Pullback attractor for a non-autonomous reaction-diffusion equation in some unbounded domains

Bol. So . Esp. Mat. Apl. n o 51(2010), 917 PULLBACK ATTRACTOR FOR A NON-AUTONOMOUS REACTION-DIFFUSION EQUATION IN SOME UNBOUNDED DOMAINS MARÍA ANGUIANO Dpto. Euaiones Difereniales y Análisis Numério Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla (Spain) anguianous.es Abstrat The existene of a pullbak attrator in L2(Ω) for the following nonautonomous reation-diusion equation 8 > < > : ∂u ∂t − △u=f(u) + h(t), in Ω×(τ, +∞) , u= 0, on ∂Ω×(τ, +∞) , u(x, τ) = uτ(x), x ∈Ω , (1) is proved in this paper, when the domain Ω is not neessarily bounded but satisfying the Poinaré inequality, and h∈L2 loc(R;H−1(Ω)) . The main onept used in the pro of is the asymptoti ompatness of the pro ess generated by the problem. Key words: pul lbak attrator, asymptoti ompatness, evolution proess, non-autonomous reation-diusion equation. AMS sub jet lassiations: 35B41, 35Q35, 35Q30, 35K90, 37L30. 1 Intro dution and setting of the problem Let Ω⊂RN b e an op en set, not neessarily bounded and suppose that Ω satises the Poinaré inequality, i.e., there exists a onstant λ1>0 suh that ZΩ|u(x)|2dx ≤λ−1 1ZΩ|∇u(x)|2dx, ∀u∈H1 0(Ω) . (2) Let us onsider the following problem for a non-autonomous reationdiusion equation with zero Dirihlet b oundary ondition in Ω ,      ∂u ∂t −△u=f(u) + h(t), in Ω×(τ, +∞) , u= 0, on ∂Ω×(τ, +∞) , u(x, τ) = uτ(x), x ∈Ω , (3) 9 10 M. Anguiano where τ∈R , uτ∈L2(Ω) , h∈L2 loc(R;H−1(Ω)) and f∈C(R) satises that there exist onstants α1>0 , α2>0 , l≥0 , and p > 2 suh that −α1|s|p≤f(s)s≤ −α2|s|p , (4) (f(s)−f(r))(s−r)≤l(s−r)2∀r, s ∈R . (5) The aim of this paper is to show the existene of a pullbak attrator in the phase spae L2(Ω) for the problem (3) in the ase of op en domains not neessarily b ounded but satisfying the Poinaré inequality. This, and the fat that the nonautonomous h b elongs to the spae L2 loc(R;H−1(Ω)), are the main novelties of our problem. The lak of ompatness of the injetion H1 0(Ω) ⊂L2(Ω) (in the ase of unb ounded domains) implies that the standard tehniques previously used, partiularly the one involving the so-alled atenning prop erty (see [6℄, [7℄, [12℄, [14℄, amongst others), whih have b een suessfully used when Ω is b ounded and h∈L2 loc(R;L2(Ω)) , do not work in our ase. Instead, we will use the asymptoti ompatness already used in the ase of non-autonomous 2D-Navier-Stokes (see [1℄ and [2℄, see also [5℄ for a lose result), and whih was previously used in [11℄ for the autonomous ase. We would like to emphasize that this tehnique seems to be the only one whih allows to prove the main result of this pap er (namely Theorem 4) onerning the existene of pullbak attrator for our problem. It is also worth mentioning that our problem has reeived muh attention over the last years in the ase of a b ounded domain or for a less general term h (see [3℄, [7℄, [12℄, [14℄). Finally, the reader an nd similar results for several variants of our model in the referenes [9℄, [10℄, among others. 