Strong Convergence of Solutions and Attractors for Reaction-Diffusion Equations Governed by a Fractional Laplacian

S ong con e gence o solu ions and a ac o s o eac ion-di usion equa ions go e ned by a ac ional Laplacian Jiaohui Xu Cen e o Nonlinea S udies, School o Ma hema ics, No hwes Uni e si y, Xi’an 710127, P. R. China Tom´ as Ca aballo1 2 Dp o. Ecuaciones Di e enciales y An´alisis Num´e ico, Facul ad de Ma em´a icas, Uni e sidad de Se illa, c/ Ta ia s/n, 41012-Se illa, Spain Depa men o Ma hema ics, Wenzhou Uni e si y, Wenzhou, Zhejiang P o ince, 325035, P. R. China Jos´e Vale o Cen o de In es igaci´on Ope a i a, Uni e sidad Miguel He n´andez de Elche, A enida de la Uni e sidad s/n, 03202-Elche, Spain Abs ac A nonlocal eac ion-di usion equa ion go e ned by a ac ional Laplace ope a o on a bounded domain is s udied in his pape . Fi s , he s ong con e gence o solu ions o he equa ions go e ned by ac ional Laplacian o he solu ions o he classical equa ions go e ned by a s anda d Laplace ope a o is p o ed, when he ac ional pa ame e g ows o 1. Second, o he au onomous case, he uppe semicon inui y o global a ac o s wi h espec o he a ac o s o he limi p oblem is es ablished. Appa en ly, hese a e he i s esul s o his kind o p oblems on bounded domains. Keywo ds: F ac ional Laplacian; S ong con e gence o solu ions; Global a ac o s. AMS subjec classi ica ions. 35R11, 35A15, 35B41, 35K65 1. In oduc ion In his pape , we s udy he p oblem        ∂u ∂ + (−∆)γu= (u) + h( ),in O× (τ, ∞), u= 0,on ∂O, u(x, τ) = uτ(x),in O, (1) whe e (−∆)γ,γ∈(0,1), s ands o he ac ional Laplace ope a o ( ac ional Laplacian), Ois a bounded subse o Rmwi h su icien ly smoo h bounda y, τ∈R,h∈L2 loc(R;L2(O)) and is a con inuous unc ion sa is ying some app op ia e assump ions as speci ied la e . Fi s o all, we need o decide which de ini ion o he ac ional Laplacian is mo e app op ia e o ou analysis. I is well known ha , in he unbounded domain case, namely in Rm, he e a e se e al equi alen de ini ions o ac ional Laplacian (see, o ins ance, [13] o en equi alen de ini ions), bu in he bounded domain case, he e exis se e al de ini ions which a e no equi alen . We e e he eade o he pape s [6, 16, 19] o mo e de ails. We will no discuss abou hem in his pape , bu will simply choose one o hose de ini ions o ou in es iga ion, mo e p ecisely he so called “ egional”de ini ion (see [6]), in such a way ha we a e able o p o e he con e gence o solu ions o (1) o he co esponding ones o he limi ing equa ion wi h s anda d Laplacian when γ→1−, ensu ing his con e gence holds in he s ong sense, wha somehow jus i ies he sui abili y and accu acy o he chosen de ini ion o ac ional Laplacian. 