Regularity for entropy solutions of parabolic p-Laplacian type equations

Publicacions Ma em`a iques, Vol 43 (1999), 665–683.
REGULARITY FOR ENTROPY SOLUTIONS OF
PARABOLIC p-LAPLACIAN TYPE EQUATIONS
S. Segu a de Le´
on and J. Toledo
Abs ac
In his no e we gi e some summabili y esul s o en opy solu-
ions o he nonlinea pa abolic equa ion u −di ap(x, ∇u)=
in ]0,T[×Ω wi h ini ial da um in L1(Ω) and assuming Di ichle ’s
bounda y condi ion, whe e ap(., .) is a Ca a h´eodo y unc ion sa -
is ying he classical Le ay-Lions hypo heses, ∈L1(]0,T[×Ω) and
Ω is a domain in RN. We find spaces o ype L (0,T;Mq(Ω)) con-
aining he en opy solu ion and i s g adien . We also include some
summabili y esul s when = 0 and he p-Laplacian equa ion is
conside ed.
1. In oduc ion
Le Ω be a domain in RN(bounded o no ) and le 1 <p<N. Le
ap:Ω×RN→RNbe a Ca a h´eodo y unc ion sa is ying he classical
Le ay-Lions condi ions in such a way ha di ap(x, ∇u(x)) defines an
ope a o om W1,p
0(Ω) on o W−1,p(Ω) (see [10]). The model example
o such unc ion is ap(x, ξ)=|ξ|p−2ξwhich de e mines he p-Laplacian
ope a o ∆pu= di (|∇u|p−2∇u).
1991 Ma hema ics subjec classifica ions: 35K65, 47H20.
This esea ch has been suppo ed by he Spanish DGICYT, P oyec o PB94-0960.
666 S. Segu a, J. Toledo
We a e in e es ed in he egula i y o en opy solu ions (see Defini-
ion 2.4 below) o he ollowing pa abolic p oblem:
(P) 




u −di ap(x, ∇u)= , in QT:=]0,T[×Ω,
u=0,on ST:=]0,T[×∂Ω,
u(0) = u0,in Ω;
whe e u0∈L1(Ω) and ∈L1(QT).
A dis ibu ional solu ion o his p oblem was ound in [9]. The no ion
o en opy solu ion o (P) is used in [1] and [11] o ob ain uniqueness.
(An equi alen concep , in e ms o eno malized solu ion, can be ound
in [7], [13], [14] and [16].) We ema k ha in [1] i is only conside ed he
case = 0, al hough he p oo o he gene al case can be easily ob ained
ollowing he same s eps and aking in o accoun he a gumen s o [6,
Chap e 4]. On he o he hand, he hypo heses in [11] a e p>2−1
N+1
and Ω bounded. In bo h a icles some summabili y esul s on he en opy
solu ion uand i s g adien a e gi en; mo e p ecisely, u∈M
(N+1)p−N
N(QT)
and |∇u|∈M
p−N
N+1 (QT) (see [1]), and |∇u|∈∩
q<p−N
N+1 Lq(QT) (see
[11]o [9]). These summabili y esul s a e ob ained join ly o ime and
space a iables; hus, as a consequence o hese pape s, i is no possible
o ge op imal egula i y when bo h a iables a e sepa a ely conside ed.
The pu pose o his no e is o gi e a p ecise summabili y esul o
he en opy solu ion and i s g adien wi h espec o space and ime.
This was s udied in [8] o weak solu ions o (P) in he amewo k o
Lebesgue and Sobole spaces when p≥2, u0= 0 and Ω is bounded.
The main esul o [8] s a es ha he e exis s a weak solu ion uo (P)
such ha u∈L (0,T;W1,q
0(Ω)) o p/2≤q<N(p−1)/(N−1) and
1≤ <q((N+1)p−2N)/((N+1)q−N), and o 1 ≤q<p/2 and
=p.
