On a parabolic-elliptic chemotactic model with coupled boundary conditions

On a pa abolic-ellip ic chemo ac ic model wi h coupled bounda y condi ions 1 Manuel Delgado1, C is ian Mo ales-Rod igo1, An onio Su´ a ez1and J. Ignacio Tello2 1. Dp o. de Ecuaciones Di e enciales y An´alisis Num´e ico Fac. de Ma em´a icas, Uni . de Se illa Calle Ta ia s/n, 41012-Se illa, Spain 2. Depa amen o de Ma em´a ica Aplicada, Escuela de In o m´a ica, Uni e sidad Poli ´ecnica de Mad id, Ca e e a de Valencia Km 7. Campus Su . 28031-Mad id, Spain E-mail add esses: [email p o ec ed], [email p o ec ed], [email p o ec ed], [email p o ec ed] Abs ac This pape deals wi h a nonlinea sys em o pa abolic-ellip ic ype wi h a logis ic sou ce e m and coupled bounda y condi ions ela ed o pa e n o ma ion. We p o e he exis ence o a unique posi i e global in ime classical solu ion. We analyze also he s a iona y p oblem associa ed. Mo eo e i is p o ed, unde he assump ion o su icien ly s ong logis ic dumping, ha he e is only one nonze o homogeneous equi- lib ium, and all he solu ions o he non-s a iona y end o his s eady-s a e o la ge imes. AMS Classi ica ion. 35K45, 35K57, 92C17. Keywo ds. Chemo axis, Global exis ence, Asymp o ic beha io , Coexis ence s a es. 1 In oduc ion In many a eas o esea ch, om he biology o he emb yonic de elopmen o he umo al g ow h, he models o he cell mo emen play a undamen al ole. B oadly speaking, he con inuous models all in wo main classes: he mechanochemical models and he chemo- ac ic models. In he o me , cells exe a ac ion on he ex acellula ma ix (ECM), which ca ies ou a key ole; in he la e , cells sec e e a chemical subs ance, which a ac s o epels, and mo e owa ds o away om he g adien o his chemical. The e exis s a la ge numbe o examples whe e bo h ypes o models a e applied o desc ibe di e en biological phenomena. Nume ical simula ions a e used o compa e he expe imen al da a wi h he ma hema ical esul s and o jus i y hese ma hema ical models (see, [7], [12]). In any case, he s udy o hese sys ems ha e in e es by i sel (see, o ins ance, [13]). On he o he hand, we mus no o ge he in luence o he bounda y condi ions on he beha io o he solu ions. Recen ly, nonlinea bounda y condi ions ha e been inco po a ed 1MD and AS ha e been suppo ed by he Spanish Minis y o Science and Technology unde G an MTM2006-07932 and JIT by Spanish Minis y o Sciences and Technology unde g an MTM2009-13655. Ap il 26, 2014 M. Delgado, C. Mo ales-Rod igo, A. Su´a ez, J. I. Tello pe mi ing a mo e accu a e conside a ion o se e al si ua ions (see, o ins ance, [6] and e e ences he ein). In [10], a gene alized chemo ac ic model is conside ed whose o igin was he spa ial pa - e n o ma ion in emb yology. I includes a gene alized bounda y condi ion which pe mi s o co e a numbe o hem a ying he con ol pa ame e s. We hink i is no di icul o apply hese models in o he ields, o example, in angiogenesis, he cell mo emen linked o he umo al g ow h whe e he endo helial cells (ECM) mo e ollowing he g adien o TAF, he chemicals sec e ed by he umo cells; o his eason, we hink i is in e es ing o s udy i . The nondimensional model is                  u =D∆u−χ∇ · (u∇ ) + µsu(1 −u) in Ω ×(0, T), = ∆ −s +su γ+uin Ω ×(0, T), ρ1D∂u ∂n −χu ∂ ∂n= (1 −ρ1)(ρ2−u), ρ3 ∂ ∂n = (1 −ρ3)ρ2 1 + γ− on ∂Ω×(0, T), u(x, 0) = u0(x) in Ω, (1) whe e Ω ⊂IRdis a bounded domain wi h egula bounda y, uis he cell densi y, is he concen a ion o chemoa ac an , D > 0 is he di usion a e, χ > 0 is he chemo ac ic coe icien o he mo ile cells, µis he linea g ow h a e o he cell popula ion, γis a cons an go e ning he a e o chemoa ac an p oduc ion and deg ada ion and sis a pa ame e which con ols spa ial and empo al scale. In [10], i is p esen ed nume ical solu ions o his model in one spa ial dimension and is discussed he beha io o he model in wo dimensions. The au ho s conside also he possibili y o gi e di e en alues o he pa ame e s ρ1, ρ2, ρ3in di e en pa s o he bounda y. Du ing his wo k, we conside he ollowing sys em o equa ions                  u = ∆u−χ∇ · (u∇ ) + µu(1 −u) in Ω ×(0, T), 0 = ∆ − +u 1 + uin Ω ×(0, T), ∂u ∂n −χu ∂ ∂n = (θ−u),∂ ∂n = 0θ 2− on ∂Ω×(0, T), u(x, 0) = u0(x) in Ω, (2) whe e µ, , 0, θ, χ deno e non-nega i e cons an s. As we can see, i is he model esul ing om (1) o D=s=γ= 1, ρ2=θ, =1−ρ1 ρ1 , 0=1−ρ3 ρ3 and supposing ha he empo al scale o chemical di usion is much la ge han he scale o di usion o cells and consequen ly we can ake = 0. We will suppose ha θ, and 0a e nonnega i e cons an s on ∂Ω. Since we a e in e es ed only in non-nega i e solu ions we assume ha u0(x)≥0 in Ω. Ou pu pose is he heo e ical s udy o (2) and i s associa ed s a iona y p oblem as- socia ed. Ou main esul s can be summa ized as ollows: a) The pa abolic p oblem has exis ence and uniqueness o global solu ion ∀ , 0, θ, µ ≥ 0, o sui able ini ial da a. b) Wi h espec o he ellip ic p oblem, we can summa ize he si ua ion as ollows: (a) I θ > 0, 2 On a pa abolic-ellip ic chemo ac ic model... Ap il 26, 2014 • he e exis s a leas a coexis ence s a e ∀ , 0, µ ≥0, • he e exis s he i ial solu ion i , and only i , = 0= 0; which is uns able o µ > 0, • he e exis s one semi i ial solu ion (0, ) i = 0, 0>0; which is uns able o µ > 0. In pa icula , i θ= 1, he sys em has he homogeneous solu ion (u, ) = (1,1/2). Then, we p o e ha •i , 0>0, hen (1,1/2) is globally s able when µ≥0, •i 0= 0, ≥0, hen (1,1/2) is globally s able when µis big enough. (b) I θ= 0, he e exis s he i ial solu ion and he e is no semi i ial solu ion (0, V ). We can ind h ee cu es, µ=h1( ), µ =h2( ), µ =h3( ), being h1( )≥h2( )≥h3( ), such ha •i µ>h1( ), hen he e exis s a coexis ence s a e, •i µ<h3( ), hen he e exis s no coexis ence s a e, •i µ<h2( ), hen he i ial solu ion is s able. I is wo hy o be ema ked ha we use a ixed poin a gumen o s udy he e olu i e p oblem due o he ac ha ou sys em has one pa abolic equa ion and one ellip ic equa ion and he gene al heo y o pa abolic equa ions (which is used, o example, in [8]) has o be applied in a non-local amewo k, less cons uc i e han he me hod we ollow. On he o he hand, he me hod o bi u ca ion o s udy he s a iona y p oblem (see also [8]) needs he knowledge o some nonnega i e solu ion o he sys em o begin he b anch o posi i e solu ions; his solu ion is usually one o he semi i ial solu ions, bu in ou case he sys em does no admi any semi i ial solu ion when , 0,and θa e posi i e and, o his eason, we use a decoupling me hod. The pape is o ganized as ollows. In Sec ions 2 and 3, we s udy he exis ence and uniqueness o he global solu ion o he pa abolic p oblem (2). This p oblem has only one possible cons an coexis ence s a e o θ= 1, (u, ) = (1,1 2); we s udy in his case he asymp o ic beha io o he solu ions o (2) in Sec ion 4. In Sec ion 5, we conside he s eady p oblem associa ed o ou sys em, by a decoupling me hod and a sub-supe solu ion me hod o nonlocal p oblems. We will ob ain some esul s o exis ence o solu ions and some esul s o s abili y o he semi i ial solu ions when hey exis . 