2 Existene and uniqueness of solution We state in this setion a result on the existene and uniqueness of solution of problem (3). By |·| we denote the norm in L2(Ω) , by |∇·| the norm in H1 0(Ω) and by k·k∗ the norm in H−1(Ω) . We will use (·,·) to denote the salar pro dut in L2(Ω) and we will use h·,·i to denote the duality pro dut b etween H−1(Ω) and H1 0(Ω) . Theorem 1 Suppose that Ω satises (2). Assume that f∈C(R) satises (4) and (5), and h∈L2 loc(R;H−1(Ω)) . Then, for al l τ∈R , uτ∈L2(Ω) , there exists a unique solution u(t) = u(t;τ, uτ) of (3) suh that u∈L2(τ, T ;H1 0(Ω)) ∩Lp(τ, T ;Lp(Ω)) ∀T > τ, d dt(u(t), v)−h∆u(t), vi=hf(u(t)), vi +hh(t), vi, in D′(τ, ∞),∀v∈H1 0(Ω) ∩Lp(Ω) , u(τ) = uτ. A reation-diusion equation in some unb ounded domains 11 Moreover, u∈C([τ, ∞); L2(Ω)), and u satises the energy equation, 1 2 d dt|u(t)|2+|∇u(t)|2=hf(u(t)), u(t)i +hh(t), u(t)i in D′(τ, ∞). (6) Proof . The pro of of this Theorem an b e done by the metho d of monotony (see [8℄).  3 Preliminaries on the theory of pullbak attrators Now, we will reall the main p oints from the theory of pullbak attrators whih will b e needed to prove our ob jetive (see [1℄ and [2℄ for more details). Let us onsider a pro ess (also alled a two-parameter semigroup) U on a metri spae X , i.e., a family {U(t, τ); −∞ < τ ≤t < +∞} of ontinuous mappings U(t, τ) : X→X , suh that U(τ, τ)x=x, and U(t, τ) = U(t, r)U(r, τ) for all τ≤r≤t. (7) Supp ose that D is a nonempty lass of parameterized sets b D={D(t); t∈R} ⊂ P(X), where P(X) denotes the family of all nonempty subsets of X . Denition 1 The proess U(·,·) is said to be pul lbak D -asymptotial ly ompat if for any t∈R , any b D∈ D, any sequene τn→ −∞, and any sequene xn∈D(τn) , the sequene {U(t, τn)xn} is relatively ompat (i.e. pre-ompat) in X . Denition 2 It is said that b B∈ D is pul lbak D -absorbing for the proess U(·,·) if for any t∈R and any b D∈ D , there exists a τ0(t, b D)≤t suh that U(t, τ)D(τ)⊂B(t) for al l τ≤τ0(t, b D). Denition 3 The family b A={A(t); t∈R} ⊂ P(X) is said to be a pul lbak D -attrator for U(·,·) if 1. A(t) is ompat for al l t∈R , 2. b A is pul lbak D -attrating, i.e., lim τ→−∞dist(U(t, τ)D(τ), A(t)) = 0, for al l b D∈ D , and al l t∈R , 3. b A is invariant, i.e., U(t, τ)A(τ) = A(t), for −∞ < τ ≤t < +∞. 12 M. Anguiano We have the following result (see [2℄ for more details). Theorem 2 Suppose that the proess U(·,·) is pul lbak D -asymptotial ly ompat and that b B∈ D is a family of pul lbak D -absorbing sets for U(·,·) . Then, the family b A={A(t); t∈R} ⊂ P(X) dened by A(t) = Λ( b B, t), t ∈ R, where for eah b D∈ D Λ( b D, t) = \ s≤t [ τ≤s U(t, τ)D(τ) , is a pul lbak D -attrator for U(·,·) whih satises in addition that A(t) = Sb D∈D Λ( b D, t), for t∈R. Furthemore, b A is minimal in the sense that if b C={C(t); t∈R} ⊂ P(X) is a family of losed sets suh that limτ→−∞ dist(U(t, τ)B(τ), C(t)) = 0, then A(t)⊂C(t) . 4 Existene of the pullbak attrator Now, we an prove our main result in this pap er. First, we need a ontinuity result whih is established in the next subsetion. 