Some ema ks on hei assump ions a e as ollows. Fi s ly, i is possi-
ble o ge a simila esul i 2 −1/(N+1)<p<2, as i is obse ed in
[8, Rema k 1.7]. Mo eo e , analogous esul s also hold o o he ini ial
da a u0= 0, as i is poin ed ou in [12]. Ne e heless, hei hypo hesis
o Ω bounded canno be emo ed i hei a gumen s a e o be ollowed,
in ac i is needed om he e y begining when hey ob ain he a p i-
o i es ima es in [8, Lemma 2.2]. We wan o show in his pape wha
he si ua ion looks like in he amewo k o Ma cinkiewicz spaces (see
Defini ion 2.1 below) since we hink hey should be mo e sui able o
L1-da a (see [3] o he ellip ic case). We ob ain an imp o emen o he
esul s o [8] since ou a gumen s wo k o small p>1 and unbounded
domains.
Regula i y o en opy solu ions 667
Theo em 1. Le Ωbe a bounded domain and le ube he en opy
solu ion o p oblem (P).
(1) I 2N
N+1 <p<N, hen
u∈L (0,T;Mq(Ω)) o 1<q<N(p−1)
N−p
and <(N+1)p−2N
N
q
q−1,
|∇u|∈L (0,T;Mq(Ω)) o p
2<q<N(p−1)
N−1
and <q(N+1)p−2N
(N+1)q−N,
o q=p
2and <p, and
o q<p
2and =p.
(2) I 1<p≤2N
N+1 , hen
|∇u|∈Lp(0,T;Mq(Ω)) o q<p
2.
Rema k 1.1. When Ω is bounded, one can eplace Mq(Ω) by Lq(Ω)
due o he ela ions be ween Ma cinkiewicz and Lqspaces in bounded
domains, and o he ac ha qsa isfies s ic inequali ies.
Theo em 2. Le Ωbe an unbounded domain and le ube he en opy
solu ion o p oblem (P).
(1) I 2N
N+1 <p<N, hen
u∈L (0,T;Mq(Ω)) o 1<q<N(p−1)
N−p
and <(N+1)p−2N
N
q
q−1,
|∇u|∈L (0,T;Mq(Ω)) o p
2<q<N(p−1)
N−1
and <q(N+1)p−2N
(N+1)q−N.
668 S. Segu a, J. Toledo
(2) I 1<p< 2N
N+1 , hen
u∈L (0,T;Mq(Ω)) o N(p−1)
N−p<q<1
and <(N+1)p−2N
N
q
q−1,
|∇u|∈L (0,T;Mq(Ω)) o N(p−1)
N−1<q<p
2
and <q(N+1)p−2N
(N+1)q−N.
Rema k 1.2. Fo p= 2, he bound on is exac ly he same one ha
can be compu ed o he undamen al solu ion o he Hea equa ion.
We imp o e hese egula i y esul s when = 0 and he p-Laplacian
ope a o is conside ed.
Theo em 3. Le Ωbe a bounded o unbounded domain, 1<p<N
and le ube he en opy solu ion o p oblem (P) wi h he p-Laplacian
ope a o and =0, hen
u∈L (0,T;MN(p−1)
N−p(Ω)) o <p−1.
|∇u|∈L (0,T;MN(p−1)
N−1(Ω)) o <p−1.
|∇u|∈L (0,T;Mp
2(Ω)) o <p.
This pape is di ided in o fi e sec ions. The nex one is on p elimi-
na ies: we include he defini ions o Ma cinkiewicz spaces and en opy
solu ions o p oblem (P) and he fi s es ima es on he solu ion and i s
de i a i e. Sec ion 3 is de o ed o p o e Theo em 1 (Theo ems 3.1 and
3.2), while in sec ion 4 we p o e Theo em 2 (Theo ems 4.1 and 4.2). Fi-
nally, in sec ion 5, we p o e Theo em 3, and we also gi e o he egula i y
esul s when = 0 and he ini ial da um u0lies in Ms(Ω) ∩L1(Ω) o
Ls(Ω) ∩L1(Ω), 1 <s<2.