2 P elimina ies Fi s we obse e ha χ∇ · (u∇ ) = χ(∇u· ∇ +u∆ ) = χ∇u· ∇ +u −u 1 + u. So, we may ew i e he sys em (2) as ollows                    u = ∆u−χ∇u· ∇ +χu u 1 + u− +µu(1 −u) in Ω ×(0, T), 0=∆ − +u 1 + uin Ω ×(0, T), ∂u ∂n =χu 0θ 2− + (θ−u),∂ ∂n = 0θ 2− on ∂Ω×(0, T), u(x, 0) = u0(x) in Ω. (3) 3 Ap il 26, 2014 M. Delgado, C. Mo ales-Rod igo, A. Su´a ez, J. I. Tello Nex , in o de o a oid he singula i y o u u+ 1 we de ine he penalized unc ion h(u) := u+ u++ 1 and we in oduce he sys em                  u = ∆u−χ∇u· ∇ +χu h(u)− +µ χ(1 −u)in Ω ×(0, T), 0=∆ − +h(u) in Ω ×(0, T), ∂u ∂n =χu 0θ 2− + (θ−u),∂ ∂n = 0θ 2− on ∂Ω×(0, T), u(x, 0) = u0(x) in Ω. (4) Obse e ha we may conside he equa ion o uas a linea equa ion U −∆U=−χa(x, )· ∇U+χb(x, )Uin Ω, ∂U ∂n =χ 0c(x, )U+ (θ−U) on ∂Ω. whe e a(x, ) := ∇ , b(x, ) := h(u)− +µ χ(1 −u), c(x, ) := θ 2− . Thus, i b(x, )∈L∞(Ω×(0, Tmax)), hen, by he maximum p inciple, U(x, ) = u(x, )≥0 in Ω ×[0, T) and a solu ion o (4) is a solu ion o (3) and ice e sa. Now, ha ing in mind ha o posi i e solu ions he sys ems (3) and (4) a e equi alen i u, ∈L∞(Ω×(0, Tmax)), we will show he local exis ence heo em o (4). P e iously we p o e he ollowing esul whe e he no a ion is aken om [2], Sec ions 6 and 7: Lemma 2.1 Le p > 1,1< β < 2α < 1 + 1 pand (W2α−2,p B, Aα−1)an elemen o he in e pola ion-ex apola ion scale gene a ed by A0:= −∆ + Iand he eal in e pola ion unc o , hen ke− Aα−1ukWβ,p ≤Ce−ν −κkukW2α−2,p B,(5) wi h κ(β)∈(0,1) and ν∈(0,1). P oo . By Theo em 7.2 o [2], we ha e (W2α,p B, W2α−2,p B)κ,p =Wβ,p B, he e o e, using his e e ence, we know ha he e exis s κ∈(0,1) such ha ke− Aα−1ukWβ,p B ≤ ke− Aα−1ukκ W2α,p B ke− Aα−1uk1−κ W2α−2,p B . Nex , we apply [2, Theo em 8.5] oge he wi h [2, (3.1)] and we ge ke− Aα−1ukWβ,p B ≤ k(I+Aα−1)e− Aα−1ukκ W2α−2,p B ke− Aα−1uk1−κ W2α−2,p B . 4 On a pa abolic-ellip ic chemo ac ic model... Ap il 26, 2014 Taking in o accoun ha I+Aα−1and Aα−1a e sec o ial ope a o s wi h Reσ(I+Aα−1) = 1 + Reσ(Aα−1) = 2 hen we in oke [9, Theo em 1.3.4]. Thus, he e exis ν∈(0,1), α∈(0,2) such ha ke− Aα−1ukWβ,p B ≤ k(I+Aα−1)e− (I+Aα−1)e Iukκ W2α−2,p B ke− Aα−1uk1−κ W2α−2,p B =eκ k(I+Aα−1)e− (I+Aα−1)ukκ W2α−2,p B ke− Aα−1uk1−κ W2α−2,p B ≤e(κ(1−α)−ν(1−κ)) −κkukW2α−2,p B. Finally, we pick 1 −α=−νand use he ac ha Wβ,p B=Wβ,p o conclude he p oo . Lemma 2.2 Le 1< β < 2α < 1+ 1 p hen he e exis s κ < 1such ha Xκ,→Wβ,p, whe e Xκ:= D((I+Aα−1)κ) P oo . A guing in he same manne as we did in he p e ious lemma we ha e ha he e exis s θ < 1 such ha kukWβ,p ≤Ck(I+Aα−1)ukκ W2α−2,p kuk1−κ W2α−2,p . Finally he lemma can be concluded wi h he use o [9, pg. 28, Exe . 11]. 3 Global exis ence in ime Theo em 3.1 Le p > d and conside he ini ial da a u0∈W1,p(Ω) wi h u0≥0. Then he e exis s τ(ku0kW1,p )such ha he sys em (4) has a unique posi i e local in ime solu ion (u, )∈C([0, τ]; W1,p(Ω)) ∩ C1((0, τ); C2+α(Ω))2, and u(x, ), (x, )≥0 o (x, )∈Ω×[0, τ]. Mo eo e , he solu ion depends con inuously on he ini ial da a, i.e. i u(u0)and u(u0)deno e he solu ions o (4) wi h inial da a u0 and u0 espec i ely hen ku(u0)−u(u0)k(C([0,τ];W1,p))2≤Cku0−u0kW1,p . P oo . The p oo o he Theo em is based on a s anda d ixed poin a gumen . Le XT:= C([0, T]; W1,p(Ω)). Fo each ∈XTwe conside he ope a o S:XT→ C([0, T]; W2,p(Ω)) 7→ S( ) = , whe e is he unique solu ion o      −∆ + =h( ) in Ω ×(0, T), ∂ ∂n = 0θ 2− on ∂Ω×(0, T). 