Asymptotic behaviour of nonlocal reaction-diffusion equations

Asymp o ic beha iou o nonlocal eac ion-di usion equa ions∗ Ma ´ıa Anguiano Dp o. de Ecuaciones Di e enciales y An´alisis Num´e ico, Uni e sidad de Se illa, Apdo. de Co eos 1160 41080 Se illa, Spain E-mail: [email p o ec ed] P.E. Kloeden and T. Lo enz Fachbe eich Ma hema ik, Johann Wol gang Goe he Uni e si ¨a D-60054 F ank u am Main, Ge many E-mail: kloeden / [email p o ec ed] Abs ac The exis ence o a global a ac o in L2(Ω) is es ablished o a eac ion- di usion equa ion on a bounded domain Ω in Rdwi h Di ichle bounda y condi ions, whe e he eac ion e m con ains an ope a o F:L2(Ω) → L2(Ω) which is nonlocal and possibly nonlinea . Exis ence o weak solu- ions is es ablished, bu uniqueness is no equi ed. Compac ness o he mul i alued low is ob ained ia es ima es ob ained om limi s o Gale kin app oxima ions. In con as wi h he usual si ua ion, hese limi s apply o all and no jus o almos all ime ins an s. 1 In oduc ion A simple popula ion model wi h spa ial dependence is gi en by he eac ion- di usion equa ion ∂u ∂ = ∆u+u(1 −u) , (1) on a bounded domain Ω in Rdwi h Di ichle bounda y condi ions. The long e m dynamics o his model is well unde s ood. In he abo e model he popula ion a a poin x∈Ω depends only on i s alue a his poin , apa om he di usi i y e m. Mo e ealis ically, i could ∗Pa ially suppo ed by he Minis e io de Ciencia y Tecnolog´ıa (Spain) and FEDER (Eu- opean Communi y) g an BFM2002-03068 as well as he DFG g an s KL1203/7, LO273/5) 1 depend on he popula ion size a o he poin s, in pa icula ly a nea by poin s. Fo example, i could depend on he a e age o e a small neighbou hood uδ( , x) = ZB(x;δ) u( , y)dy ZB(x;δ) dy , o some small δ > 0 ins ead o on u( , x) i sel . This leads o a nonlocal PDE ∂u ∂ = ∆u+uδ(1 −uδ) . (2) Al e na i ely, uδ( , x) could be some o he unc ional o he solu ion u(·, ) e alua ed a he poin x. The e a e many applica ions o nonlocal e ec s in pa ial di e en ial equa- ions in he li e a u e, e.g., in combus ion heo y [13] and he Na ie -S okes equa ions [5]. The nonlocal e m is o en an in eg al ope a o and he equa ions a e hen called “in eg o-di e en ial” equa ions. Bol zmann equa ions a e a e y well known class o in eg o-di e en ial, bu a e i s o de unlike hose o in e es he e. Howe e , he nonlocal e m could be di e en , see, e.g., he e iew a icle by Ba es [3]. The e is a la ge li e a u e on he exis ence, egula i y and blow–up o solu ions o nonlocal e olu ion equa ions, see o example [15, 16, 17] and he pape s ci ed he ein. Global a ac o s o nonlocal e olu ion equa ions ha e been in es iga ed ecen ly o he globally modi ied Na ie -S okes equa ions by Ca aballo e al. [5], o m-Laplacian pa