Combining linear and nonlinear diffusion

Combining linea and nonlinea di usion Manuel Delgado1 Dp o. de Ecuaciones Di e enciales y An´alisis Num´e ico Uni e sidad de Se illa Calle Ta ia s/n 41012-Se illa, Spain e-mail: [email p o ec ed] Juli´an L´opez-G´omez2 Depa amen o de Ma em´a ica Aplicada Uni e sidad Complu ense de Mad id 28040-Mad id, Spain e-mail: [email p o ec ed] An onio Su´a ez1 Dp o. de Ecuaciones Di e enciales y An´alisis Num´e ico Uni e sidad de Se illa Calle Ta ia s/n 41012-Se illa, Spain e-mail: [email p o ec ed] 1Suppo ed by he Spanish Minis y o Science and Technology unde G an s BFM2000-0797 and BFM2003-06446. 2Suppo ed by he Spanish Minis y o Science and Technology unde G an s BFM2000-0797 and REN2003-00707. Abs ac In his pape we s udy a gene alized po ous medium equa ion whe e he di usion a e, say m(x) —spa ially he e ogeneous—, is assumed o be linea , m= 1, on a piece o he suppo domain, Ω1, and slow nonlinea , m(x)>1, in i s complemen , Ωm:= Ω ¯ Ω1. Mos p ecisely, we cha ac e ize he exis ence o posi i e solu ions and cons uc he co esponding global bi u ca ion diag am as one o he pa ame e s o he model changes, showing ha a con inuous ansi ion occu s be ween he diag ams o he comple ely linea case (Ω = Ω1) and o he comple ely nonlinea case (Ωm= Ω). As a esul , he e ec o a localized slow di usion a e wi h a ying suppo is comple ely cha ac e ized. Ou analysis is impe a i e in o de o design po ous media mul i-componen s sys ems wi h changing di usion a es. Key Wo ds. Nonlinea di usion. Spa ial he e ogenei ies. F om linea o nonlinea di usion. AMS Classi ica ion. 35B32, 35J25, 35J60, 35K57. 2M. Delgado, J. L´opez-G´omez and A. Su´a ez 1 In oduc ion In his pape we s udy he posi i e solu ions o he ollowing bounda y alue p oblem    −∆¡wm(x)¢=λw in Ω , w= 0 on ∂Ω,(1.1) whe e Ω ⊂IRN,N≥1, is a bounded domain o class C2,λ∈IR , and m= 1 + p χΩm whe e Ωmis an open smoo h subdomain o Ω such ha ¯ Ωm⊂Ω and p∈ C(¯ Ωm) sa is ies p(x)>0 o each x∈Ωm. Finally, we deno e by Ω1:= Ω ¯ Ωm, he open se whe e m= 1. Gi en any measu able se M⊂Ω, χMs ands o he cha ac- e is ic unc ion o M, i.e., χM(x) = 1 o each x∈M, and χM(x) = 0 o each x∈Ω M. An admissible choice o pwould be aking a cons an p > 0. Then, m=χΩ1+ (1 + p)χΩm. In Figu e 1, we ha e ep esen ed an admissible con igu a ion. The da k egion s ands o Ωm, whe e m>1, and he whi e egion is he subdomain o Ω whe e m= 1. Ω Ω 1 m Linea di usion Nonlinea di usion Figu e 1. An admissible con igu a ion. In he special case Ωm=∅, (1.1) educes o he classical linea eigen alue p oblem o he Laplacian unde Di ichle bounda y condi ions in Ω. Subsequen ly, o any po en ial V∈L∞(Ω) we shall deno e by σ[−∆ + V; Ω] he p incipal eigen alue o −∆ + Vin Ω unde homogeneous Di ichle bounda y condi ions. Acco ding o K ein–Ru man heo em, (1.1) possesses a posi i e solu ion i , and only, i λ=σ[−∆; Ω]. Ac ually, in such case, all posi i e solu ions a e mul iples o a p incipal