1Co esponding au ho 2E-mail add esses: [email protected] (J. Xu), [email p o ec ed] (T. Ca aballo), j [email p o ec ed] (J. Vale o). 1 2 The p ope ies o solu ions and a ac o s o p oblem (1) when γis ixed ha e been s udied widely in he li e a u e. Fo example, he well-posedness and he exis ence o global a ac o s (in bo h de e minis ic and andom se ing, wi h bounded o unbounded domains) o p oblem (1) ha e been ex ensi ely in es iga ed in he li e a u e (see, e.g., [8, 17, 27, 28, 29, 30, 31] and he e e ences he ein. Addi ionally, o a s ochas ic e sion o (1), he uppe semicon inui y o he a ac o s was es ablished in [27, 28, 29] when a pa ame e in he noise e m a ies. I is ema kable ha he de ini ion o ac ional Laplacian used in hese s ochas ic pape s is exac ly he “ egional”one we will use in ou cu en analysis, wha ein o ces he idea ha he de ini ion we ha e chosen is app op ia e. Ou aim in his manusc ip is o analyze he beha io o solu ions o p oblem (1) when he pa ame e γgoes o 1−, and he mo i a ion is based on he p ope ies o ac ional Laplace ope a o (−∆)γas γ→1−, (see, o example, [8] and [21]). In [21, P oposi ion 4.4], he au ho s p o ed ha (−∆)γu(x) con e ges o −∆u(x) as γ→1 o all x∈Rmwhen u∈C∞ 0(Rm). In [8, P oposi ion 2.3], he au ho s p o ed ha , o e e y u∈C∞ 0(Rm) and ∈W1,2 0(O), i holds lim γ→1−ZO (−∆)γudx =−ZO ∆udx. No ice ha in he pape s [3], [4] his ques ion was s udied o an abs ac pa abolic equa ion and he Sch ¨odinge equa ion. Howe e , as a as we a e awa e, he e a e no esul s in he li e a u e conce ning he con e gence o solu ions o (1) o he solu ions o he ollowing limi p oblem,        ∂u ∂ −∆u= (u) + h( ),in O× (τ, ∞), u= 0,on ∂O, u(x, τ) = uτ(x),in O, (2) as he pa ame e γgoes o 1−. In [33], we comple ed a i s s ep by p o ing he con e gence o solu ions wi h espec o he weak opology in L2(Rm) in he case ha he ex e nal unc ion is sublinea . In his manusc ip , we will go much mo e u he o s udy he con e gence o solu ions in he s ong opology sense o mo e gene al unc ions . Mo eo e , he uppe semicon inui y o he global a ac o s will be es ablished as well. The in e es in s udying p oblem (1) and i s s a iona y e sion comes om di e en ields as has been desc ibed in ou pape [33] (see