abolic equa ions wi h a nonlocal nonlinea i y by Chen [6] and by Hilho s e al [8] o a nonlocal Ku amo o–Si ashinsky equa ion. Se e al aspec s o eac ion-di usion equa ions a e being analyzed o e he las yea s, pa icula ly, hei asymp o ic beha iou , see o example [14,15] and [17]. In his pape we conside gene al nonlinea nonlocal e ms in au onomous eac ion- di usion equa ions, which gene a e s ic mul i alued semi lows. In pa icula , we es ablish he exis ence o a global a ac o a e i s p o ing weak solu ions and he compac ness o a ainabili y se s o he mul i alued semi low. Fo his we use es ima es ob ained as limi s o Gale kin app oxima ions which hold o e e y ime ins an and no jus o almos all ime ins an s. The p oblem is o mula ed in he nex sec ion and dissipa i i y es ima es a e p esen ed in Sec ion 3 wi h some longe p oo s gi en a he end o he pape in Sec ion 7. The exis ence o weak solu ions is es ablished in Sec ion 4, while he gene a ion o a s ic mul i alued semi low and he exis ence o a global a ac o a e shown in Sec ion 5. Finally an explici example is p esen ed in Sec ion 6. 2 Se ing o he p oblem Le Ω ⊂RNbe a bounded open se , i sa is ies he Poinca ´e inequali y, i.e., he e exis s a cons an λΩ>0 such ha ZΩ u2(x)dx ≤λ−1 ΩZΩ (∇u(x))2dx,∀u∈H1 0(Ω).(3) 2 Le (·,·) deno e he scala p oduc in L2(Ω) and k·kL2(Ω) he co esponding no m in L2(Ω). In addi ion, le h·,·i deno e he duali y p oduc be ween spaces H1 0(Ω) and H−1(Ω). Conside he ollowing ini ial bounda y alue p oblem o a nonlocal eac ion– di usion equa ion wi h ze o Di ichle bounda y condi ion in Ω,      ∂u ∂ +A u =F(u) in Ω ×(0, T), u= 0 on ∂Ω×[0, T], u(x, 0) = u0(x), o x∈Ω, (4) whe e Ais a uni o mly pa abolic ope a o in di e gence o m wi h aij =aji ∈ L∞(Ω), 1 ≤i, j ≤N, o which he e exis cons an s λA, ΛA>0 such ha λA|η|2≤ N X i,j=1 aij(x)ηiηj≤ΛA|η|2 and Au(x) := − N X i,j=1 ∂xjaij(x)∂xiu(x)(5) o all x∈Ω and η∈RN. Rema k 1 The ope a o induced by Acan be in e p e ed as A∈ LH1 0(Ω), H−1(Ω) and is symme ic wi h hA , i ≥ λAk∇ k2 L2(Ω) o all ∈H1 0(Ω). Since H1 0(Ω) is included in L2(Ω) wi h compac injec ion, as a consequence o he Hilbe -Schmid Theo em he e exis s a nondec easing sequence o posi i e eal numbe s, 0< λ1≤λ2≤... ≤λk≤......, wi h limn→∞ λn= +∞and he e exis s an o hono mal basis {wk:k≥1}o L2(Ω). Mo eo e , {wk:k≥1}is an o hogonal basis o H1 0(Ω) wi h A wk= λkwk o all k≥1, whe e (u, )H1 0(Ω) De . =hAu, i. The ope a o F:L2(Ω) →L2(Ω) ul ills he ollowing assump ions: a) Fis con inuous wi h espec o he L2no m. 3 b) he e exis β∈(0, λAλΩ) and Cβ>0 wi h (u, F(u)) ≤βkuk2 L2(Ω) +Cβ o e e y u∈L2(Ω),(6) c) he e is some nondec easing Ψ : [0,∞)→Rsuch ha o all u∈L2(Ω), kF(u)k2 L2(Ω) ≤ΨkukL2(Ω)(7) In special cases Fwill also be