eigen unc ion. Combining linea and nonlinea di usion 3 On he o he hand, when Ωm= Ω and mis cons an , (1.1) p o ides us wi h he classical po ous medium equa ion, which gene a ed a huge indus y in Pa ial Di e en ial Equa ions since he pionee ing s udies o G. I Ba enbla [2] and A. G. A onson & L. A. Pele ie [1]. In ac , one o he esul s o [1] es ablishes ha (1.1) possesses a posi i e solu ion i , and only i , λ > 0, and ha i is unique and asymp o ically s able i i exis s. Ac ually, i we deno e i by wλi u ns ou ha limλ↓0wλ= 0 and ha λ7→ wλis inc easing (c . [4] o u he de ails). In Figu e 2, we ha e ep esen ed a bi u ca ion diag am scheme o he posi i e solu ions o (1.1) in hese ex eme opposi e cases. Figu e 2(a) shows he bi u ca ion diag am o he linea eigen alue p oblem, and Figu e 2(b) ep esen s he bi u ca ion diag am o posi i e solu ions o he classical po ous medium equa ion. In Figu e 2(a) we ha e deno ed σ1:= σ[−∆; Ω]. Ou main in e es in his pape is ocused in o he p oblem o analyzing how change hese diag ams when di usion is nonlinea in some piece o Ω, Ωm, whe eas i is linea in he complemen , Ω1, ying o asce ain all possible in e media e e en ual ansi ions be ween he p e ious wo limi ing cases. Such an analysis is impe a i e in o de o s udy he e ec o local nonlinea di usi i ies in he global dynamics o po ous media. Consequen ly, we will h oughou assume ha Ωm, and so Ω1, a e p ope subdomains o Ω. I should be no ed ha , hough he non-linea i y is discon inuous, i is o Ca a heodo y in L∞and, hence, all solu ions li e in W1,p o all p > 1. The analysis o his p oblem i s in o ou gene al p og am o analyzing eac ion di u- sion equa ions in he p esence o spa ial he e ogenei ies; hose he e ogenei ies migh a ise in nonlinea di usion a es, o cou se. As i will become clea la e he global na u e o he co esponding bi u ca ion diag am o posi i e solu ions o he gene al p oblem we a e dealing wi h is a he di e en . ww 00 σ1 λ λ (a) (b) Figu e 2. Bi u ca ion diag am in he limi ing cases. Since he change o a iable u=wm(x) ans o ms (1.1) in o    −∆u=λu 1 m(x)in Ω , u= 0 on ∂Ω.(1.2) 4M. Delgado, J. L´opez-G´omez and A. Su´a ez mos o ou a en ion will be ocused in o (1.2). By ellip ic egula i y heo y, i is olklo e ha any weak non-nega i e solu ion u6= 0 is an s ong solu ion almos e e ywhe e wice di e en iable and, as a esul o he maximum p inciple, u(x)>0 o each x∈Ω and ∂u ∂n (x)<0 o each x∈∂Ω, whe e ns ands o he ou wa d no mal ec o - ield o Ω. The e o e, a necessa y condi ion o he exis ence o a posi i e weak solu ion is λ > 0. The ollowing unc ion will play a c ucial ole in ou exposi ion µ(λ) := σ[−∆−λχΩ1; Ω] , λ ∈[0,∞).(1.3) I sa is ies µ(0) >0, and, due o he mono onici y o he p incipal eigen alue wi h espec o he po en ial, i is dec easing in λ. Ac ually, i sa is ies µ0(λ)<0 o each λ > 0, since λ7→ µ(λ) is conca e; by a celeb a ed heo em o P. Hess and T. Ka o [6] (c . [8] o u he de ails). Mo eo e , by he mono onici y o he p incipal eigen alue wi h espec o he domain, gi en a ball B⊂Ω1 o each λ≥0 we ha e ha µ(λ)< σ[−∆−λχΩ1;B] = σ[−∆; B]−λ . Thus, lim λ↑∞ µ(λ) = −∞ and, hence, he e exis s λ0=λ0(Ω1)∈(0, σ[−∆; Ω1]) such ha µ−1(0) ∩[0,∞) = {λ0}.