5 Ap il 26, 2014 M. Delgado, C. Mo ales-Rod igo, A. Su´a ez, J. I. Tello Mo eo e o each ∈[0, T], hanks o [1], he ollowing es ima e is sa is ied k ( )kW2,p ≤C( )kh( )kp+kθ/2 0kW1−1/p,p(∂Ω).(6) Nex , we conside he ope a o H:XT→XT 7→ H( ) = u, whe e uis he unique solu ion o he linea pa abolic p oblem      u −∆u=−χ∇ · ∇ +χ (h( )− ) + µ (1 − ) in Ω ×(0, T ), ∂u ∂n =χ 0θ 2− + (θ− ) on ∂Ω×(0, T). (7) Le 2α∈1,1 + 1 p. Thanks o he gene alized a ia ions o cons an s o mula, see [2] pg. 63, we can ew i e (7) in he ollowing manne u( ) = e− Aα−1u0+Z 0 e−( −τ)Aα−1(F( , ) + Aα−1Bcγg( , ))dτ, (8) whe e γ:W1,p(Ω) →W1−1/p,p(∂Ω) deno es he ace ope a o and F( , ) := −χ∇ · ∇ +χ (h( )− ) + µ (1 − ), g( , ) := χ 0θ 2− + (θ− ). Le us poin ou ha , since Aα−1Bc∈ L(W2α−1−1/p,p(∂Ω), W2α−2,p B) oge he wi h he embedding W1−1/p,p(∂Ω) ,→W2α−1−1/p,p(∂Ω), hen we can asse ha Aα−1Bcγis well de ined o g( , )∈W1,p(Ω). Nex , we de ine he closed se BT R:= { ∈ C([0, T]; W1,p(Ω)) : k kXT≤R}. Now, we ha e o e i y ha he condi ions o he Banach ixed poin Theo em a e sa is ied o he map H. S ep 1. The e exis R, T > 0 such ha o any ∈BT R, i holds ha H( )∈BT R. F om (8), hanks o (5) and he embedding Wβ,p(Ω) ,→W1,p(Ω) we ge ku( )kW1,p ≤Cku0kW1,p +CZ 0 e−ν( −τ)( −τ)−κ(kF( , )kW2α−2,p B+ +kAα−1Bcγg( , )kW2α−2,p B)dτ. Taking in o accoun he embedding W1−1/p,p(∂Ω) ,→W2α−1−1/p,p(∂Ω) we ha e kAα−1Bcγg( , )kW2α−2,p(∂Ω) ≤Ckγg( , )kW2α−1−1/p,p(∂Ω) ≤Ckγg( , )kW1−1/p,p(∂Ω) ≤Ckg( , )kW1,p also, ha ing in mind [2, (7.5),(7.8)], we ha e Lp:= W0,p B,→W2α−2,p B. Thus, we in e ku( )kW1,p ≤Cku0kW1,p +CZ 0 e−ν( −τ)( −τ)−κ(kF( , )kp+kg( , )kW1,p )dτ. (9) 6 On a pa abolic-ellip ic chemo ac ic model... Ap il 26, 2014 On one hand we ha e kF( , )kLp≤χ(k∇ · ∇ kp+k h( )kp+k kp) + µ(k kp+k 2kp).(10) Now, we es ima e each e m o (10) sepa a ely k∇ · ∇ kp≤ k∇ k∞k∇ kp ≤Ck kW1,∞k kW1,p ≤Ck kW2,p k kW1,p ≤C(kh( )kp+C( 0))k kW1,p ≤C(kh( )k∞+C( 0))k kW1,p ≤C(k kW1,p +C( 0))k kW1,p . The emaining e ms o (10) can be es ima ed in a simila way o ob ain kF( , )kp≤C(χ, µ, 0,k kW1,p ),(11) wi h C(χ, µ, 0,k kW1,p ) an inc easing unc ion on i s a gumen s. On he o he hand we ha e kg( , )kW1,p ≤χθ 0 2k kW1,p +χ 0k kW1,p +k θkW1,p + k kW1,p .(12) The e m k kW1,p is es ima ed as ollows k kW1,p ≤Ck kW1,p k kW1,∞ ≤Ck kW1,p (kh( )kp+C( 0)) ≤Ck kW1,p (k kW1,p +C( 0)). So, we ob ain kg( , )kW1,p ≤C(χ, , 0, θ, k kW1,p ),(13) whe e C(χ, , 0, θ, k kW1,p ) is an inc easing unc ion on i s a gumen s. Now, we plug (11) and (13) in (9) o ob ain ku( )kW1,p ≤Cku0kW1,p +C(R)Z 0 Ce−ν( −τ)( −τ)−κdτ ≤Cku0kW1,p +C(R)T1−κ. Thus, choosing R > Cku0kW1,p and τ0=T(R) su icien ly small hen kukXτ0≤R. Mo eo e , kukXT≤R o all T≤τ0. Now, we ix R > Cku0kW1,p and T≤τ0is ee o ou disposal. S ep 2. H is con ac i e. Le 1, 2∈BT R hen, u1=H( 1)∈BT Rand u2=H( 2)∈BT Rand u1( )−u2( ) = Z 0 e−( −τ)Aα−1((F( 1, 1)−F( 2, 2)) + Aα−1Bcγ(g( 1, 1)−g( 2, 2)))dτ. So, we ob ain ku1( )−u2( )kW1,p ≤Z 0 e−ν( −τ)( −τ)−κ(kF( 1, 1)−F( 2, 2)kp+ +kg( 1, 1)−g( 2, 2)kW1,p )dτ. (14) 7 Ap il 26, 2014 M. Delgado, C. Mo ales-Rod igo, A. Su´a ez, J. I. Tello On one hand we ha e kF( 1, 1)−F( 2, 2)kp≤χk∇ 2· ∇ 2− ∇ 1· ∇ 1kp+χk 1h( 1)− 2h( 2)kp+ +µk 1− 2kp+µk 2 2− 2 1kp. (15) Now, we es ima e each e m o (15) sepa a ely k∇ 2· ∇ 2− ∇ 1· ∇ 1kp≤ k∇( 2− 1)· ∇ 2kp+k∇ 1· ∇( 2− 1)kp ≤C(k 2kW2,p k 1− 2kW1,p +Rk 1− 2kW2,p ) ≤C(C(R)k 1− 2kW1,p +Rkh( 1)−h( 2)kp). Taking in o accoun ha kh( 1)−h( 2)kp≤ k( 1)+−( 2)+k∞≤ k 1− 2k∞we ge k∇ 2· ∇ 2− ∇ 1· ∇ 1kp≤C(R)k 1− 2kW1,p . Fo he emaining e ms o (15) we can a gue in a simila way o ob ain kF( 1, 1)−F( 2, 2)kp≤C(R)k 1− 2kW1,p .