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.