(1.4) Ac ually, λ0(Ω1)> σ[−∆; Ω] since µ(σ[−∆; Ω]) >0. Indeed, i we deno e by ϕa p incipal eigen unc ion associa ed wi h σ[−∆,Ω], hen ¡−∆−σ[−∆; Ω]χΩ1¢ϕ=σ[−∆; Ω] ¡1−χΩ1¢ϕ > 0, and, hanks o [8, Theo em 2.5], i is appa en ha µ(σ[−∆; Ω]) >0. Mo eo e , as a esul o he classical heo y o P. Hess and T. Ka o, µ(λ) is eal analy ic and conca e (e.g., [6] and [8]). Once in oduced hese no a ions, we can s a e ou main esul s. The nex one p o ides us wi h he bi u ca ion diag am o posi i e solu ions. Theo em 1.1 P oblem (1.2) possesses a posi i e solu ion i , and only i , 0< λ < λ0,(1.5) and i is unique i i exis s. Mo eo e , i we deno e i by θλ, hen, o each α∈(0,1), he map λ7→ θλis inc easing and o class C1((0, λ0); Cα 0(¯ Ω)). Fu he , lim λ↓0kθλkC1+α(¯ Ω) = 0 and lim λ↑λ0 kθλkC(K)=∞,(1.6) o any compac subse K⊂Ω. Combining linea and nonlinea di usion 5 In Figu e 3 we ha e ep esen ed he co esponding diag am o posi i e solu ions o (1.2). 0λλ λ 0 u θ Figu e 3. Bi u ca ion diag am in he gene al case. The bi u ca ion diag am consis s o an inc easing di e en iable cu e emana ing om u= 0 a λ= 0 and blowing-up o in ini y, e e ywhe e in Ω, as λ↑λ0. I should be no ed ha he u-bi u ca ion diag ams o (1.2) look simila o hose shown in Figu e 2 o (1.1). The nex esul es ablishes he exis ence o an homo opy be ween he wo limi ing bi u ca ion diag ams o Figu e 2 and he bi u ca ion diag am o Figu e 3. The concep o domain con e gence used in i s o mula ion is he one in oduced in [8], o which he e is con inuous dependence o he p incipal eigen alue and o he no malized p incipal eigen unc ion in W1,2 0. Theo em 1.2 Le m∈(1,∞)and {Ωε m}{0<ε≤1}a mono one amily o C2subdomains o Ωsuch ha Ω1 m= Ωmand Ωε 1:= Ω ¯ Ωε m,0< ε ≤1. Se mε=χΩε 1+m χΩε m,0< ε ≤1,(1.7) and deno e by θ[λ,ε],0< λ < λ0(Ωε 1),0< ε ≤1, he unique posi i e solu ion o ½−∆u=λu 1 mε(x)in Ω, u= 0 on ∂Ω.(1.8) Then, he ollowing asse ions a e ue: (a) I limε↓0Ωε 1= Ω, hen lim ε↓0λ0(Ωε 1) = σ[−∆; Ω] ,(1.9) and, o each λ∈(σ[−∆; Ω], λ0(Ω1)), he e exis s a unique ε0∈(0,1) such ha λ0(Ωε0 1) = λ. Mo eo e , limε↓ε0θ[λ,ε]=∞uni o mly on compac subse s o Ωand θ[λ,ε]=kθ[λ,ε]kC(¯ Ω)Φλ+o(kθ[λ,ε]kC(¯ Ω))as ε↓ε0in C1+α(¯ Ω) ,(1.10) whe e Φλs ands o he p incipal eigen unc ion o σ[−∆−λχΩε0 1 ; Ω] no malized so ha kΦλkC(¯ Ω) = 1, while lim ε↓0kθ[λ,ε]kC(¯ Ω) = 0 (1.11) i λ∈(0, σ[−∆,Ω]). 6M. Delgado, J. L´opez-G´omez and A. Su´a ez (b) I limε↓0Ωε m= Ω, hen lim ε↓0λ0(Ωε 1) = ∞(1.12) and, o each λ∈(0,∞), lim ε↓0kθ[λ,ε]−ΘλkC(¯ Ω) = 0 (1.13) whe e Θλs ands o he unique posi i e solu ion o he classical po ous media equa- ion ((1.2) wi h Ωm= Ω). The dis ibu ion o his pape is he ollowing. In Sec ion 2 we include he p oo o Theo em 1.1 and analyze he asymp o ic beha io o he posi i e solu ions o he pa abolic coun e pa o (1.2). Finally, in Sec ion 3 we p o e Theo em 1.2. 