Regula i y o en opy solu ions 669
2. P elimina ies
Th oughou his no e he Lebesgue measu e on Ω will be deno ed by
µN.
Defini ion 2.1. Fo 0 <q<∞, he se o all measu able unc ions u:
Ω→Rsuch ha he unc ional [u]q:= supk>0kµN{|u|>k}1/q is fini e
is called a Ma cinkiewicz space and is deno ed by Mq(Ω).
I is s aigh o wa d ha , o bounded Ω, we ha e Mq(Ω) ⊂M
(Ω)
o <q. The connec ion be ween Ma cinkiewicz and Lebesgue spaces
is easy: Lq(Ω) ⊂M
q(Ω) ⊂L
loc(Ω) o <q(see, o ins ance, [17]); le
us poin ou ha Ma cinkiewicz spaces a e also known as weak-Lebesgue
spaces.
When q>1, he Ma cinkiewicz space Mq(Ω) is a Banach space wi h
he no m defined by uq= sup >0 1−q
q
0u∗(τ)dτ, whe e u∗(τ)=
in k>0:µN{|u|>k}≤τdefines he non-inc easing ea ange-
men o u(see, o ins ance, [17, Defini ion 1.8.6]). As a consequence o
his defini ion one has ha K|u|≤µN{K}q−1
quq o any K⊂Ω wi h
fini e measu e. No e ha , endowed wi h his no m, i u, ∈M
q(Ω),
hen 
0u∗(τ)dτ ≤
0 ∗(τ)dτ o all >0 implies uq≤ q; ha is,
Mq(Ω) is a no mal space in he sense o [5, Defini ion 2.8].
Defini ion 2.2. I ,q∈]0,+∞[, we will say ha a measu able unc-
ion u:]0,T[×Ω→Rbelongs o L (0,T;Mq(Ω)) i T
0[u( )]
qd is fini e.
The ollowing esul is well known; a p oo may be seen, o ins ance,
in [2, Lemma 1.3].
Lemma 2.3. Le I⊂Rbe an in e al and deno e by Λ he se o
all measu able unc ions λ:[0,T]→I.Le :[0,T]×I→]0,+∞[
be a unc ion such ha o each λ∈Λ he unc ion → ( , λ( )) is
measu able. Then T
0supk∈I ( , k)d = supλ∈ΛT
0 ( , λ( )) d .
Be o e in oducing he concep o en opy solu ion, we will define he
ollowing unc ional spaces (see [3]). Gi en k>0, define he unca u e
ope a o by Tk(ζ)=(ζ∧k)∨(−k), whose p imi i e is Jk(ξ)=ξ
0Tk(s)ds.
The space o all measu able unc ions u:Ω→Rsuch ha Tku∈
W1,1
loc (Ω) and ∇Tku∈Lp(Ω) o all k>0 is deno ed by T1,p(Ω) while
T1,p
0(Ω) deno es he space o all unc ions u∈T1,p(Ω) such ha o e e y
k>0 he e exis s a sequence (φn)inC∞
0(Ω) sa is ying ∇φn→∇Tkuin

670 S. Segu a, J. Toledo
Lp(Ω) and φn→Tkuin L1
loc(Ω). I u∈T1,p(Ω), a de i a i e ∇ucan be
defined as he unique measu able unc ion such ha ∇Tku=∇u·χ{|u|<k}
o all k>0 (see [3, Lemma 2.1]).
Defini ion 2.4. We will say ha a unc ion u∈C[0,T]; L1(Ω)is
an en opy solu ion o (P) i u( )∈T1,p
0(Ω) o almos all ,∇Tk(u)∈
Lp(QT) o all k>0 and

0Ω
ϕTk(u−ϕ)+
0Ω
ap(x, ∇u),∇Tk(u−ϕ)
≤Ω
Jku0−ϕ(0)−Ω
Jku( )−ϕ( )+
0Ω
Tk(u−ϕ)
o all k>0, ∈[0,T] and ϕ∈L∞(QT)∩Lp0,T;W1,p
0(Ω)∩
W1,10,T;L1(Ω).