also [1, 2, 7, 9, 10, 11, 12, 18, 20, 23, 24, 25, 26, 32, 34, 35] and he e e ences he ein). This pape is o ganized as ollows. In Sec ion 2, we ecall he basic de ini ions and p ope ies conce ning he ac ional Laplace ope a o s, es ablish he se ing o he p oblem we a e ackling and impose he condi ions ha ensu e he exis ence and uniqueness o weak solu ions o p oblem (1). In Sec ion 3, we imp o e i s he known esul s abou he con e gence o he ope a o (−∆)γ by showing ha limγ→1−(−∆)γu=−∆us ongly in Lp((τ, τ +T)×Rm), o any p≥1 and u∈C∞ 0((τ, τ +T)×Rm), whe e τ∈R,T > 0. Also, limγ→1−(−∆)γu=−∆us ongly in Lp(Rm), o any p≥1 and u∈C∞ 0(Rm). A e ha , we p o e he main esul o he pape s a ing ha he solu ions o p oblem (1) con e ge o he ones o he limi p oblem (2) in C([τ, τ +T]; L2(O)), o any T > 0 as γ→1−. Finally, in Sec ion 4, we s udy he uppe semicon inui y o he global a ac o s as he pa ame e γgoes o 1−in he au onomous si ua ion, ha is, when he unc ion h belongs o L2(O). 3 2. Se ing o he p oblem We conside he ollowing ac ional eac ion-di usion equa ion,        ∂u ∂ + (−∆)γu= (u) + h( ),in O× (τ, ∞), u= 0,on ∂O, u(x, τ) = uτ(x),in O, (3) whe e (−∆)γ,γ∈(0,1), s ands o he ac ional Laplace ope a o , Ois a smoo h bounded subse o Rm,τ∈Rand h∈L2 loc(R;L2(O)). Th oughou his pape , we assume ha ∈C(R) and he e exis posi i e cons an s C ,κ,β1,β2and p≥2, such ha ( (s)− ( ))(s− )≤C (s− )2,∀s, ∈R,(4) −κ−β1|s|p≤ (s)s≤κ−β2|s|p,∀s∈R.(5) F om (5), we deduce ha he e exis s a cons an β3>0 such ha | (s)| ≤ β3(|s|p−1+ 1),∀s∈R.(6) We ix some µ∈(0, β2) and de ine he unc ion (u) = (u) + µu. Then, he equa ion in (3) eads as ∂u ∂ + (−∆)γu+µu = (u) + h( ).(7) I ollows om (5), (6) and he Young inequali y ha (s)s≤κ−β2|s|p+µs2≤κ−β2|s|p,(8) | (s)| ≤ β3(|s|p−1+ 1),(9) o some posi i e cons an s κ, β2, β3. Le Sdeno e he Schwa z space o apidly decaying C∞ unc ions on Rm. The ac ional Laplace ope a o (−∆)γo u∈ S a poin x, o 0 < γ < 1, is de ined by, (−∆)γu(x) = −1 2C(m, γ)ZRm u(x+y) + u(x−y)−2u(x) |y|m+2γdy, x ∈Rm,(10) whe e C(m, γ) is he ollowing posi i e cons an C(m, γ) = γ4γΓ(m+2γ 2) πm 2Γ(1 −γ).(11) I is well known [21] ha o any u∈ S, we ha e (−∆)γu=F−1|ξ|2γFu, whe e Fis he Fou ie ans o m gi en by Fu=1 (2π)m/2ZRm e−ix·ξu(x)dx, and F−1is he in e se Fou ie ans o m. Fo 0 < γ < 1, we de ine he ac ional Sobole space Wγ,2(Rm) := Hγ(Rm) by: Hγ(Rm) = u∈L2(Rm) : ZRmZRm |u(x)−u(y)|2 |x−y|m+2γdxdy < ∞, endowed wi h he no m kukHγ(Rm)=ZRm |u(x)|2dx +ZRmZRm |u(x)−u(y)|2 |x−y|m+2γdxdy1 2 . 