 3+q u 1 4+q. Applying again the Young inequality, we obtain 3.9. 8

Note that there are positive c, c 3 such that for u = u 1, u, v = v 1, v R we have fu fv c u 1 v 1 1 + u 1 4 + v 1 4 + c 3 u v. Thus both assumptions 3.4 and 3.5 are satisfied with p 1 = 4, p = 0 and q 1 = q = q, where q 0 is arbitrary. We also consider the following chemical reaction nonlinearity Observe that by the Young inequality we have and f 1 u 1, u = u u 3 1, f u 1, u = u 3 1 u. 3.10 u u 3 1u 1 u 1 4 + u 3 1 u u u 3 u u 1 5 u 1 8 + u 1 3 u 5 3 u 8 3 0 fu fv 18 u 1 v 1 u 1 4 + v 1 4 + 4 u v. This means that assumptions 3.4 and 3.5 are satisfied with p 1 = 4, p = 0 and q 1 = 4, q = 3. Note also that the usual dissipativity assumption q 1 = q = 0 is not satisfied in this case, since the expression u u 3 1u 1 + u 3 1 u u = u u 1 u 3 1 u can be made arbitrarily large. We consider 3.1 as an abstract semilinear parabolic Cauchy problem 1.1 in the space X = [L ] with F τ, u = fu + gτ. Note that A is a sectorial operator in X with the domain DA = [H H0] 1 see [7, Example 1.33], [1, Theorem 1.6.1], [4, Proposition 1..3] and has a compact resolvent and the fractional power spaces are described as follows X α = [X, DA] α = [H α 0 ] = [{φ H α : φ = 0}], α 1 4, 1 cf. [1, Proposition.3.3], [4, Section 1.3]. We fix α = 1 and have X 1 = [H 1 0]. Below we show that F : R X 1 X is well defined and assumption F1 is satisfied in X 1 when we suitably restrict the range of constants p l. Proposition 3.. If 0 p l 4, l = 1,...,, then there exists θ 0, 1] such that for any bounded subset G of X 1 there exists L > 0 such that for any u, v G and τ 1, τ R we have F τ 1, u F τ, v [L ] L τ 1 τ θ + u v X 1. 9

Proof. We have F τ 1, u F τ, v [L ] fu fv [L ] + gτ 1 gτ [L ]. 3.11 Since by assumption we now that gτ 1 gτ [L ] L 1 τ 1 τ θ, τ 1, τ R, it is enough to estimate the first term in 3.11. Indeed, using 3.4 and the Hölder inequality in case p l > 0, we obtain fu fv [L ] c Hence we have fu fv [L ] c u v [L p l + ] u l v l L p l + 1 + u l p l + v L p l + l p l. 3.1 L p l + 1 + u l p l L p l + + v l p l L p l +, If 0 p l 4, l = 1,...,, then H0 1 L pl+ and in consequence for any bounded subset G of X 1 = [H0] 1 we have This proves the claim. fu fv [L ] L G u v X 1, u, v G. Thus if 0 p l 4, l = 1,...,, then for any R and u 0 X 1 there exists a unique forward X 1 solution to 3.1 defined on the maximal interval of existence [, τ max, i.e. u C[, τ max, [H 1 0] C, τ max, [H H 1 0] C 1, τ max, [L ] and either τ max = or τ max < and in the latter case lim uτ [H 1 τ τ 0 ] =. 3.13 max Note that assumption F is also clearly satisfied, since by 3.7 we have τ T F τ, 0 [L ] f0 [L ] + gτ [L ] <. τ T Now we will show that under certain constraints on p l and q l assumptions F3a F3c also hold. To this end, we develop some a priori estimates following [6]. 10