assumed o sa is y a local Lipschi z condi ion: d) Fo all R > 0 he e exis s LRsuch ha i , w ∈L2(Ω) wi h k kL2(Ω) ≤R, kwkL2(Ω) ≤R, hen kF( )−F(w)kL2(Ω) ≤LRk −wkL2(Ω) . (8) Rema k 2 In he subsequen s a emen s and p oo s, condi ion (c) can be easily eplaced by he sligh ly weake assump ion ha he e is a nondec easing unc ion e Ψ : [0,∞)→Rsuch ha o e e y u∈H1 0(Ω) wi h Au ∈H1 0(Ω), |hF(u), A ui| ≤ e ΨkukL2(Ω).(9) kF(u)k2 L2(Ω) ≤e ΨkukH1,2(Ω).(10) 3 Dissipa i i y es ima es Bo h he exis ence o weak solu ions and o bounded abso bing se s in L2(Ω), H1 0(Ω), espec i ely, a e based on he ollowing a p io i es ima es. Thei p oo s do no equi e he uniquenss o weak solu ions o a gi en ini ial alue. Some o he auxilia y esul s a e o mula ed o Gale kin app oxima ions, and hei (qui e echnical) p oo s a e pos poned o §7. P oposi ion 3 I Fsa is ies hypo hesis (b), hen e e y weak solu ion u∈ L20, T;H1 0(Ω)o (4) wi h u0∈L20, T;H−1(Ω) ul ills he es ima es  u( ) 2 L2(Ω) ≤M L+e−L  u(0) 2 L2(Ω) λAZ 0k∇uk2 L2(Ω) ds ≤1 2+β L u(0) 2 L2(Ω) +Cβ+βM L o e e y ∈[0, T]wi h he cons an s M:= 2 Cβ>0and L:= 2 λΩλA−2β > 0. The p oo is gi en in §7. 4 P oposi ion 4 Suppose ha condi ions (a)–(c)hold o Fand ha T < ∞. Then he e exis s posi i e cons an s C0,C1and C2depending only on β,Cβ, ΛA,λAand λΩsuch ha e e y weak solu ion u∈L20, T ;H1 0(Ω)o (4) wi h u0∈L20, T;H−1(Ω)and ku0kL2(Ω) ≤ρsa is ies he a p io i es ima es ku(s)k2 L2(Ω) ≤C1+e−C0sρ2  ∇u( ) 2 L2(Ω) ≤C1·max{1,1 }· 1+e−C2 ρ2+ ΨC1·(1 + e−C2 ρ2) o e e y 0≤s < ≤T. The p oo , gi en below, uses analogous es ima es o he Gale kin app oxima- ions in he ollowing lemma, which is p o ed in he §7. Lemma 5 Le {wk:k≥1}be an o hogonal basis o H1 0(Ω) as in Rema k 1. Fo each n∈N, suppose ha un( ) = n X k=1 unk( )·wkis a solu ion o    d d (un( ), wk) + hAun( ), wki= (F(un( )), wk) (un(0), wk) = (u0, wk), k = 1 . . . n. (11) I F:L2(Ω) →L2(Ω) sa is ies he hypo heses (b)and (c), hen he e exis posi i e cons an s C1,C2and C3depending only on β,Cβ,ΛA,λAand λΩ such ha whene e ku0kL2(Ω) ≤ρ, he ollowing es ima es holds o e e y s0,s, ∈(0, T]wi h 0< s0≤s≤  ∇un( ) 2 L2(Ω) ≤C1·max 1,1 ·1+e−C2 ρ2+ ΨC1(1 + e−C2 ρ2) Z sku0 n(ξ)kL2dξ ≤C3·√ −s·cons (s0, T, ρ). ku0 n( )kH−1(Ω) ≤cons (β, Cβ,ΛA, λA, λΩ, ρ). P oo o P oposi ion 4. The inclusions u∈L20, T;H1 0(Ω)and u0∈ L20, T;H−1(Ω)always imply ha u∈C0[0, T ]; L2(Ω), see [7, §5.9]. Le {wk:k≥1}be an o hogonal basis o H1 0(Ω) as in Rema k 1. Then o each n∈N, un: [0, T]→H1 0(Ω), 7−→ n X k=1 u( ), wkwk is induced by he o hogonal p ojec ion o u( ) on span{w1. . . wn}in L2(Ω). I sol es he pe u bed nonlocal Gale kin p oblem (d d (un( ), wk) + hAun( ), wki= (G( ), wk) (un(0), wk) = (u0, wk), k = 1 . . . n, (12) wi h he map G: [0, T]→L2(Ω) de ined