2 P oo o Theo em 1.1 Subsequen ly, we deno e by P he cone o posi i e unc ions o C1+α 0(¯ Ω); ◦ Ps anding o i s in e io . Gi en u, ∈ C1+α(¯ Ω), i is said ha u> i u− ∈P {0}, and uÀ i u− ∈◦ P. We al eady know ha λ > 0 is necessa y o he exis ence o a posi i e solu ion. Now, le ϕλÀ0 deno e a p incipal eigen unc ion associa ed o µ(λ) (c . (1.3)) and asume ha (1.2) possesses a posi i e solu ion, u. Then, mul iplying (1.2) by ϕλ, and in eg a ing in Ω i is appa en ha µ(λ)ZΩ uϕλ=λZΩm u1 mϕλ. Thus, µ(λ)>0 and, he e o e, λ<λ0(Ω1). Recall ha µ(λ)>0 i and only i 0 <λ< λ0(Ω1). This shows ha (1.5) is necessa y o he exis ence. To show ha (1.5) implies he exis ence o a posi i e solu ion we use he sha p e sion o he me hod o sub and supe solu ions de eloped by P. Hess [5] which demands no egula i y assump ions. Suppose (1.5) and conside ˜ ψ:=    ψin ¯ B , 0 in Ω B , (2.1) whe e Bis a ball wi h ¯ B⊂Ωmand ψs ands o he posi i e eigen unc ion associa ed o σ[−∆; B] no malized so ha kψkC(¯ B)= 1. I is ou ine o check ha he unc ion u:= ε˜ ψ is a weak subsolu ion o (1.2) i 0< ε ≤min (1,µλ σ[−∆; B]¶in Bm in Bm−1),(2.2) Combining linea and nonlinea di usion 7 since ∂ψ ∂n <0 on ∂B, whe e nis he ou wa d uni no mal ec o - ield o B. I should be no ed ha in B m>1. Ac ually, up o ides us wi h a subsolu ion o any λ > 0. Now, pick λ∈(0, λ0) and, o each su icien ly small δ > 0, conside Ωm,δ := {x∈Ωm: dis (x, ∂Ωm)> δ }, and µδ(λ) := σ[−∆−λχΩ1,δ ; Ω] , whe e Ω1,δ := Ω ¯ Ωm,δ . By he con inuous dependence o he p incipal eigen alue wi h espec o he po en ial, µδ(λ)>0 i δ > 0 is su icien ly small. Assume δhas been chosen in ha way. Le ϕδ λ deno e he posi i e eigen unc ion associa ed o µδ(λ) no malized so ha kϕδ λkC(¯ Ω) = 1. Then, he unc ion ¯u:= Kϕδ λ p o ides us wi h a posi i e supe solu ion o (1.2) i K≥max          1,  λ µδ(λ)µin Ωm,δ ϕδ λ¶ 1−supΩm,δ m supΩm,δ m  in Ωm,δ m in Ωm,δ m−1 ,µin Ωm ϕδ λ¶−1         .(2.3) No e ha in Ωm,δ m>1. Finally, by choosing ε > 0 su icien ly small and K > 1 su icien ly la ge, i is clea ha u≤¯uand, he e o e, (1.2) possesses a weak posi i e solu ion in he in e al [u, ¯u]; necessa ily s ong, by ellip ic egula i y. This concludes he p oo o he exis ence. To p o e he uniqueness we will adap he a gumen gi en in he p oo o [4, Theo em 3.2]. Suppose uis a posi i e solu ion o (1.2). Then,   ³−∆−λ u 1 m−1´u= 0 in Ω , u= 0 on ∂Ω, and, hence, by he uniqueness o he p incipal eigen alue, we ind ha σ[−∆−λ u 1 m−1; Ω] = 0 .(2.4) Suppose (1.2) possesses a u he posi i e solu ion 6=u. Then, −∆(u− ) = λ³u1 m− 1 m´=λ mZ1 0 [ u + (1 − ) ]1 m−1d (u− ). 8M. Delgado, J. L´opez-G´omez and A. Su´a ez Thus, se ing W:= −λ mZ1 0 [ u + (1 − ) ]1 m−1d , gi es    (−∆ + W) (u− ) = 0 in Ω , u− = 0 on ∂Ω.