Lemma 3.3. For any γ > 0 there exists C γ > 0 such that for any h > 0, any real τ + h and any nonnegative integrable function z on [, τ] we have t ztdt C γ e γ τ t h zsds. 3.14 t [+h,τ] Proof. Observe that ztdt e γ 1 + e γ + e γ +... + e γ [ τ h ] t [+h,τ] t e γ τ t h zsds, since e γ τ h 1 e γ [ τ h ] e γ. This leads to 3.14 with C γ = e γ 1 e γ 1. We also adapt the following lemma from [10, Proposition 3]. Lemma 3.4. Assume that a continuous function z : [a, b [0,, a < b, satisfies zτ D 0 e βτ a + D 1 + µ {e γτ s zs}, a τ < b 3.15 s [a,τ] with β γ > 0, D 0, D 1 0 and 0 µ < 1. Then we have zτ D 0 1 µ 1 e γτ a + D 1 1 µ 1, a τ < b. 3.16 Proof. Fix any a < T < b. From 3.15 it follows that zτ D 0 e βτ a + D 1 + µ {e γ τ s zs}, τ [a, T ]. 3.17 s [a,t ] Let us fix ρ [a, T ]. Then we multiply the above equation by e γ ρ τ and tae the remum with respect to τ [a, T ] τ [a,t ] e γ ρ τ zτ D 0 Note that we have Concluding, we get τ [a,t ] e βτ a γ ρ τ + D 1 + µ e βτ a γ ρ τ = e γρ a and τ [a,t ] τ [a,t ] s [a,t ] {e γ τ s + ρ τ zs} = τ [a,t ] s [a,t ] s [a,t ] e γ ρ s zs. e γ ρ s zs D 0 e γρ a + D 1 + µ e γ ρ s zs. s [a,t ] s [a,t ] 11 {e γ τ s + ρ τ zs}.

Since 0 µ < 1 and e γ ρ s zs <, we obtain s [a,t ] e γ ρ s zs D 0 1 µ 1 e γρ a + D 1 1 µ 1, ρ [a, T ]. s [a,t ] We apply this estimate to 3.17. From the arbitrary choice of T < b we get 3.16. Proposition 3.5. Let u = u 1,..., u be an X 1 solution of 3.1 on [, τ max. If τ max <, then with h > 0 such that < + h < τ max we have for τ < τ max u l τ +q λ1ν l e L +q l +C 8 h u l +q λ1ν l L +q e τ + l ql + gs [L ] + 1, 3.18 and for + h τ < τ max ν τ h u l s q l + L ds e λ 1 ν h +C 8 u l +q λ1ν l L +q e τ + l ql + gs [L ] + 1, 3.19 with E = [, τ max, where C 8 = C 8 h is a positive constant. If τ max =, then we choose h = 1 and 3.18 holds with E =, τ 0 + for τ, τ T, whereas 3.19 holds with E =, τ 0 + for + 1 τ, τ T. If τ max =, then for any T > 0 we choose 0 < h < T and 3.18 holds with E = [, +T ] for τ + T, while 3.19 holds with E = [, + T ] for + h τ + T. Proof. For each l = 1,..., we multiply the l-th equation in 3.1 by u l u l q l and integrate over t u l u l u l q l dx + A l u l u l u l q l dx = f l uu l u l q l dx + g l tu l u l q l dx. Note that t u l u l u l q l dx = 1 q l + t u l +q l L +q l 1

and by integration by parts and 3.3 we have Thus if q l > 0 then A l u l u l u l q l dx = q l + 1 A l u l u l u l q l dx = 4q l + 1 q l + 4q l + 1 q l + ν and if q l = 0 then we have Since 4q l+1 q l +, we obtain i=1 t u l +q l L +q l + ν ul ql+ 3 i,j=1 3 i,j=1 a l ij xj u l u l q l xi u l dx. a l ij xi u l q l + xj u l q l + dx 3 xi u l q l + 4q l + 1 dx = q l + ν ul ql+ A l u l u l u l q l dx ν u l L. q l + f l uu l u l q l dx + L omitting the modulus under the gradient when q l = 0. We set F u t = u l +q l L +q l, Φ ut = u l q l + L We add the obtained inequalities and use 3.5 to get, G u t = L g l tu l u l q l dx g l tu l u l q l dx. t F u t + νφ u t q + C + G u t, 3.0 where q = max{q 1,..., q }. We use the Poincaré inequality λ 1 φ L φ L with φ = u l q l + if q l > 0 or φ = u l if q l = 0 and thus obtain t F u t + λ 1 νf u t q + C + G u t. We multiply by e λ 1νt and integrate from to τ to get with C 1 > 0 F u τ F u e λ 1ντ + C 1 + q + 13 G u te λ 1ντ t dt, τ < τ max. 3.1