(16) On he o he hand we ha e kg( 1, 1)−g( 2, 2)kW1,p ≤χθ 0 2+ k 1− 2kW1,p +χ 0k 2 2− 1 1kW1,p . We deduce k 2 2− 1 1kW1,p ≤ k 2( 2− 1)kW1,p +k 1( 2− 1)kW1,p ≤ k 2kW1,∞k 1− 2kW1,p +k 1kW1,p k 2− 1kW1,∞ ≤C(R)k 1− 2kW1,p . Thus, we ge kg( 1, 1)−g( 2, 2)kW1,p ≤C(R)k 1− 2kW1,p .(17) Finally, we pu he es ima es (16) and (17) in (14) o ob ain ku1−u2kXT≤C(R)T1−κk 1− 2kXT. Hence, aking Tsu icien ly small we p o e ha His con ac i e. Now we deal wi h he egula i y o he solu ion. Le us ix any ∈(0, τ) hen he equa ion o uhas he abs ac ep esen a ion du d + (I+Aα−1)u= (x, ), u(0) = u0. Thus, hanks o [9, Theo em 3.5.2] du d ( )∈Xκwi h κ < 1. In pa icula , by Lemma 2.2,we ha e du d ( )∈Wβ,p o some β > 1, p > d. So, we ob ain u∈ C1((0, τ); W1,p(Ω)) and since he -equa ion p ese es he egula i y in ime hen ∈ C1((0, τ); W1,p(Ω)). We obse e ha    −∆ ( ) + ( ) = h(u)( ) in Ω, ∂ ∂n( ) + 0 ( ) = 0θ 2on ∂Ω. 8 On a pa abolic-ellip ic chemo ac ic model... Ap il 26, 2014 Taking in o accoun ha h(u)( )∈ Cα(Ω) hen he ellip ic egula i y assu es ( )∈ C2+α(Ω). So, we ha e p o ed ha ∈ C1((0, τ); C2+α(Ω)). Now we ew i e he u-equa ion as ollows      −∆u( ) + ∇u( )· ∇ ( ) = ( ) in Ω, ∂u ∂n( ) +  −χ 0θ 2− u( ) = θ on ∂Ω, whe e ( ) := (uh(u)−u +µu(1 −u)−u )( ). Since ( )∈ Cα(Ω), −χ 0θ 2− ( )∈ C1+α(∂Ω) and ∇ ( )∈ Cα(Ω) hen ellip ic egu- la i y en ails u( )∈ C2+α(Ω). Nex we obse e ha he posi i i y o (u, ) is consequence o he maximum p inciple o pa abolic equa ions. A he end we show he con inui y espec o he ini ial da a, o his pu pose we a gue in he ollowing manne . Le R > C(ku0kW1,p +ku0kW1,p ). We ha e u(u0)( ) = e− Aα−1u0+Z 0 e−( −τ)Aα−1(F(u(u0), (u0)) + Aα−1Bcγg(u(u0), (u0)))dτ =e− Aα−1u0+H(u(u0)) −e− Aα−1u0. Hence, we in e k(u(u0)−u(u0))( )kW1,p ≤2ke− Aα−1(u0−u0)kW1,p +kH(u(u0)−H(u(u0))kW1,p . Taking sup emum on ime, hanks o he con ac i i y o H, we ob ain ku(u0)−u(u0)kXT≤Cku0−u0kW1,p +kku(u0)−u(u0)kXT, wi h k < 1. Also we ha e k( (u0)− (u0))( )kW2,p ≤C( )kh(u(u0))( )−h(u0(u0))( )kLp ≤C( )ku(u0)−u(u0)kW1,p .(18) F om (18) he p oo o he con inui y can be easily concluded. Now we deal wi h he issue o global in ime solu ions. To his end we ha e jus o show ha ku( )kW1,p ≤C( ) o all <Tmax whe e Tmax s ands o he maximal in e al o exis ence. We obse e ha      −∆ + =u 1 + uin Ω ×(0, Tmax), ∂ ∂n + 0 = 0θ 2on ∂Ω×(0, Tmax). Since    u( ) 1+u( )  ∞≤1 hen k ( )kW2,p ≤C o all ∈[0, Tmax). Nex we pu he o mula o gene alized a ia ions o cons an s o ob ain u( ) = e− Aα−1u0+Z 0 e−( −τ)Aα−1(F(u, ) + Aα−1Bcg(u, ))dτ. 9 Ap il 26, 2014 M. Delgado, C. Mo ales-Rod igo, A. Su´a ez, J. I. Tello whe e in he las inequali y we used ha +=−(− )−. We no e ha −∆ + −u 1 + u=U1 (1 + ξ2)2 o some ξ2(x, )∈(min{u, u},max{u, u}). A e mul iplying he p e ious exp ession by ( −u 1+u)−we ge ZΩ ∇ −u 1 + u−! 2 +ZΩ −u 1 + u2 − =ZΩ U1 (1 + ξ2)2 −u 1 + u− ≤ZΩ U− 1 (1 + ξ2)2 −u 1 + u− ≤1 2ZΩ U2 −+1 2ZΩ −u 1 + u2 − Taking in o accoun ha he bounda y e m is non-posi i e hen, om he abo e inequali y we deduce 1 2ZΩ −u 1 + u2 − ≤1 2ZΩ U2 −.