(2.5) In Ω1,W=−λ, while, in Ωm, Z1 0 [ u + (1 − ) ]1 m−1d < u 1 m−1Z1 0 1 m−1d =mu1 m−1, and, hence, W > −λu 1 m−1. Thus, by he mono onici y o he p incipal eigen alue wi h espec o he po en ial, we ind om (2.4) ha σ[−∆ + W; Ω] >0. As he p incipal eigen alue is dominan , om (2.5) i is appa en ha u= . This con adic ion ends he p oo o he uniqueness. Subsequen ly, o each λ∈(0, λ0), we deno e by θλ he unique posi i e solu ion o (1.2). The ac ha he map (0, λ0)−→ Cα 0(¯ Ω) λ7→ θλ (2.6) is o class C1 ollows easily om he implici unc ion heo em applied o he ope a o (0, λ0)×◦ PT −→ Cα 0(¯ Ω) (λ, u)7→ u−λ(−∆)−1³u1 m´ whose ze os a e in one- o-one co espondence wi h he posi i e solu ions o (1.2). Tis o class C1and, o each λ∈(0, λ0), DuT(λ, θλ) : C1+α 0(¯ Ω) −→ Cα 0(¯ Ω) is he linea con inuous compac ope a o de ined by DuT(λ, θλ)u:= u−λ(−∆)−1µ1 mθ 1 m−1 λu¶, u ∈ C1+α 0(¯ Ω) . Combining linea and nonlinea di usion 15 Fix ε < ε0. Thanks o he con inuous dependence o he p incipal eigen alue wi h espec o he domain, he e exis s δ0=δ(λ, ε)>0 such ha µδ ε(λ) := σ[−∆−λχΩε 1; Ωδ]>0 i δ∈[0, δ0). Fix one o hose δ’s and le ϕδdeno e he p incipal eigen unc ion o µδ ε(λ) no malized so ha kϕδkC(¯ Ω) = 1. A di ec calcula ion shows ha he unc ion ¯u:= Kϕδ p o ides us wi h a posi i e supe solu ion o (1.8) in Ω o each su icien ly small ε > 0 and su icien ly la ge K > 1, which can be chosen o be independen o ε. No ice ha hose supe solu ions a e bounded away om ze o all o e Ω. Also, hanks o (2.2), all he co esponding posi i e solu ions a e bounded bellow by a uni e sal posi i e unc ion —bellow he supe solu ion. By adap ing he compac ness a gumen o he p oo o Pa (a), one can easily see ha Θλ:= limε↓0θ[λ,ε]À0 is well de ined and ha i p o ides us wi h a posi i e solu ion o he po ous medium equa ion (i.e., (1.2) wi h Ωm= Ω). This concludes he p oo . Re e ences [1] A. G. A onson & L. A. Pele ie , La ge ime beha iou o solu ions o some po ous medium equa ion in bounded domains, J. Di . Eqns. 39 (1981), 378–412. [2] G. I. Ba enbla , On some uns eady mo ions o a liquid o a gas in a po ous medium, P ikl. Ma . Meh. 16 (1952), 67–68. [3] H. B ezis & L. Oswald, Rema ks on sublinea ellip ic equa ions, Nonl. Anal. T.M.A. 10 (1986), 55–64. [4] M. Delgado, J. L´ opez-G´ omez & A. Su´ a ez, Non-linea e sus linea di usion. F om classical solu ions o me asolu ions, Ad . Di . Eqns. 7(2002), 1101–1124. [5] P. Hess, On he sol abili y o nonlinea ellip ic bounda y alue p oblems, Ind. Uni . Ma h. J.,25 (1976) 461-466. [6] P. Hess & T. Ka o, On some linea and nonlinea eigen alue p oblems wi h an inde ini e weigh unc ion, Comm. Pa . Di . Eqns.,5(1980), 999-1030. [7] T. Ka o,Pe u ba ion Theo y o Linea Ope a o s, Classics in Ma hema ics, Sp inge , Be lin, 1995. [8] J. L´ opez-G´ omez, The maximum p inciple and he exis ence o p incipal eigen alues o some linea weigh ed bounda y alue p oblems, J. Di . Eqns.,127 (1996) 263- 294. 16 M. Delgado, J. L´opez-G´omez and A. Su´a ez [9] D. Sa inge ,Topics in S abili y and Bi u ca ion Theo y, Lec u es No es in Ma h- ema ics, 309, Sp inge , Be lin, 1973.