Taking ϕ= 0 in he o mula ion o en opy solu ion i ollows ha
1
kT
0Ω
ap(x, ∇u),∇Tku
≤Ω
Jk(u0)
k+T
0Ω
Tku
k≤Ω
|u0|+T
0Ω
| |
o all k>0. Consequen ly,
(2.1) T
0Ω
|∇Tku( )|p
k≤M, ∀k>0,
being M=Ω|u0|+T
0Ω| |. (Recall ha one o he Le ay-Lions as-
sump ions asse s ha he e exis s α>0 sa is ying α|ξ|p≤ap(x, ξ),ξ;
we conside α= 1 since he e is no loss o gene ali y in doing so.)
F om now on uwill deno e he en opy solu ion o (P).
We will finish his sec ion gi ing he basic es ima es in o de o p o e
ou egula i y esul s. We will use he ac ha u∈C([0,T],L
1(Ω)) and
he es ima e gi en in (2.1).
P oposi ion 2.5. Fo e e y δ>0 he e is g∈L1(0,T)such ha
Ω
|∇Tku( )|p
k=kδg( ),i ∈[0,T]and k≥1;
k−δg( ),i ∈[0,T]and k≤1.
Regula i y o en opy solu ions 671
P oo : We jus p o e he fi s asse ion, he second one can be p o ed
simila ly. Conside a measu able unc ion λ:[0,T]→[1,+∞[ and he
se s Aj:= { ∈[0,T]:2
j≤λ( )<2j+1},j≥0, and compu e o ge he
ollowing inequali ies
T
0Ω
|∇Tλ( )u( )|p
λ( )1+δ≤
∞

j=0 AjΩ
|∇T2j+1 u( )|p
2j(1+δ)
=
∞

j=0
2
2jδ AjΩ
|∇T2j+1 u( )|p
2j+1 ≤2M
∞

j=0
1
2jδ .
By Lemma 2.3,
T
0
sup
k≥1Ω
|∇Tku|p
k1+δ<∞,
and now Fubini’s heo em allows us o ake g( )≥supk≥1Ω
|∇Tku( )|p
k1+δ.
Obse e ha he e is no loss o gene ali y in aking g( )≥MN−p
Nas
we will do in ou ollowing es ima es.
P oposi ion 2.6. Le δ∈]0,p−1[. Then he e exis s a cons an C>0
such ha o almos all ∈[0,T],
sup
k≥1
kµN{|u( )|>k}N−p
N(p−1−δ)≤(Cg( )) 1
p−1−δ
(1)
sup
k≤1
kµN{|u( )|>k}N−p
N(p−1+δ)≤(Cg( )) 1
p−1+δ.(2)
P oo : P oposi ion 2.5 and Sobole ’s inequali y imply wo ac s: on
he one hand, o all k≥1 and almos all ,
µN{|u( )|≥k}≤Ω
|Tku( )|p∗
kp∗
≤C
kp∗Ω
|∇Tku( )|pN
N−p
≤Cg( )N
N−p
kN(p−1−δ)/(N−p),
and, on he o he hand, o all k≤1 and almos all ,
µN{|u( )|≥k}≤Ω
|Tku( )|p∗
kp∗
≤C
kp∗Ω
|∇Tku( )|pN
N−p
≤Cg( )N
N−p
kN(p−1+δ)/(N−p),
om whe e he esul ollows.
672 S. Segu a, J. Toledo
P oposi ion 2.7. Le δ∈]0,p−1[. Then he e exis s a cons an C>0
such ha o almos all ∈[0,T],
sup
h≥g( )1/pM−1/p
hµN{|∇u( )|>h}2+δ
p≤(Cg( ))1
p.(1)
sup
h≥g( )−1/(N−p)
hµN{|∇u( )|>h}N−1−δ
N(p−1−δ)≤(Cg( )) 1
p−1−δ.(2)
sup
h≤g( )1/pM−1/p
hµN{|∇u( )|>h}2−δ
p≤(Cg( ))1
p.(3)
sup
h≤g( )−1/(N−p)
hµN{|∇u( )|>h}N−1+δ
N(p−1+δ)≤(Cg( )) 1
p−1+δ.(4)
P oo : A simila a gumen o he one in [3, Lemma 4.2] can be applied.