4 In wha ollows, we deno e by k · kp he no m in Lp(Rm) o p > 2,whe eas by k · k and (·,·), we deno e he no m and he inne p oduc o L2(Rm), espec i ely. By abusing o he no a ion, (·,·) will be also used o he pai ing be ween Lq(Rm) and Lp(Rm). Mo eo e , we will conside he Gaglia do semi-no m o Hγ(Rm),deno ed by k·k ˙ Hγ(Rm),which is gi en by, kuk2˙ Hγ(Rm)=ZRmZRm |u(x)−u(y)|2 |x−y|m+2γdxdy, u ∈Hγ(Rm). The e o e, kuk2 Hγ(Rm)=kuk2+kuk2˙ Hγ(Rm) o all u∈Hγ(Rm). No ice ha Hγ(Rm) is a Hilbe space wi h inne p oduc , (u, )Hγ(Rm)=ZRm u(x) (x)dx +ZRmZRm (u(x)−u(y))( (x)− (y)) |x−y|m+2γdxdy, o u, ∈Hγ(Rm).Thanks o [21], he no m kukHγ(Rm)is equi alen o kuk2+k(−∆)γ 2uk21 2 o u∈Hγ(Rm). Namely, i ollows ha kuk2 Hγ(Rm)=kuk2+2 C(m, γ)k(−∆)γ 2uk2,∀u∈Hγ(Rm). Since he ac ional Laplace ope a o (−∆)γgi en by (10) is nonlocal, we he e in e p e he homo- geneous Di ichle bounda y p oblem (3) as ollows,        ∂u ∂ + (−∆)γu= (u) + h( ),in O× (τ, ∞), u= 0,on Rm O, u(x, τ) = uτ(x),in O. (12) Na u ally, he limi p oblem (2) will be w i en as        ∂u ∂ −∆u= (u) + h( ),in O× (τ, ∞), u= 0,on Rm O, u(x, τ) = uτ(x),in O, (13) so ha hey keep o mally consis en . Thus, we de ine he spaces Vγ={u∈Hγ(Rm) : u= 0 a.e. on Rm O}, H={u∈L2(Rm) : u= 0 a.e. on Rm O}, V={u∈H1(Rm) : u= 0 a.e. on Rm O}, wi h hei dual spaces deno ed by V∗ γ,H∗and V∗, espec i ely. Mo eo e , we will use kuk2 Vγ=kuk2+k(−∆)γ 2uk2, o deno e he no m o space Vγ. The no m in Vwill be deno ed by k·kV. Fu he mo e, le b:Vγ×Vγ→Rdeno e a bilinea o m gi en by, bγ( 1, 2) = µ( 1, 2) + 1 2C(m, γ)ZRmZRm ( 1(x)− 1(y))( 2(x)− 2(y)) |x−y|m+2γdxdy, 1, 2∈Vγ,(14) whe e C(m, γ) is he same as in (11). Fo con enience, we associa e wo ope a o s Aγ,Aγ:Vγ→V∗ γ wi h b, such ha < Aγ( 1), 2>(V∗ γ,Vγ)=µ( 1, 2)+ < Aγ( 1), 2>(V∗ γ,Vγ)=bγ( 1, 2),(15) 5 o all 1, 2∈Vγ,whe e <·,·>(V∗ γ,Vγ)is he duali y pai ing o V∗ γand Vγ. I is known [21] ha he embeddings V⊂Vγ2⊂Vγ1a e con inuous o 0 < γ1≤γ2<1. Iden i ying Hwi h i s dual space, we ob ain he chain o con inuous embeddings V⊂Vγ⊂H⊂V∗ γ⊂V∗,∀γ∈(0,1) . Pai ing be ween spaces Vand V∗will be deno ed by <·,·>(V∗,V ). We conside wi hou loss o gene ali y ha h∈L2 loc(R;H) by se ing h( , x) = 0 o x∈Rm O. De ini ion 1. Le τ∈R,uτ∈Hand γ∈(0,1). The unc ion u∈C([τ, ∞); H)is said o be a weak solu ion o p oblem (12) i u(τ) = uτ,u∈L2 loc(τ, ∞;Vγ)∩Lp loc(τ, ∞;Lp(Rm)),du d ∈ L2 loc(τ, ∞;V∗ γ) + Lq loc(τ, ∞;Lq(Rm)) and d d (u, ξ) + 1 2C(m, γ)ZRmZRm (u( , x)−u( , y))(ξ(x)−ξ(y)) |x−y|m+2γdxdy =ZO ( (u( , x)) + h( , x)) ξ(x)dx, o any ξ∈Vγ∩Lp(Rm), in he sense o scala dis ibu ions in (τ, ∞). In [27, Theo em 2.3], one can see ha p oblem (12) possesses a unique weak solu ion o any uτ∈Hand γ∈(0,1). In addi ion, his solu ion is con inuous wi h espec o he ini ial da um uτ in H, and sa is ies he ene gy equali y, d d kuk2+C(m, γ)kuk2˙ Hγ(Rm)= 2 ZO ( (u( , x)) + h( , x)) u( , x)dx, o a.a. ≥τ. De ini ion 2. Le τ∈Rand uτ∈H. The unc ion u∈C([τ, ∞); H)is said o be a weak solu ion o p oblem (13) i u(τ) = uτ,u∈L2 loc(τ, ∞;V)∩Lp loc(τ, ∞;Lp(O)),du d ∈L2 loc(τ, ∞;V∗) + Lq loc(τ, ∞;Lq(O)) and d d (u, ξ) + ZO ∇u·∇ξdx =ZO ( (u( , x)) + h( , x)) ξ(x)dx, o any ξ∈V∩Lp(O), in he sense o scala dis ibu ions in (τ, ∞). Recall ha (see, e.g., [5] o [22]) p oblem (2) possesses a unique weak solu ion o any uτ∈H, which is con inuous wi h espec o he ini ial da um uτin H. 3. Con e gence o solu ions Ou aim now is o p o e ha he solu ions o p oblem (12) con e ge, as γ→1−, o he unique solu ion o he limi p oblem wi h γ= 1, ha is, o he unique solu ion o he s anda d eac ion- di usion equa ion (13). I is well known [21, P oposi ion 4.4] ha i u∈C∞ 0(Rm), hen lim γ→1−(−∆)γu(x) = −∆u(x), o all x∈Rm. (16) Le C∞ 0(X) be he space o in ini ely di e en iable unc ions u:X→Rwi h compac suppo , whe e Xis a Banach space. Deno e by D0 he space o dis ibu ions o D=C∞ 0((τ, τ +T)×Rm) and by <·,·>(D0,D) he duali y pai ing o D0and D. Deno e by BR he ball in Rmcen e ed a 0 wi h adius R. Now, we ecall a esul p o ed in [33] which ex ends he con e gence esul (16) o a mo e gene al one. 6 Lemma 3. (See [33, Lemma 3.1]) Fo any τ∈R,T > 0and u∈C∞ 0((τ, τ +T)×Rm), he ollowing s a emen holds, lim γ→1−(−∆)γu=−∆u, s ongly in Lp((τ, τ +T)×Rm),∀p≥1. In pa icula , limγ→1−(−∆)γu=−∆uin he sense o dis ibu ions in D0. Now, in a simila way we s a e he ollowing esul . Lemma 4. (See [33, Lemma 3.2]) Fo any u∈C∞ 0(Rm),limγ→1−(−∆)γu=−∆us ongly in Lp(Rm) o any p≥1. We also need he ollowing echnical lemma which is c ucial o ou analysis. Lemma 5. (i) Fo any γ∈1 2,1and ∈Vγ, we ha e k kV1 2 ≤ 3 2k kV γ.(17) (ii) The e exis s a posi i e cons an Ksuch ha , o any ∈Vγ,  Aγ  V∗≤Kk kV γ, o any γ∈(0,1) .