(38) Plugging he es ima e (38) in o (37) we ob ain χZΩ U+u 1 + u− ≤χ−1 0ZΩ U2 ++ZΩ U2 −. The p e ious inequali y p o ides wi h he ollowing bound in (36) d 2d ZΩ U2 +≤ZΩ U2 +g(u, , u) + χ−1 0ZΩ U2 ++ZΩ U2 −. Since g(u, , u)≤C hen d 2d ZΩ U2 +≤CZΩ U2 ++ZΩ U2 −. In he same ashion we ha e d 2d ZΩ U2 −≤CZΩ U2 ++ZΩ U2 −. Adding he abo e inequali ies and aking in o accoun ha (U0)+= (U0)−= 0, we may in oke G onwall’s Lemma o achie e U+=U−= 0 (39) which p o es he s ep. S ep 5. Since u≤u≤uand u≤1≤u hen ku( )−1k∞≤u−u≤ −−1 0ln(0)eγ00 ∀ > 0.(40) 16 On a pa abolic-ellip ic chemo ac ic model... Ap il 26, 2014 Nex we obse e ha ξ=1 2is he unique solu ion o he p oblem      −∆ξ+ξ=1 2in Ω ×(0,+∞), ∂ξ ∂n = 0 on ∂Ω×(0,+∞). The e o e z:= −ξsa is ies      −∆z+z=u 1 + u−1 2in Ω ×(0,+∞), ∂z ∂n = 0 on ∂Ω×(0,+∞), (41) and ellip ic egula i y asse s kz( )kW2,p ≤C    u( )−1 2(1 + u( ))   ∞ ≤Cku( )−1k∞, concluding he esul . Co olla y 4.4 I µ≥2χ hen he solu ion (u, ) o (2) is globally exponen ially asymp- o ically s able and con e ges o he homogeneous s eady-s a e (1,1 2). Mo eo e , i µ≥2χ, hen he p e ious homogeneous s eady-s a e is he only posi i e solu ion o he s eady-s a e p oblem associa ed o (2). P oo . Assume min u0= 0 hen, by he s ong maximum p inciple, min u(τ)>0 o a bi a y τ > 0 small as desi ed. Nex , we obse e ha γ0<2χ−µ≤0 and hanks o Theo em 4.2 we conclude he i s pa . The second pa is a di ec consequence o he global s abili y. Ano he consequence o Theo em 4.2 is he nex Co olla y Co olla y 4.5 I µ > χ 2 hen he solu ion (1,1 2) o (2) is locally exponen ially asymp o i- cally s able. 5 The s a iona y p oblem In his sec ion, we analyze he s a iona y p oblem associa ed o (2), ha is                      −∆u=−χ∇ · (u∇ ) + µu(1 −u) in Ω, −∆ =− +u 1 + uin Ω, ∂u ∂n −χu ∂ ∂n = (θ−u) on ∂Ω, ∂ ∂n = 0θ 2− on ∂Ω. (42) Fi s , in o de o p esen and p o e ou main esul , we need in oduce some no a ion. Gi en unc ions a, b ∈ C(Ω) wi h a≥a0>0, ∈ C(∂Ω), we deno e by λ1(a;b;N+ ) he p incipal eigen alue o he p oblem    −di (a(x)∇u) = λb(x)uin Ω, ∂u ∂n + (x)u= 0 on ∂Ω. (43) 17 Ap il 26, 2014 M. Delgado, C. Mo ales-Rod igo, A. Su´a ez, J. I. Tello Recall ha λ1(a;b;N+ ) is inc easing in aand and dec easing in b. On he o he hand, deno e by 0 he unique posi i e solu ion o    −∆ + = 1 in Ω, ∂ ∂n + 0 = 0θ 2on ∂Ω. (44) I is clea ha 0= 1 i 0= 0. Mo eo e , i 0>0 min{1, θ/2} ≤ 0≤max{1, θ/2}. Finally, obse e ha since u/(u+ 1) ≤1 we ge ≤ 0in Ω. (45) Now, we a e eady o s a e ou main esul : Theo em 5.1 a) Assume ha θ > 0and 0≥0. Then, he e exis s a leas a posi i e solu ion o (42) i µ≥0and ≥0. b) Assume ha θ= 0 and 0≥0. Then, he e exis s a leas a posi i e solu ion o (42) i µ>λ1(eχ 0; 1; N+ ), and (42) does no possess a posi i e solu ion i 0≤µ≤λ1(1; eχ 0;N+ ). Co olla y 5.2 Assume ha θ= 0= 0. Then, he e exis s a leas a posi i e solu ion o (42) i , and only i , µ>λ1(1; 1; N+ ), In o de o p o e he main esul we a e going o use a decoupling me hod and a sub- supe solu ion me hod o non-local p oblems. Le us begin showing he alidi y o he his las me hod. Conside a con inuous map B:C(Ω) 7→ C(Ω) and he non-linea equa ion    −∆u= (x, u, B(u)) in Ω, ∂u ∂n + (x)u=h(x) on ∂Ω, (46) whe e : Ω ×IR × C(Ω) 7→ IR is a egula unc ion; , h ∈ C(∂Ω). De ini ion 5.3 We say ha u, u ∈ C2(Ω) ∩ C(Ω) is a sub-supe solu ion o (46) i u≤u in Ωand a) −∆u− (x, u, B(u)) ≤0≤ −∆u− (x, u, B(u)) in Ω,∀u∈[u, u], b) ∂u ∂n + (x)u≤h(x)≤∂u ∂n + (x)uon ∂Ω. 18 On a pa abolic-ellip ic chemo ac ic model... Ap il 26, 2014 P oposi ion 5.4 Assume ha he e exis s a sub-supe solu ion o (46), u, u, in he sense o De ini ion 5.3. Then, he e exis s a solu ion u∈[u, u]o (46). P oo . Being he p oo s anda d, we ou line only i . Take M > 0 la ge enough such ha u7→ (x, u, ξ) + Mu is inc easing o all x∈Ω and ξ∈ C(Ω) and λ1(1; 1; N+ ) + M > 0. Conside he map T: [u, u]2→[u, u]2 w7→ u:= T(w), being u he unique solu ion o he ollowing p oblem    −∆u+Mu = (x, w, B(w)) + Mw in Ω, ∂u ∂n + (x)u=h(x) on ∂Ω, (47) whe e [u, u]2:= {u∈L2(Ω) : u≤u≤u}. I is no ha d o show ha we can apply he Schaude ixed poin heo em o Tand conclude he esul . To s udy sys em (42) we a e going o apply he change o a iable u=eχ w which ans o ms he i s equa ion o (42) in o    −di (eχ ∇w) = µeχ w(1 −eχ w) in Ω, ∂w ∂n + w = θe−χ on ∂Ω. (48) Wi h espec o his equa ion, we ge : P oposi ion 5.5 Fix ∈ C(Ω) and deno e by L:= min x∈Ω (x), M:= max x∈Ω (x). a) Assume ha θ > 0. Then, he e exis s a unique posi i e solu ion, deno ed w, o (48) o µ≥0. Mo eo e , min{1, θ}e−χ M≤w≤max{1, θ}e−χ Li µ > 0, θe−χ M≤w≤θe−χ Li µ= 0.(49) b) Assume ha θ = 0. Then, i µ > 0 he e exis s a unique posi i e solu ion o (48) i , and only i , µ>λ1(eχ ;eχ ;N+ ). Mo eo e , µ−λ1(eχ ;eχ ;N+ ) µeχ Mkϕk∞ ϕ≤w≤e−χ L(50) whe e ϕis a posi i e eigen unc ion associa ed o λ1(eχ ;eχ , N + ). I µ= 0 he e exis s a posi i e solu ion i , and only i , = 0. In such case, any posi i e cons an is solu ion. 19 Ap il 26, 2014 M. Delgado, C. Mo ales-Rod igo, A. Su´a ez, J. I. Tello P oo . We will apply he sub-supe solu ion me hod. a) Assume ha θ > 0. Take w:= ε > 0 and w:= K > 0 wi h ε, K o be chosen. Then, εand Kmus sa is y 0≤µ(1 −eχ ε) in Ω, ε≤θe−χ on ∂Ω, and 0≥µ(1 −eχ K) in Ω, K≥θe−χ on ∂Ω. This p o es (49). The uniqueness ollows because he map w7→ µ(1 −eχ w) is dec easing when µ > 0, see [5]. In he case µ= 0 he equa ion (48) is linea , and hence i is clea he uniqueness esul . b) Assume θ = 0. Le wbe a posi i e solu ion o (48). Then, −di (eχ ∇w)< µeχ win Ω, ∂w ∂n + w = 0 on ∂Ω, and so, mul iplying by ϕand in eg a ing we ge ha µ>λ1(eχ ;eχ ;N+ ).Fo he exis ence o solu ion ake in his case w:= εϕ wi h ε=µ−λ1(eχ ;eχ ;N+ ) µeχ Mkϕk∞ . Again, he uniqueness ollows simila ly. Finally, he case µ= 0 ollows easily. We ix now µ > 0, he case µ= 0 will be ea ed sepa a ely. Fix ∈ C(Ω) and conside he equa ion (48). Deno e by w( ) := (w i θ > 0 o i θ = 0 and µ>λ1(eχ ;eχ ;N+ ) , 0 in o he case, being w he unique posi i e solu ion o (46), which exis s by P oposi ion 5.5. Lemma 5.6 The ope a o ∈ C(Ω) 7→ w( )∈ C(Ω) is con inuous. P oo . Conside sequences n→ in C(Ω) and wn:= w( n). Obse e ha wnis solu ion o −di (an(x)∇wn) + wn=hn,in Ω, an(x)∂wn ∂n =gnon ∂Ω, being an(x) := eχ n, hn:= wn+µwn·(1 −eχ nwn), gn:= an·(− wn+ θe−χ n). Obse e ha 0< α ≤an≤β < ∞ and hanks o (49) and (50) we ge ha khnk∞≤Cand kgnkL∞(∂Ω) ≤C, and hen, by Theo em 2.1 in [3], i ollows ha kwnkCν(Ω) ≤C, 20 On a pa abolic-ellip ic chemo ac ic model... Ap il 26, 2014 o ν∈(0,1) and some cons an s depending on αand β. Hence, o a subsequence, wn→win C(Ω), being wa weak solu ion o (46). The ellip ic egula i y p o es ha wis a classical solu ion o (46). Now, we can ollow he lines o he p oo o Lemma 3.1 o [4] o conclude ha in ac w=w( ). P oo o Theo em 5.1. Now, we ha e o s udy he ollowing non-local and nonlinea equa ion        −∆ + =eχ w( ) 1 + eχ w( )in Ω, ∂ ∂n + 0 = 0θ 2on ∂Ω. (51) To s udy his equa ion we apply P oposi ion 5.4 wi h = 0being 0 he unique posi i e solu ion o (44). We ake as subsolu ion = 0 . Obse e ha = 0 is sub-solu ion and no solu ion o (51) i θ 0≥0 and w( )≥0∀ ∈[0, 0] and some inequali