We jus p o e (1) and (2), since he o he asse ions can be p o ed in an
analogous manne .
Be o e showing (1), le us poin ou ha he ope a o associa ed wi h
he ellip ic e m −di ap(x, ∇u) + he Di ichle bounda y condi ion, is
acc e i e in L1(Ω) (see [3, Theo em 7.1] o [1]); hus, u( )1≤u01+
 1 o all ∈[0,T] and so
(2.2) µN{|u( )|≥k}≤1
kΩ
|u( )|≤1
kΩ
|u0|+T
0Ω
| |=M
k.
This ac and P oposi ion 2.5 imply he ollowing inequali ies.
µN{|∇u( )|>h}≤µN{|∇u( )−∇Tku( )|>h/2}+µN{|∇Tku( )|>h/2}
≤µN{|u( )|≥k}+2pΩ
|∇Tku( )|p
hp≤M
k+2pk1+δg( )
hp.
By aking k=hpM
g( )1/(2+δ)≥1, we ge
µN{|∇u( )|>h}≤(Ch−pg( ))1/(2+δ),
and (1) ollows.
To show (2), apply P oposi ion 2.6 (1) o ob ain kp−1−δµN{|u( )|≥
k}N−p
N≤Cg( ) o all k≥1. Then, using P oposi ion 2.5,
µN{|∇u( )|>h}≤µN{|u( )|≥k}+2
pΩ
|∇Tku( )|p
hp
≤Cg( )N
N−p
kN(p−1−δ)/(N−p)+Ck1+δg( )
hp.
Taking k=g( )1/(N−1−δ)h(N−p)/(N−1−δ)≥1, (2) is ob ained.
Regula i y o en opy solu ions 673
3. Case Ωbounded
Theo em 3.1. I 2N
N+1 <p<N, hen
u∈L (0,T;Mq(Ω)) o 1<q<N(p−1)
N−p
and <(N+1)p−2N
N
q
q−1.
P oo : Le 2N
N+1 <p<Nand qbe as abo e and ake δ>0 such
ha p>N(2+δ)
N+1 and 1 <q<N(p−1−δ)
N−p. I will be enough o show ha
u∈L 0,T;Mq(Ω) o =p(N+1)−N(2+δ)
N
q
q−1.
Fi s obse e ha he abo e inequali ies imply ha >p−1−δ; hus,
o k≥1, we ob ain ha
k µN{|u( )|≥k} /q
=kp−1−δµN{|u( )|≥k}N−p
N·k −p+1+δµN{|u( )|≥k} −p+1+δ.
On he one hand, applying P oposi ion 2.6, kp−1−δµN{|u( )|≥k}N−p
N≤
Cg( ) and on he o he hand, by (2.2), kµN{|u( )|≥k}≤M, so ha
sup
k≥1
k µN{|u( )|≥k} /q ≤CM −p+1+δg( ),
gbeing an in eg able unc ion.
Since we also ha e, o 0 <k≤1, ha k µN{|u( )|≥k} /q ≤
µN{Ω} /q; he unc ion defined by [u( )]
qis in eg able and consequen ly
u∈L (0,T;Mq(Ω)).
Theo em 3.2.
(1) I 2N
N+1 <p<N, hen
|∇u|∈L (0,T;Mq(Ω)) o p
2<q<N(p−1)
N−1
and <q(N+1)p−2N
(N+1)q−N,
o q=p
2and <p, and
o q<p
2and =p.
(2) I 1<p≤2N
N+1 , hen
|∇u|∈Lp(0,T;Mq(Ω)) o q<p
2.
680 S. Segu a, J. Toledo
Theo em 5.2. Le ube he en opy solu ion o (P) and le 1<s<2.