Let h > 0 be such that < + h < τ max. Assume now that + h τ < τ max. We integrate 3.0 from τ h to s τ and in consequence we get F u s + ν s [τ h,τ] τ h Combining this estimate with 3.1 we obtain F u s + ν s [τ h,τ] τ h Φ u tdt F u τ h + q + C h + q + τ h G u t dt. Φ u tdt F u e λ1ντ h + C + C 3 G u t e λ1ντ t dt, 3. where C = C h and C 3 = C 3 h are positive constants. We estimate the last term using Lemma 3.3 with γ = λ 1 νh and get with C 4 = C 4 h > 0 t C 3 G u t e λ1ντ t dt C 4 e λ 1 ν τ t G u s e λ1ντ s ds. 3.3 t [+h,τ] Moreover, it follows that e λ 1 ν τ t t G u s e λ 1ντ s ds e λ 1 ν t τ t g l su l u l q l dx ds. 3.4 Observe that by Schwarz and Young inequalities we have t g l su l u l q l dx ds gs [L ] u l q l+1 L q l +1 [,t],l q l +1 s [,τ] ql µ u l q l+ + C L q l +1 [,t],l q l +1 µ gs [L ] +, s [,τ] where µ > 0 and C µ is independent of l. Note that u l q l+ L q l +1 [,t],l q l +1 C u l q l+ L rq l + [,t],l rq l + = C u l q l + L r [,t],l r since q l + 1 < 7 6 q l + = r q l + with r = 7 3 and C does not depend on l and t. Observe that by interpolation inequalities we have u l q l + Ĉ u l q l + L r [,t],l r 1 θ 0 L [,t],l 14 u l q l + θ 0, L [,t],h0 1,

where θ 0 = 6, since by [9, 4.3.1, Theorem ] 7 and by [9, 1.18.410] [L, H 1 ] θ0 = H θ 0 L 6 3 θ 0 = L r [L [t h, t], L, L [t h, t], H 1 ] θ0 = L r [t h, t], [L, H 1 ] θ0. Hence we get with C 5 = C 5 h > 0 u l q l+ L q l +1 [,t],l q l +1 C 5 F u s + ν s [,t] where we used the Young inequality again. Summarizing, we get t g l su l u l q l dx ds µc 5 s [,t] F u s + ν t t Φ u sds Φ u sds + C µ s [,τ] gs [L ] ql+. Applying this estimate to 3.4 we obtain with C 6 = C 6 h > 0 e λ 1 ν τ t t where G u s e λ1ντ s ds µc 6 e λ 1 ν τ t Z u t + C µ gs [L ] ql+, s [,τ] Z u t = Therefore, it follows from 3.3 that C 3 G u t e λ1ντ t dt µc 7 e λ 1 ν t [+h,τ] t F u s + ν Φ u sds. s [,t] τ t Z u t + C µ, gs [L ] q l+ s [,τ] with C 7 = C 7 h > 0. Applying this estimate to 3. we finally obtain for any µ > 0 Z u τ F u e λ1ντ h + µc 7 e λ 1 ν τ t Z u t + t [+h,τ] +Ĉµ gs [L ] ql+ + 1, + h τ < τ max. s [,τ] 15 3.5

If τ max <, then we choose µ = 1 C 7 Z u τ F u e λ 1 ν τ h + C 8 and use Lemma 3.4 to see that for + h τ < τ max gs [L ] ql+ + 1, 3.6 where E = [, τ max. It follows immediately that 3.19 holds with E = [, τ max and + h τ < τ max. Moreover, we now in particular that s [,+h] F u s F u + C 8 gs [L ] ql+ + 1 3.7 with E = [, τ max. This implies 3.18 with E = [, τ max and for τ < τ max. If τ max =, then we set h = 1 and apply Lemma 3.4 to 3.5 and in case + 1 < τ 0 we obtain 3.6 with E =, τ 0 + for + 1 τ, τ T and 3.7 with E =, τ 0 +. This implies that 3.19 holds with h = 1, E =, τ 0 + for + 1 τ, τ T and 3.18 with E =, τ 0 + for τ, τ T. Moreover, in case + 1 τ 0 and τ, τ T, we now that 3.6 holds with h = 1, E =, τ 0 + for + 1 τ < τ 0 + and hence 3.18 holds with E =, τ 0 + for τ, τ T also in this case. Finally, pose that τ max = and let T > 0. We choose 0 < h < T and apply Lemma 3.4 to 3.5 in order to obtain 3.6 and thus 3.19 with E = [, + T ] for + h τ + T. Moreover, 3.7 holds with E = [, + T ] and hence 3.18 with E = [, + T ] for τ + T. As follows from the above proposition we will obtain below a priori estimates in the following three cases: 1 τ max <, < + h < τ max, E = J = [, τ max, J h = [ + h, τ max, τ max =, T > 0, 0 < h < T, E = J = [, + T ], J h = [ + h, + T ], 3 τ max =, h = 1, E =, τ 0 +, J = {τ R: τ, τ T }, J h = {τ R: + 1 τ, τ T }. 3.8 Proposition 3.6. Let q l p l, l = 1,..., and u = u 1,..., u be an X 1 solution of 3.1 on [, τ max. We have for τ J F τ, uτ [L ] c 4 u l +q λ1ν l L +q e τ + P gs l [L ], 3.9 where c 4 = c 4 h > 0 is a constant and P = P h is a nondecreasing positive function in any of the three cases stated in 3.8. 16