by G( ) := F(u( )) o each ∈[0, T], which depends only on ime in combina ion wi h he weak solu ion u(·) (bu no 5 on unexplici ly). In pa icula , G ul ills he condi ions (a)–(c) (uni o mly wi h espec o ime). Hence, Lemma 5 p o ides a p io i es ima es o each Gale kin app oxima ion un(·) depending essen ially only on he L2no m o he ini ial alue u0. Since {wk, k ≥1}is an o hono mal basis o L2(Ω), he sequence (un( ))n∈N con e ges o u( ) in L2(Ω) a each ime ∈[0, T]. The L2bound o he H1 0 no m implies a weakly con e gen subsequence o (un)n∈Nin L20, T;H1 0(Ω) and, i s weak limi is u∈L20, T ;H1 0(Ω)(again). Fo each s0∈(0, T), we e en ha e a L∞bound o he H1 0no ms o (un)n∈Nin [s0, T] and so Lemma 7 below gua an ees he same es ima es holds o  ∇u( ) 2 L2(Ω) a e e y ime ∈ [s0, T], no jus o Lebesgue-almos all such . Fixing s0∈(0, T) a bi a ily, he Gale kin app oxima ions o any weak solu ion u(·) wi h ku(0)kL2(Ω) ≤ρa e equi–con inuous w. . . L2(Ω) in [s0, T ] due o he second es ima e in Lemma 5. Now he poin wise con e gence o he Gale kin app oxima ions o u(·) implies he ollowing s a emen di ec ly: Lemma 6 Suppose ha he condi ions (a)–(c)hold o Fand ha T < ∞. Fo e e y ρ > 0, he subse o C0[0, T ]; L2(Ω)consis ing o all weak solu ions u∈L20, T;H1 0(Ω)o (4) wi h u0∈L20, T;H−1(Ω)and ku(0)kL2(Ω) ≤ρ is equi-con inuous in he subin e al [s0, T] o e e y s0∈(0, T). Lemma 7 Le X, Y be Banach spaces such ha Xis e lexi e, and he in- clusion X⊂Yis con inuous. Assume ha (un)n∈Nis a bounded sequence in L∞( 0, T;X)such ha un* u weakly in Lp( 0, T;X) o some p∈[1,+∞) and u∈C0([ 0, T]; Y). Then, o e e y ∈[ 0, T ],u( )belongs o Xand sa is ies ku( )kX≤sup n≥1kunkL∞( 0,T ;X). P oo o Lemma 7. We deno e C:= sup n≥1kunkL∞( 0,T ;X). As (un)n∈Nis a bounded sequence in L∞( 0, T ;X), he e exis a subsequence (uµ) and ∈L∞( 0, T;X) such ha uµ ∗ * in L∞( 0, T;X), i.e., ZT 0hw∗( ), uµ( )id −→ ZT 0hw∗( ), ( )id ∀w∗∈L1( 0, T ;X0), whe e by h·,·i we deno e he duali y p oduc be ween X0and X. In pa icula , we ha e his con e gence o all w∗∈Lp0( 0, T ;X0). Then, uµ* weakly in Lp( 0, T;X), and as we also ha e un* u weakly in Lp( 0, T ;X), hen =u. 6 Then, uµ ∗ * u in L∞( 0, T;X) and by he ∗−weak lowe semicon inui y o he no m, we ob ain kukL∞( 0,T ;X)≤lim in µ→∞ kuµkL∞( 0,T ;X)≤C. (13) Now ix ∈[ 0, T]. By (13) he e exis s a sequence ( n)n∈Nin [ 0, T ] such ha n→ and u( n)∈Xwi h ku( n)kX≤C o all n∈N. As Xis e lexi e, he e exis a subsequence ( µ) and x∈Xsuch ha u( µ)* x weakly in X. The inclusion X⊂Yis assumed o be con inuous and so, u( µ)* x weakly in Y. (14) Due o u∈C0([ 0, T]; Y), we ha e in addi ion ha u( n)→u( ) in Y. This implies u( ) = x∈X. Finally he weak lowe semi-con inui y o he no m implies o e e y ∈[ 0, T ] ku( )kX=kxkX≤lim in µ→∞ ku( µ)kX≤C 4 Exis ence o weak solu ions P oposi ion 8 Suppose ha hypo heses (a)–(c)hold o F:L2(Ω) →L2(Ω). Then, o e e y u0∈L2(Ω), he e exis s a weak solu ion u∈L20, T;H1 0(Ω) o p oblem (4). Mo eo e , ubelongs o C0[0, T]; L2(Ω)and, u|(0,T ]is locally bounded in H1 0(Ω). Fu he mo e, i Fis locally Lipschi z as in hypo hesis (d), hen he weak solu- ion is unique. P oo . The p oo is based on a sequence o Gale kin app oxima ions and he a p io i es ima es in §3. Le {wk, k ≥1}be an o hogonal basis o H1 0(Ω) as in Rema k 1 and o each n∈N, conside a solu ion un( ) = n X k=1 unk( )wko (11). Fix s0∈(0, T) a bi a ily. By P oposi ion 3, he sequence (un)n∈Nis bounded in L20, T ;H1 0(Ω)Due o Lemma 5, (u0 n)n∈Nis bounded bo h in L2s0, T ;L2(Ω)and L2(0, T ;H−1(Ω)). Hence Alaoglu’s Theo em p o ides a subsequence (again deno ed by) (un)n∈N and unc ions u∈L20, T;H1 0(Ω), ∈L2s0, T;L2(Ω),w∈L20, T;H−1(Ω) wi h          un−→ uweakly in L20, T;H1 0(Ω) u0 n−→ weakly in L2s0, T;L2(Ω) u0 n−→ wweakly in L20, T;H−1(Ω). 7 In pa icula , u0= =wholds Lebesgue-almos e e ywhe e, which a simple check o he dis ibu ional de i a i e p ope y e eals. This implies ha u∈ L2(0, T;H−1(Ω)) ∩C0[0, T]; L2(Ω). S anda d a gumen s conclude u(0) = u0 om un(0) −→ u0in L2(Ω) o n→ ∞ (see e.g. [7, §7.1]). The sequence (un)n∈Nis equi–con inuous in C0[s0, T ]; L2(Ω)by Lemma 6. Mo eo e , he se un( ) ∈[s0, T], n ∈Nis ela i ely compac in L2(Ω) as a consequence o Lemma 5 and he Sobole Embedding Theo em. Hence he A zel`a–Ascoli Theo em p o ides a u he subsequence (again deno ed by) (un)n∈Nwi h un( )−→ u( ) in L2(Ω) uni o mly o ∈[s0, T]. Hypo hesis (a) on he con inui y o Fimplies ha F(un( )) → F(u( )) in L2(Ω) o e e y ∈[s0, T]. Taking he limi as n→∞gi es ha u∈L2s0, T;H1 0(Ω)is a weak solu ion o he pa ial di e en ial equa ion in p oblem (4) wi h u0∈L2s0, T;H−1(Ω). Finally, u|[s0,T ]: [s0, T ]→H1 0(Ω) is bounded due o P oposi ion 4. I Fsa is ies he local Lipschi z condi ion (d) in addi ion he uniqueness o he weak solu ion ollows by a s anda d a gumen . 5 The mul i alued semi low o weak solu ions De ine ΦF: [0,∞)×L2(Ω) → PL2(Ω), whe e PL2(Ω)consis s o all nonemp y subse s o L2(Ω), by ΦF( , u0) := u( )∈L2(Ω) ∃u(·)∈L20, ;H1 0(Ω): u0∈L20, ;H−1(Ω), u(0) = u0and uis weak solu ion o (4) in Ω ×(0, ). (15) I is clea ha his mul i alued map ΦF o ms a s ic mul i alued semi low on L2(Ω) as in he he ollowing de ini ion o Kapus yan e al. [9, De ini ion 2.1]. De ini ion 9 Le (X, d)be a me ic space and, P(X)consis s o all i s nonemp y subse s. A map Φ : [0,∞)×X→ P(X)is called s ic mul i alued semi low (m–semi- low) on Xi i sa is ies he ollowing condi ions: (A) Φ(0, x) = {x} o all x∈X (B) Φ( +s, x) = Φ( , Φ(s, x)) o all s, ≥0, x ∈X. Rema k 10 Kapus yan e al. [9, De ini ion 2.1] de ine, in ac , a mo e gen- e al m–semi low Φ : [0,∞)×X→ P(X), which sa is ies condi ion (A), bu 8 ins ead o he equali y condi ion (B), sa is ies he ollowing weake inclusion condi ion (C) Φ( +s, x)⊂Φ( , Φ(s, x)) o all s, ≥0, x ∈X. The eason is ha he coun e pa o he se alued mapping ΦFabo e o many sys ems, in pa icula he 3-dimensional Na ie –S okes equa ions, is es ic ed o weak solu ions ha sa is y an ene gy inequali y. Howe e , such ene gy in- equali ies only hold (o can only be p o ed o hold) o almos all ime ins an s. Hence a conca en a ion o weak solu ions sa is ying he ene gy inequali y on ad- jacen ime in e als may no sa is y he ene gy inequali y on he conca ena ed ime in e al, which means ha he s ic p ope y (B) need no hold. see also Mo illas & Vale o [12]. This si ua ion does no a ise o he mapping ΦFde ined abo e o he p oblem (4) unde conside a ion in his pape . E e y s ic mul i alued semi low Φ : [0,∞)×X→ P(X) (in he sense o De ini ion 9) induces a map ˜ Φ : [0,∞)× P(X)→ P(X) in he ollowing canonical way: ˜ Φ( , M)De . =[ x∈M Φ( , x) o all nonemp y M⊂X. Ob iously, i sa is ies he co esponding condi ions (A∗)˜ Φ(0, M) = M o all M∈ P(X) (B∗)˜ Φ( +s, M) = ˜ Φ( , ˜ Φ(s, M)) o all s, ≥0, M ∈ P(X) o in he gene al nons ic case (C∗)˜ Φ( +s, M)⊂˜ Φ( , ˜ Φ(s, M)) o all s, ≥0, M ∈ P(X) Hence one usually does no dis inguish be ween ˜ Φ and Φ. 5.1 Global a ac o s o mul i alued semi lows: Gene al esul s De ini ion 11 Aglobal a ac o o a s ic mul i alued semi low Φon a me ic space (X, d)is a nonemp y subse Ao Xwhich is Φ-in a ian , i.e. Φ( , A) = A o all ≥0, and a ac s e e y bounded subse Bo X, i.e. dis X(Φ( , B), A)→0as → ∞, whe e dis Xis he Hausdo semi–dis ance on P(X). Fo a gene al non-s ic mul i alued semi low he a ac o Ais only equi ed o be Φ-nega i ly in a ian , i.e. A⊂Φ( , A) o all ≥0 [11, De ini ion 6]. The exis ence o a global a ac o is usually concluded om he compac ness o asymp o ic compac ness o he mul i alued mapping Φ( , ·) o > 0: 9 we obse e ha i we conside m≥1, we ha e ZΩ|u|mdx ≤|Ω| |B(0; δ)|m/2kukm L2(Ω) . (18) We also ob ain ZΩ (F(u))2dx ≤C|Ω| |B(0; δ)|kuk2 L2(Ω) +C|Ω| |B(0; δ)|2kuk4 L2(Ω) +2C|Ω| |B(0; δ)|3/2kuk3 L2(Ω) , and hen we ha e (7) wi h ΦkukL2(Ω)=C|Ω| |B(0;δ)|kuk2 L2(Ω) +C|Ω| |B(0;δ)|2kuk4 L2(Ω) + 2C|Ω| |B(0;δ)|3/2kuk3 L2(Ω). We obse e ha i u∈L2(Ω), hen F(u)∈L∞(Ω) ⊂L2(Ω) . Now, we p o e ha Fis con inuous. We suppose ha un−→ us ongly in L2(Ω), hen we ha e o p o e ha F(un)−→ F(u) s ongly in L2(Ω). I is su icien o p o e ha he e exis s {uµ}⊂{un}such ha F(uµ)−→ F(u) s ongly in L2(Ω). As un−→ us ongly in L2(Ω), hen (see [4]) he e exis s {uµ}⊂{un}such ha uµ(x)−→ u(x) a.e. in Ω. Then, as is con inuous