(18) P oo . (i) Since 1 2<1 2γ<1,0<1−1 2γ<1 2. By he Fou ie ans o m and Young inequali y, he es ima e (17) ollows om k k2 V1 2 =k k2+k(−∆)1 4 k2=ZRm (1 + |ξ|)|F (ξ)|2dξ ≤ZRm2−1 2γ+1 2γ|ξ|2γ|F (ξ)|2dξ ≤3 2ZRm1 + |ξ|2γ|F (ξ)|2dξ =3 2k k2 Vγ. (i) Fo he second s a emen , le us i s p o e he exis ence o posi i e cons an Ksuch ha   (−∆)γ 2u  ≤KkukV,∀u∈V. (19) On he one hand, by Fubini’s Theo em, we ha e ZRmZ|x−y|<1 |u(x)−u(y)|2 |x−y|m+2γdxdy ≤ZRmZB1 |u(y+z)−u(y)|2 |z|2|z|m+2(γ−1) dzdy ≤ZRmZB1 1 |z|m+2(γ−1) Z1 0 |∇u(y+ z)|d 2 dzdy ≤ZB1 1 |z|m+2(γ−1) Z1 0ZRm |∇u(y+ z)|2dyd dz ≤ kuk2 VZB1 1 |z|m+2(γ−1) dz ≤ kuk2 Vωm−1Z1 0 1 ρ2γ−1dρ =ωm−1 2 (1 −γ)kuk2 V. 7 On he o he hand, he change o a iable implies ZRmZ|x−y|≥1 |u(x)−u(y)|2 |x−y|m+2γdxdy ≤4ZRmZ|x−y|≥1 |u(y)|2 |x−y|m+2γdxdy = 4 kuk2Z|z|≥1 1 |z|m+2γdz ≤4kuk2ωm−1Z∞ 1 1 ρ2γ+1 dρ =2ωm−1 γkuk2 V. Thus, combining he abo e wo es ima es, we ob ain   (−∆)γ 2u   2=C(m, γ) 2kuk2˙ Hγ(Rm) =C(m, γ) 2ZRmZRm |u(x)−u(y)|2 |x−y|m+2γdxdy ≤ωm−1C(m, γ) 4 (1 −γ)+C(m, γ) γkuk2 V ≤K2kuk2 V, o some cons an s K > 0, whe e he las inequali y ollows om he ac lim γ→0+ C(m, γ) γ= lim γ→0+ C(m, γ) γ(1 −γ)=2 ωm−1 , see [21, Co olla y 4.2] o mo e de ails. Immedia ely, by he Young inequali y and (19), we ind  Aγ  V∗= sup u∈V, kukV≤1< Aγ( ), u >(V∗,V ) = sup u∈V, kukV≤1< Aγ( ), u >(V∗ γ,Vγ) = sup u∈V, kukV≤1 1 2C(m, γ)ZRmZRm ( (x)− (y))(u(x)−u(y)) |x−y|m+2γdxdy ≤sup u∈V, kukV≤1 1 2C(m, γ)k k˙ Hγ(Rm)kuk˙ Hγ(Rm) = sup u∈V, kukV≤1  (−∆)γ 2     (−∆)γ 2u   ≤Ksup u∈V, kukV≤1  (−∆)γ 2   kukV≤K  (−∆)γ 2   ≤Kk kV γ. We inish he p oo o his lemma.  We can now es ablish he main esul o his pape abou he con e gence o solu ions. Theo em 6. Le un τ→uτin H, le un(·)be he solu ion o p oblem (12) o γ=γnwi h ini ial alue un τ. Then un→uin C([τ, τ +T]; H)as n→ ∞ o any T > 0, whe e u(·)is he unique solu ion o p oblem (13). I un τ→uτweakly in H, hen un→uin C([τ+ε, τ +T]; H)as n→ ∞ o any 0< ε < T. 8 P oo . Conside a sequence γn→1−. Le un(·) be he unique solu ion o p oblem (12) wi h ini ial alue un τ. Mul iplying equa ion (7) by un, using he Young inequali y and on accoun o (8), we de i e 1 2 d d kunk2+µkunk2+1 2C(m, γn)kunk2˙ Hγn(Rm) ≤κ−β2kunkp p+kh( )k kunk ≤κ−β2kunkp p+1 2µkh( )k2+µ 2kunk2,(20) which is equi alen o d d kunk2+µkunk2+C(m, γn)kunk2˙ Hγn(Rm)+ 2β2kunkp p≤2κ+1 µkh( )k2.(21) Mul iplying (21) by eµs and in eg a ing i o e he in e al (τ, ), we deduce kun( )k2≤ kun τk2e−µ( −τ)+2κ µ1−e−µ( −τ)+1 µZ τ e−µ( −s)kh(s)k2ds. (22) Then o T > 0, we in e ha sup ∈[τ,τ+T] kun( )k2≤MT,(23) whe e MTdepends on T, and Zτ+T τC(m, γn)kun(s)k2˙ Hγn(Rm)+ 2β2kun(s)kp pds ≤MT.