4.1 Weak Continuity Assume that the funtion f∈C(R) satises (4) and (5), and that h∈ L2 loc(R;H−1(Ω)) . Thanks to Theorem 1, we an dene a pro ess {U(t, τ) , τ≤t} in L2(Ω) , as U(t, τ)uτ=u(t;τ, uτ)∀uτ∈L2(Ω) ,∀τ≤t . (8) From the uniqueness of solution to problem (3), it follows that (8) denes a pro ess in L2(Ω) . In addition, it an b e proved that the pro ess dened by (8) is ontinuous in L2(Ω) . Moreover, U is weakly ontinuous, and more exatly the following result holds true. We will denote by  ⇀  the weak onvergene in the orresp onding indiated spae, while  →  will denote the strong onvergene, as usual. Prop osition 3 Let {uτn} ⊂ L2(Ω) be a sequene onverging weakly in L2(Ω) to an element uτ∈L2(Ω) . Then, for al l T > τ , it fol lows U(t, τ)uτn⇀ U (t, τ)uτ in L2(Ω) ∀t≥τ , (9) U(·, τ)uτn⇀ U (·, τ)uτ in L2(τ, T ;H1 0(Ω)), (10) U(·, τ)uτn⇀ U (·, τ)uτ in Lp(τ, T ;Lp(Ω)) , (11) f(U(·, τ)uτn)⇀ f (U(·, τ)uτ) in Lp′(τ, T ;Lp′(Ω)). (12) If Ω is a bounded set, then U(·, τ)uτn−→ U(·, τ)uτ in L2(τ, T ;L2(Ω)) . (13) A reation-diusion equation in some unb ounded domains 13 Proof . This result may be proved in muh the same way as Theorem 1, and using similar arguments to [11℄.  4.2 The existene of the global pullbak attrator Let Rλ1 b e the set of all funtions r:R→(0,+∞) suh that lim t→−∞eλ1tr2(t) = 0 , and denote by Dλ1 the lass of all families b D={D(t) : t∈R} ⊂ P(L2(Ω) ) suh that D(t)⊂B(0, r b D(t)) , for some rb D∈ Rλ1 , where B(0, r b D(t)) denotes the losed ball in L2(Ω) entered at zero with radius rb D(t) . Now, we an prove the following result. Theorem 4 Suppose that Ω satises (2), and suppose that f∈C(R) satises (4) and (5) with l= 0 . Let h∈L2 loc(R;H−1(Ω)) be suh that Zt −∞ eλ1skh(s)k2 H−1(Ω) ds < +∞ ∀t∈R . Then, there exists a unique global pul lbak Dλ1 -attrator for the proess U , whih belongs to Dλ1 , and is dened by (8). Proof . We only give the main ideas of the pro of. Let τ∈R , and uτ∈L2(Ω) b e xed, and denote u(t) = u(t;τ, uτ) = U(t, τ)uτ∀t≥τ . Let b D∈ Dλ1 b e given. Taking into aount (2), (4), the energy equality and integrating b etween τ and t , |U(t, τ)uτ|2≤e−λ1tZt −∞ eλ1skh(s)k2 H−1(Ω) ds +eλ1(τ−t)r2 D(τ) , (14) for all uτ∈D(τ) and for all t≥τ . Denote by Rλ1(t) the nonnegative number given for eah t∈R by R2 λ1(t) = e−λ1tZt −∞ eλ1skh(s)k2 H−1(Ω) ds + 1 . (15) Observe that Rλ1∈ Rλ1 . Now, onsider the family b Bλ1 of losed balls in L2(Ω) , b Bλ1={Bλ1(t) : t∈R} , dened by Bλ1(t) = v∈L2(Ω) : |v| ≤ Rλ1(t) . It is straightforward to hek that b Bλ1∈ Dλ1, and moreover, by (14), the family b Bλ1 is pullbak Dλ1 -absorbing for the pro ess U . Aording to Theorem 2, to nish the pro of of the theorem we only have to prove that U is pullbak Dλ1 -asymptotially ompat. 14 M. Anguiano Let us x b D∈ Dλ1 , a sequene τn→ −∞ , a sequene uτn∈D(τn) , and t∈R . We have to prove that from the sequene {U(t, τn)uτn} we an extrat a subsequene that onverges in L2(Ω) . As the family b Bλ1 is pullbak Dλ1 -absorbing, by a diagonal pro edure, it is not diult to onlude that there exist a subsequene τn′, uτn′⊂ {(τn, uτn)} , and a sequene {wk;k≥0} ⊂ L2(Ω) suh that for all k≥0 , and wk∈ Bλ1(t−k) , U(t−k, τn′)uτn′⇀ wk in L2(Ω) , (16) and |w0| ≤ lim inf n′→∞ U(t, τn′)uτn′ . (17) If we now prove that also lim sup n′→∞ U(t, τn′)uτn′≤ |w0| , (18) then we will have lim n′→∞U(t, τn′)uτn′=|w0| . And this, together with the weak onvergene, will imply the strong onvergene in L2(Ω) of U(t, τn′)uτn′ to w0 . In order to prove (18), onsider [u] := |∇u|2−λ1 2|u|2−hf(u), ui . Taking into aount (2), (4), the energy equality and integrating b etween τ and t , it is immediate that for all k≥0 and all τn′≤t−k , U(t, τn′)uτn′2 (19) =U(t−k, τn′)uτn′2e−λ1k + 2 Zt t−k