we ha e (uµ(x)) −→ (u(x)) a.e. in Ω. (19) We obse e ha a guing as be o e we ob ain |uµ(x)−u(x)|=1 |B(x;δ)|ZB(x;δ)∩Ω (uµ(y)−u(y)) dy ≤1 |B(0; δ)|1/2kuµ−ukL2(Ω) −→ 0, and hen uµ(x)−→ u(x) a.e. in Ω. (20) F om (19) and (20) we ha e F(uµ)(x)−→ F(u)(x) a.e. in Ω. (21) 16 On he o he hand, we obse e ha kuµkL2(Ω) is bounded because Ω is a bounded domain and uµ−→ us ongly in L2(Ω). Then, we ha e |F(uµ(x))| ≤ C|uµ(x)||1−uµ(x)|(22) ≤C1 |B(0; δ)|1/2kuµkL2(Ω) 1 + 1 |B(0; δ)|1/2kuµkL2(Ω)! ≤e C, whe e e Cis a posi i e cons an . Then, om (21), (22) and by he Domina ed Con e gence Theo em, we oba in F(uµ)−→ F(u) s ongly in L2(Ω) , hen Fis con inuous. F om Theo em 8 we ha e ha he e exis s a weak solu ion u∈L20, T ;H1 0(Ω) o p oblem (16) wi h u0∈L20, T;H−1(Ω).Acco ding o Theo em 15, he p oblem (16) de ines a mul i alued semi low in L2(Ω), which possesses a com- pac global a ac o A. Also, Ais he minimal closed a ac ing se . 7 P oo s o he dissipa i i y es ima es P oo o P oposi ion 3. The uni o m ellip ici y condi ion o he linea ope a o Aand assump ion (b) on he nonlocal ope a o Fused in he ene gy equali y, 1 2 d d kuk2 L2(Ω) +ZΩ u(A u)dx =ZΩ uF(u)dx (23) gi e 1 2 d d kuk2 L2(Ω) +λAk∇uk2 L2(Ω) ≤βkuk2 L2(Ω) +Cβ.(24) Hence, by he Poinca ´e inequali y (3), 1 2 d d kuk2 L2(Ω) +λΩλAkuk2 L2(Ω) ≤βkuk2 L2(Ω) +Cβ. i.e., d d kuk2 L2(Ω) ≤M−Lkuk2 L2(Ω) , (25) whe e M:= 2 Cβ>0 and L:= 2 λΩλA−2β > 0. Mul iplying by eL , and in eg a ing be ween 0 and , i ollows eL ku( )k2 L2(Ω) ≤ ku(0)k2 L2(Ω) +MZ 0 eLs ds ≤ ku(0)k2 L2(Ω) +MZ −∞ eLs ds, 17 i.e., o e e y ∈[0, T], ku( )k2 L2(Ω) ≤M L+e−L ku(0)k2 L2(Ω) . (26) Finally, in eg a ing equa ion (24) wi h espec o ime, we ob ain λAZ 0k∇uk2 L2(Ω) ds ≤1 2ku0k2 L2(Ω) −ku( )k2 L2(Ω)+βZ 0kuk2 L2(Ω) ds +Cβ ≤ ku0k2 L2(Ω) 1 2+βZ 0 e−L s ds+Cβ+βM L . P oo o Lemma 5. Le {wk:k≥1}deno e an o hogonal basis o H1 0(Ω) as men ioned in Rema k 1. Fo each n≥1, le un( ) = n X k=1 unk( )wk, be a Gale kin app oxima ion o p oblem (4) sa is ying he ini e dimensional sys em o o dina y di e en ial equa ions in he sense ha (d d (un( ), wk) + hAun( ), wki= (F(un( )), wk) (un(0), wk)=(u0, wk), k = 1, . . . , n. (27) We obse e ha Aun( ) = n X k=1 unk( )Awk= n X k=1 unk( )λkwk∈span {w1, . . . , wn}. Then, mul iplying equa ion (27) by unk( )λk, summing om k= 1 o n, and using p ope y (c) o F, gi es hu0 n( ), Aun( )i+kAun( )k2 L2(Ω) = (F(un( )), Aun( )) ≤1 2kF(un( ))k2 L2(Ω) +kAun( )k2 L2(Ω) ≤Ψ(un( )) + 1 2kAun( )k2 L2(Ω) (28) 18 o Lebesgue-almos all ∈[0, T ]. On he o he hand, hu0 n( ), Aun( )i=−ZΩ N X i,j=1 ∂xjaij(x)∂xiunu0 ndx (29) = N X i,j=1 ZΩ aij(x)∂xiun(∂ ∂xjun)dx =d d 1 2·A[un, un](30) o he symme ic bilinea o m A[u, ] := ZΩ N X i,j=1 aij(x)∂xiu ∂xj dx,u, ∈H1 0(Ω). since i is assumed