Proof. We have We estimate using 3.1 fu [L ] c F τ, u [L ] fu [L ] + gτ [L ]. 3.30 c 1 1 + u l L +p l u l +p l L +p l 1 + u l p l + f0 L +p l [L ] c 1 + u l +q l L +q l Combining it with 3.30 and applying 3.18 on an appropriate interval J with the corresponding set E, we obtain 3.9. Proposition 3.7. Let q l p l, l = 1,..., and u = u 1,..., u be an X 1 solution of 3.1 on [, τ max. Then we have for τ J. uτ [L ] c 9 u l +q λ1ν l L +q e 4 τ + Q gs l [L ], 3.31 and for τ J h τ h u l s L ds c 9 u l +q λ1ν l L +q e 4 τ + Q gs l [L ], 3.3 where c 9 = c 9 h > 0 is a constant, Q = Qh is a nondecreasing positive function and J, J h and E come from each of the three cases in 3.8. Proof. For each l = 1,..., we multiply the l-th equation in 3.1 by u l, integrate over and add the equations. Integrating by parts and using the Schwarz inequality we obtain 1 t ut [L ] 3 i,j=1 a l ij xi u l xj u l dx By 3.3 and the Cauchy inequality we get for any ε > 0 1 t ut [L ] + ν F l t, u L u l L. u l t L ε ut [L ] + 1 ε F t, u [L ]. 3.33 17

By the Poincaré inequality we obtain t ut [L ] + λ 1 ν ut [L ] ε ut [L ] + 1 ε F t, u [L ] Taing ε = 7λ 4 1ν, multiplying by e λ 1 ν 4 t and integrating over [, τ] gives uτ [L ] u [L ] e λ 1 ν 4 τ + 4 7λ 1 ν F t, u [L ] e λ 1 ν 4 τ t dt. Let τ J and E be the corresponding set from 3.8. We apply 3.9 and obtain uτ [L ] u [L ] + c 5 u l +q l L +q l e λ 1 ν 4 τ + c 6 P gs [L ] with c 5 = c 5 h > 0, c 6 = c 6 h > 0. This yields 3.31, since u l L u l +q l L +q l +. Assume now that τ J h. Integrating 3.33 with ε = 1 over [τ h, τ] we get uτ [L ] + ν uτ h [L ] + τ h Using 3.31 and 3.9 we obtain for τ J h u l t L dt c 7 τ h + Q gs [L ] c 8 τ h u l t L dt ut [L ] + F t, u [L ] dt. u l +q l L +q l u l +q λ1ν l L +q e 4 τ + Q l e λ 1 ν 4 τ + e λ 1 ν 4 t dt + τ h gs [L ], where c 7 = c 7 h > 0, c 8 = c 8 h > 0 and Q = Qh is a nondecreasing positive function. This gives 3.3. Proposition 3.8. Let p l q l 4, l = 1,..., and u = u 1,..., u be an X 1 solution of 3.1 on [, τ max. Then we have for τ J u l τ L R 1 u l L e λ 1 ν 4 τ + R gs [L ], 3.34 where R 1 = R 1 h, R = R h are both nondecreasing positive functions and h, J and E come from each of the three cases in 3.8. 18

Proof. For each l = 1,..., we multiply the l-th equation in 3.1 by A l u l, integrate over and add the obtained equations. Since a l ij do not depend on time and a l ij = a l ji, integration by parts and the Schwarz inequality imply 1 3 i,j=1 t a l ij xi u l xj u l dx + We add to both sides a term with and obtain z u t = A l u l L 1 F t, u [L ] + 1 3 i,j=1 a l ij xi u l xj u l dx A l u l L. t z u t + λ 1 νz u t + A l u l L λ 1νz u t + F t, u [L ]. 3.35 Since the functions a l ij are continuous on, we now that a l ijx max a l ijx = C a. Therefore, it follows from 3.3 that i,j=1,...,3,..., x ν u l L z ut 3C a u l L. 3.36 Applying 3.36 to 3.35, multiplying by e λ 1νt and integrating over [, τ], we obtain z u τ z u e λ 1ντ + 3λ 1 νc a u l t L e λ 1ντ t dt+ + F t, u [L ] e λ1ντ t dt. Let τ J and E be the corresponding set from 3.8. We apply 3.9, 3.36 and get u l τ L 3C a ν u l L e λ 1ντ + 1 λ 1 ν P gs [L ] + 19