eλ1(s−t)h(s), U(s, t −k)U(t−k, τn′)uτn′ds −2Zt t−k eλ1(s−t)U(s, t −k)U(t−k, τn′)uτn′ds . Now we will prove that Zt t−k eλ1(s−t)[U(s, t −k)wk]ds (20) ≤lim inf n′→∞ Zt t−k eλ1(s−t)U(s, t −k)U(t−k, τn′)uτn′ds . Denote Jk(v) = J(1) k(v) + J(2) k(v) , where J(1) k(v) = Zt t−k eλ1(s−t)|∇v(s)|2−λ1 2|v(s)|2ds , A reation-diusion equation in some unb ounded domains 15 and J(2) k(v) = −Zt t−k eλ1(s−t)hf(v), vids , for all v∈L2(t−k, t;H1 0(Ω)) ∩Lp(t−k, t;Lp(Ω)) . We also obtain from (16) and using Prop osition 3 lim inf n′→∞ (J(1) k(U(·, t −k)U(t−k, τn′)uτn′) ≥J(1) k(U(·, t −k)wk) . (21) Using (5) with l= 0 , from (16) and Prop osition 3 we easily obtain lim n′→∞inf J(2) k(U(·, t −k)U(t−k, τn′)uτn′) ≥J(2) k(U(·, t −k)wk) . Therefore (20) is easily obtained from the last inequality and (21). Then, taking into aount that the family b Bλ1 is pullbak Dλ1 -absorbing, from (16), using Prop osition 3 and thanks to (19) and (20), we obtain lim n′→∞sup U(t, τn′)uτn′2 ≤R2 λ1(t−k)e−λ1k+|w0|2 , for all k≥1. Taking into aount (15), we easily obtain lim n′→∞sup U(t, τn′)uτn′2≤ |w0|2 .  Aknowledgements. This work has b een partially supp orted by Junta de Andaluía under pro jet P07-FQM-02468. Referenes [1℄ T. Caraballo, G. Lukaszewiz, and J. Real, Pul lbak attrators for asymptotial ly ompat non-autonomous dynamial systems . Nonlinear Analysis, 64 (2006), 484498. [2℄ T. Caraballo, G. Lukaszewiz, and J. Real, Pul lbak attrators for nonautonomous 2D Navier-Stokes equations in unbounded domains . C. R. Math. Aad. Si. Paris, 342 (2006), 263268. [3℄ T. Caraballo, J. A. Langa, and J. Valero, Dimension of attrators of nonautonomous reation-diusion equations . ANZIAM J., 45 (2003), 207 222. [4℄ H. Crauel, A. Debusshe, and F. Flandoli, Random attrators . J. Dyn. Di. Eq., 9 (1995), no. 2, 307341. 16 M. Anguiano [5℄ M.J. Garrido-Atienza and P. Marín-Rubio, Navier-Stokes equations with delays on unbounded domains . Nonlinear Anal., 64 (2006), no. 5, 1100 1118. [6℄ P.E. Klo eden and J.A. Langa, Flatenning, squeezing and the existene of random attrators . Pro . Roy. So . Lond. Series A, 463 (2007), 163181. [7℄ Y. Li and C.K. Zhong, Pul lbak attrators for the norm-to-weak ontinuous proess and appliation to the nonautonomous reation-diusion equations . Applied Mathematis and Computation, 190 (2007), 10201029. [8℄ J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires . Duno d, Paris 1969. [9℄ F. Morillas and J. Valero, Attrators for reation-diusion equations in RN with ontinuous nonlinearity . Asymptoti Analysis, 44 (2005), 111130. [10℄ M. Prizzi, A remark on reation-diusion equations in unbounded domains . Disrete Contin. Dyn. Syst., 9 (2003), 281286. [11℄ R. Rosa, The Global Attrator for the 2D Navier-Stokes ow on some unbounded domains . Nonlinear Anal., 32 (1998), no. 1, 7185. [12℄ H. Song and H. Wu, Pul lbak attrators of nonautonomous reationdiusion equations . J. Math. Anal. Appl., 325 (2007), 12001215. [13℄ R. Temam, Navier-Stokes equations and nonlinear funtional analysis . CBMS-NSF Regional Conferene Series in Applied Mathematis, Vol. 66. SIAM, Philadelphia, 1983 (2nd edition, 1995). [14℄ Y. Wang and C. Zhong, On the existene of pul lbak attrators for nonautonomous reation-difusion equations . Dynamial Systems, 23 (2008), 116.