y s ic . I is clea ha his holds i θ 0>0 and pa ag aph a) ollows o µ > 0. Assume now ha 0= 0 and θ > 0. I > 0 hen θ > 0 and by P oposi ion 5.5 we ha e ha w( )≥min{1, θ}e−χ M>0 o all µ > 0. I = 0, by P oposi ion 5.5 µ−λ1(eχ ;eχ ;N+ ) µeχ Mkϕk∞ ϕ≤w( ) i µ>λ1(eχ ;eχ ;N) = 0. Hence, w( )>0 i µ > 0. Finally assume ha 0≥0 and θ= 0. I = 0, by a simila easoning, we need ha µ > 0. I > 0 we need ha µ>λ1(eχ ;eχ ;N+ )∀ ∈[0, 0]. Thanks o (45) we ge λ1(eχ ;eχ ;N+ )≤λ1(eχ 0; 1; N+ ), hence i is enough ha µ>λ1(eχ 0; 1; N+ ). Finally, we show he non-exis ence esul . Assume ha θ = 0. Obse e ha w( ) = 0 i µ > 0 and µ≤λ1(eχ ;eχ ;N+ ). Bu λ1(eχ ;eχ ;N+ )≥λ1(1; eχ 0;N+ ) whence we deduce he esul . Conside now he case µ= 0. I θ > 0 hen he e exis s a unique w( )>0 solu ion o (48) and he esul ollows in a simila way. I θ= 0 and > 0 hen w( ) = 0, and hen u= 0. Howe e , i = 0, we ob ain ha w=C o any posi i e cons an C, and hen u=eχ C. 21 Ap il 26, 2014 M. Delgado, C. Mo ales-Rod igo, A. Su´a ez, J. I. Tello I is enough now o s udy he equa ion o −∆ + =Ceχ 1 + Ceχ in Ω,∂ ∂n + 0 = 0θ 2on ∂Ω. We can a gue as in he abo e case and conclude he esul . Wi h espec o he semi i ial and i ial solu ions, we ge : P oposi ion 5.7 a) The i ial solu ion (u, ) = (0,0) exis s i and only i θ = 0θ= 0. In such case, he solu ion is uns able o µ > λ1(1; 1; N+ )and s able o µ<λ1(1; 1; N+ ). b) The semi i ial solu ion (0, V 0)exis s i and only i = 0 and 0, θ > 0, being V 0 he unique solu ion o    −∆V+V= 0 in Ω, ∂V ∂n + 0V= 0θ 2on ∂Ω.(52) In such case, (0, w 0)is uns able o µ > 0. P oo . a) The exis ence esul is no ha d o show. On he o he hand, he s abili y o (0,0) is gi en by he eal pa s o he eigen alues o which he ollowing p oblem admi s a solu ion (ξ, η)6= (0,0)                  −∆ξ−µξ =σξ in Ω, −∆η+η=ξ+ση in Ω, ∂ξ ∂n + ξ = 0 on ∂Ω, ∂η ∂n + 0η= 0 on ∂Ω. (53) I ξ= 0, hen σ=λj(1; 1; N+ 0)+1>0. Assume now ha ξ6= 0, so σ=λj(1; 1; N+ )−µ≥λ1(1; 1; N+ )−µ. Then, i µ<λ1(1; 1; N+ ) we ob ain ha σ > 0 and (0,0) is s able. Assume now ha µ>λ1(1; 1; N+ ). Then, σ1:= λ1(1; 1; N+ )−µ < 0. Deno e by ξa posi i e eigen unc ion associa ed o σ1, ha is −∆ξ−µξ =σ1ξin Ω, ∂ξ ∂n + ξ = 0 on ∂Ω. Since σ1<0, hen λ1(1; 1; N+ 0)+1−σ1>0, and so he e exis s η6= 0 such ha −∆η+η−σ1η=ξin Ω,∂η ∂n + 0η= 0 on ∂Ω. 22 On a pa abolic-ellip ic chemo ac ic model... Ap il 26, 2014 Then, σ1<0 is an eigen alue o (53) wi h associa ed eigen unc ion (ξ, η), so (0,0) is uns able. b) Again he exis ence esul is di ec . Now, he linea iza ion a ound (0, V 0) is                    −∆ξ=−di (χξ∇V 0) + µξ +σξ in Ω, −∆η+η=ξ+ση in Ω, ∂ξ ∂n −χξ ∂V 0 ∂n = 0 on ∂Ω, ∂η ∂n + 0η= 0 on ∂Ω. (54) The i s equa ion, a e he change o a iable ξ=eχV 0ψ, is ans o med in o −di (eχV 0∇ψ)=(µ+σ)eχV 0ψin Ω, ∂ψ ∂n = 0 on ∂Ω. I ξ= 0 hen, since η6= 0, hen σ=λj(1; 1; N+ 0) + 1 >0. On he o he hand, i ξ6= 0 we ge σ=λj(eχV 0;eχV 0;N)−µ≥λ1(eχV 0;eχV 0;N)−µ > 0, since λ1(eχV 0;eχV 0;N) = 0 and µ < 0. Assume now ha µ > 0, hen σ1:= λ1(eχV 0;eχV 0;N)−µ < 0 and conside ψa posi i e eigen unc ion associa ed o σ1, ha is −di (eχV 0∇ψ) = (µ+σ1)eχV 0ψin Ω, ∂ψ ∂n = 0 on ∂Ω. Again, conside he change o a iable ξ=eχV 0ψ, and η he solu ion o −∆η+η=ξ+σ1ηin Ω, ∂η ∂n + 0η= 0 on ∂Ω, which exis s because λ1(1; 1; N+ 0)+1−σ1>0. Then, σ1<0 is an eigen alue o (54) wi h associa ed eigen unc ion (ξ, η), so (0, V 0) is uns able. Re e ences [1] S. Agmon, A. Douglis and L. 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