+c 4 λ 1 ν u l +q λ τ 1ν l L +q e τ + 3λ l 1 C a u l t L e λ 1ντ t dt. We consider now two cases. In the first case we assume that τ belongs to J h corresponding to the appropriate case in 3.8. We use Lemma 3.3 and 3.3 to estimate τ t u l t L e λ 1ντ t dt c 10 e λ 1 ν τ t u l s L ds c 11 t [+h,τ] u l +q λ1ν l L +q e 4 τ + c l 10 Q gs [L ], τ Jh, where c 10 = c 10 h > 0 and c 11 = c 11 h > 0. In the second case when τ + h, τ J, we have by 3.3 u l t L e λ 1ντ t dt c 9 u l +q λ1ν l L +q e 4 τ +Q gs l [L ]. Combining the two cases we obtain u l τ L c 1 u l L + u l +q l e λ 1 ν L +q 4 τ + l +R gs [L ], τ J, where c 1 = c 1 h is a positive constant and R = Rh is a nondecreasing positive function. Since q l 4, l = 1,..., and u l H 1 0, it follows from the Sobolev embedding and the Poincaré inequality that u l L +q l D u l L. This ends the proof of 3.34. Propositions 3.7 and 3.8 imply global forward in time solvability of 3.1, thus F3a holds. Moreover, assumptions F3b and F3c are also satisfied. Corollary 3.9. If p l q l 4, l = 1,..., and u = u 1,..., u is an X 1 solution of 3.1 on [, τ max, then τ max = and for τ, τ T we have uτ [H 1 0 ] Q 1 u [H 1 0 ] e λ 1 ν 8 τ + Q 0 s,τ 0 + gs [L ], 3.37

where Q 1, Q are both nondecreasing positive functions and for any T > 0 there exist nondecreasing positive functions Q 1 = Q 1 T, Q = Q T such that for τ + T uτ [H 1 0 ] Q 1 u [H 1 0 ] e λ 1 ν 8 τ + Q s [,+T ] gs [L ]. 3.38 Proof. The fact that X 1 solutions of 3.1 exist globally forward in time follows from 3.13 and Propositions 3.7 and 3.8 in the context of the first case in 3.8, while 3.37 and 3.38 correspond to the second and the third case in 3.8, respectively. Therefore, we can apply Theorem 1.1 and obtain Theorem 3.10. Under assumptions 3.4, 3.5 with 0 p l q l 4, l = 1,..., and assumptions 3.6, 3.7 the problem 3.1 generates an evolution process {Uτ, : τ } in [H0] 1 and for any β 1, 1 there exists a family {Mτ: τ R} of nonempty compact subsets of [H β 0 ] with the following properties i {Mτ: τ R} is positively invariant under the process Uτ,, i.e. Uτ, M Mτ, τ, ii Mτ has a finite fractal dimension in [H β 0 ] uniformly w.r.t. τ R, i.e. d [H1 0 ] f Mτ d [Hβ 0 ] f Mτ d <, τ R, iii {Mτ: τ R} has the property of pullbac exponential attraction, i.e. ϕ>0 B1 [H β 0 ], bounded τ R lim t e ϕt dist [H β 0 ] Uτ, τ tb 1, Mτ = 0 and if τ 0 =, the pullbac attraction is uniform with respect to τ ϕ>0 B1 [H β 0 ], bounded lim t e ϕt τ R dist [H β 0 ] Uτ, τ tb 1, Mτ = 0. Furthermore, the pullbac exponential attractor {Mτ: τ R} contains a finite dimensional pullbac global attractor {Aτ: τ R}, i.e. a family of nonempty compact subsets of [H β 0 ], invariant under the process {Uτ, : τ } Uτ, A = Aτ, τ, and pullbac attracting all bounded subsets of [H β 0 ] B1 [H β 0 ], bounded τ R lim t dist [H β 0 ] Uτ, τ tb 1, Aτ = 0. Remar 3.11. Observe that the assumptions of the above theorem are satisfied in case of the FitzHugh-Nagumo system and the chemical reaction system considered in Remar 3.1. 1

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