ha aij =aji, wi h i, j = 1, .., N, and hese coe icien s do no depend on . In eg a ing om sand , hen gi es 1 2·A[un( ), un( )] ≤1 2·A[un(s), un(s)] + Z s Ψkun(ξ)kL2(Ω)dξ. (31) Hence, i ollows om he inequali ies o coe ci i y and con inui y λAk∇un(s)k2 L2(Ω) ≤ A[un(s), un(s)] ≤ΛAk∇un(s)k2 L2(Ω) ha λA 2·k∇un( )k2 L2(Ω) ≤ΛA 2·k∇un(s)k2 L2(Ω) +Z s Ψkun(ξ)kL2(Ω)dξ. (32) In eg a ing now he a iable sbe ween 0:= max{0, −1}and , gi es λA 2·min{ , 1}·k∇un( )k2 L2(Ω) ≤ΛA 2Z max{0, −1}k∇un(s)k2 L2(Ω) ds + Z max{0, −1} Ψkun(s)kL2(Ω)ds. On he o he hand, in eg a ing (24) wi h espec o ime in [ 0, ], i ollows ha kun( )k2 L2(Ω) −kun( 0)k2 L2(Ω) + 2 λAZ 0k∇un(s)k2 L2(Ω) ds ≤2βZ 0kun(s)k2 L2(Ω) ds + 2 Cβ. 19 Then, using P oposi ion 3, ΛAZ 0k∇un(s)k2 L2(Ω) ds ≤ΛA 2β+ 1 2λA sup s∈[ 0, ]kun(s)k2 L2(Ω) +CβΛA λA ≤C2·1 + e−L ku0k2 L2(Ω)(33) o all ≥0 wi h a cons an C2=C2(β, ΛA, λA, λΩ, Cβ)>1. Thus, whene e ku0kL2(Ω) ≤ρ, k∇un( )k2 L2(Ω) ≤2C2 λAmin{ , 1}1 + e−L ρ2+ ΨM L+e−L ρ2.(34) Mo eo e , his uppe bound holds o e e y ∈(0, T] — in con as o he immedia e consequence o inequali y (28) and he coe ci i y o A. The nex s ep is o es ablish an es ima e o ku0 nkL∞(0,T ;H−1(Ω)). This ol- lows essen ially he a gumen s o he s anda d ene gy es ima e, aking he p eceding a p io i bounds in o conside a ion. Fo e e y ∈H1 0(Ω), he e is a unique ep esen a ion = 1+ 2wi h 1∈ span{w1, . . . , wn}and ( 2, wk) = 0 o k= 1, . . .,nsince {wk, k ≥1}is an o hono mal basis o L2(Ω). Fo Lebesgue-almos e e y ∈[0, T ] i ollows ha u0 n( ), 1+Aun( ), 1=F(un( )), 1 hu0 n( ), i= (u0 n( ), ) = u0 n( ), 1 =F(un( )), 1−Aun( ), 1 ≤(kF(un( ))kL2+ ΛAk∇un( )kL2)k 1kH1 0(Ω). Then, inequali y (7) in assump ion (c) o Fimplies ha hu0 n( ), i ≤ pΨ (kun( )kL2)+ΛAk∇un( )kL2k 1kH1 0(Ω). Hence, in iew o he es ima es (26) and (34), ku0 n( )kH−1(Ω) ≤pΨ (kun( )kL2)+ΛAk∇un( )kL2 ≤cons β, Cβ,ΛA, λA, λΩ,ku0kL2(Ω). Finally, mul iplying (27) by u0 nk( ) and summing om k= 1 o n, i ollows 20 om inequali y (10) in assump ion (c) o F ha hu0 n( ), u0 n( )i+hAun( ), u0 n( )i= (F(un( )), u0 n( )) ≤ kF(un( ))k2 L2(Ω) +1 4ku0 n( )k2 L2(Ω) ≤Ψkun( )kL2(Ω)+1 4ku0 n( )k2 L2(Ω). Using (30), his can be e o mula ed as ku0 n( )k2 L2(Ω) +d d 1 2A[un, un]≤Ψkun( )kL2(Ω)+1 4ku0 n( )k2 L2(Ω) i.e., o Lebesgue-almos e e y ∈[0, T ], ku0 n( )k2 L2(Ω) +d d A[un, un]≤2 Ψkun( )kL2(Ω). Now ix s0∈(0, T[ a bi a ily and in eg a e be ween s0and T o ob ain ZT s0ku0 n(ξ)k2 L2(Ω) dξ +A[un(T), un(T)] ≤A[un(s0), un(s0)] + 2 ZT s0 Ψkun(ξ)kL2(Ω)dξ. The gene al inequali ies o coe ci i y and con inui y 0≤λAk∇un( )k2 L2(Ω) ≤ A[un( ), un( )] ≤ΛAk∇un( )k2 L2(Ω) , imply ha ZT s0ku0 n(ξ)k2 L2(Ω) dξ ≤ΛAk∇un(s0)k2 L2(Ω) + 2 ZT s0 Ψkun(ξ)kL2(Ω)dξ. 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