(24) The e o e, he sequence {un}is bounded in L∞(τ, τ +T;H)∩Lp(τ, τ +T;Lp(Rm)). By (23) and (24), we ha e Zτ+T τ  (−∆)γn 2un(s)   2ds ≤Zτ+T τ C(m, γn)kun(s)k2˙ Hγn(Rm)ds ≤MT,(25) Zτ+T τ kun(s)k2 Vγnds ≤MT+TMT.(26) Hence, in iew o (26) and Lemma 5(i), he e exis s n0>0, such ha γn∈[1 2,1) o n≥n0. Then, we ha e Zτ+T τ kunk2 V1 2 ds ≤3 2MT+TMT, which implies {un}is bounded in L2(τ, τ +T;V1 2). Also, (26) and (18) imply ha {Aγn(un)}is bounded in L2(τ, τ +T;V∗). Mo eo e , by (24) and (9), we in e ha { (un)}is bounded in Lq(τ, τ +T;Lq(Rm)). The e o e, {dun d }is bounded in Lq(τ, τ +T;Lq(Rm))+L2(τ, τ +T;V∗). Then he e exis unc ions u, χ, ξ such ha , up o a subsequence which we elabel he same, we deduce he ollowing con e gences, un→uweak-s a in L∞(τ, τ +T;H),(27) un→uweakly in L2(τ, τ +T;V1 2),(28) un→uweakly in Lp(τ, τ +T;Lp(Rm)),(29) (un)→χweakly in Lq(τ, τ +T;Lq(Rm)),(30) Aγn(un)→ξweakly in L2(τ, τ +T;V∗),(31) dun d →du d weakly in Lq(τ, τ +T;Lq(Rm)) + L2(τ, τ +T;V∗).(32) 9 Since he embedding V1 2⊂His compac and he embedding H⊂(V∩Lp(Rm))∗is con inuous, a s anda d Compac ness Theo em [22, Theo em 8.1] implies ha un→us ongly in L2(τ, τ +T;H),(33) un( , x)→u( , x) o a.a. ( , x)∈(τ, τ +T)× O.(34) F om he con inui y o and [15, Lemma 1.3], we also ob ain ha χ= (u). Fu he , we need o p o e ha ξ=−∆u. By Lemma 3, o a bi a y ϕ∈ D, we ha e < Aγn(un), ϕ >(D0,D)=Zτ+T τ < Aγn(un), ϕ >(V∗,V )d =Zτ+T τ < un, Aγn(un)>(V,V ∗)d =Zτ+T τZRm un(−∆)γnϕdxd γn→1− −→ Zτ+T τZRm u(−∆)ϕdxd =Zτ+T τ < u, −∆ϕ >(V,V ∗)d =<−∆u, ϕ >(D0,D), he abo e con e gence holds hanks o he ac s ha un→uin L2(τ, τ+T;H) and (−∆)γnϕ→ −∆ϕ in L2(τ, τ +T;H). Consequen ly, Aγn(un)→ −∆uas γn→1−in he dis ibu ional sense, which implies ha ξ=−∆u. Thus, o any η∈L2(τ, τ +T;V)∩Lp(τ, τ +T;Lp(Rm)), he abo e con e gences yield ha 0 = Zτ+T τ <dun d , η >(V∗+Lq(Rm),V ∩Lp(Rm)) d +Zτ+T τ < Aγn(un), η >(V∗,V )d −Zτ+T τZO ( (un) + h)ηdxd −→ Zτ+T τ <du d , η >(V∗+Lq(Rm),V ∩Lp(Rm)) d +Zτ+T τ <−∆u, η >(V∗,V )d −Zτ+T τZO ( (u) + h)ηdxd . Mo eo e , u∈L∞(τ, τ +T;H)∩Lp(τ, τ +T;Lp(Rm)), u∈L2(τ, τ +T;V1 2), du d ∈Lq(τ, τ +T;Lq(Rm)) + L2(τ, τ +T;V∗), −∆u∈L2(τ, τ +T;V∗). Since −∆u( )∈V∗, we de i e ha u( )∈V o a.a. ∈(τ, τ +T). Hence, k∇u( )k2=<−∆u( ), u( )>(V∗,V )≤ k∆u( )kV∗k∇u( )k, and, consequen ly, u∈L2(τ, τ +T;V). The e o e, u(·) is he weak solu ion o p oblem (13). Since e e y con e ging sequence has he same limi , i ollows ha he con e gences (27)-(34) hold o he whole sequence. Also, we deduce om (23) ha o n, 0∈[τ, τ +T] such ha n→ 0, we ha e un( n)→u( 0) weakly in H. (35)