(1) I u0∈Ls(Ω) ∩L1(Ω), hen u∈L (0,T;Mq(Ω)), whe e ≤
q(N+s)p−2N
N(q−s)and s<q≤N(p+s−2)
N−p, and |∇u|∈L (0,T;Mq(Ω)), whe e
≤q(N+s)p−2N
(N+s)q−sN and qlies be ween N(p+s−2)
N+s−2and sp
2.
(2) I u0∈M
s(Ω) ∩L1(Ω), hen u∈L (0,T;Mq(Ω)), whe e <
q(N+s)p−2N
N(q−s)and s<q<N(p+s−2)
N−p, and |∇u|∈L (0,T;Mq(Ω)), o q
s ic ly be ween N(p+s−2)
N+s−2and sp
2, and <q(N+s)p−2N
(N+s)q−sN .
To finish his pape , we p o e ha he abo e es ima es a e sha p
in he amewo k o Ma cinkiewicz spaces. We will show ha , in he
case p= 2 and u0∈M
s(Ω) ∩L1(Ω), we ha e u∈L (0,T;Mq(Ω)),
wi h <q 2s
N(q−s)and s<q< Ns
N−p, bu u/∈L (0,T;Mq(Ω)), wi h
=q2s
N(q−s)and s<q< Ns
N−p.
Example 5.3. Le us conside Ω = RNand he p oblem o finding
u∈C[0,T],L
1(Ω)such ha
u −∆u=0,in ]0,T[×RN;
u(0,.)=u0,in RN;
whe e u0(x)= 1
|x|αχB(0,1)(x), wi h N/2<α<N. Deno ing s=N/α,
i is s aigh o wa d ha 1 <s<2 and u0∈M
s(RN)∩L1(RN), bu
u0/∈Ls(RN). We know ha he solu ion o his linea p oblem is gi en
by
u( , x)= 1
(4π )N/2RN
e−|x−y|2
4 u0(y)dy.
Changing a iables: x=η /2and y=ξ /2, we ob ain
T
0
[u( )]
qd
=T
0
sup
k>0
k µ{x∈RN:|u( , x)|>k} /q d
=T
0
sup
k>0
k RN
χ1/(4π)N/2RNe
−|x−y|2
4 u0(y)dy>k N/2(x)dx /q
d
=T
0
N
2qsup
k>0
k RN
χ1/(4π)N/2RNe
−|η−ξ|2
4u0(ξ 1/2)dξ>k(η)dη /q
d
=T
0
N
2qsup
k>0
k µ


η∈RN:1
(4π)N/2B(0, −1/2)
e−|η−ξ|2
4
|ξ|αdξ >k α/2


/q
d .

Regula i y o en opy solu ions 681
By deno ing now h=k 1/2and g( , η)= 1
(4π)N/2B(0, −1/2)
e
−|η−ξ|2
4
|ξ|αdξ,
we ge
T
0
[u( )]
qd =T
0
2(N
q−α)sup
h>0
h µ{η∈RN:g( , η)>h} /q d .
Define
g1(η)= 1
(4π)N/2B(0,T −1/2)
e−|η−ξ|2
4
|ξ|αdξ,
g2(η)= 1
(4π)N/2RN
e−|η−ξ|2
4
|ξ|αdξ
and ci= suph>0h µ{η∈RN:gi(η)>h} /q,i=1,2 (obse e ha bo h
cons an s a e fini e). Since g1(η)≤g( , η)≤g2(η) o all ∈[0,T], i
ollows ha
c2T
0
2(N
q−α)d ≤T
0
[u( )]
qd ≤c1T
0
2(N
q−α)d .
Thus, i <q 2s
N(q−s), hen
2N
q−α>−1 and so T
0[u( )]
qd <
+∞; howe e , i =q2s
N(q−s), hen
2N
q−α=−1 and consequen ly
T
0[u( )]
qd =+∞. The e o e, in he fi s case, he solu ion ubelongs
o L (0,T;Mq(RN)), bu i is no so in he second one.
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Regula i y o en opy solu ions 683
17. W. P. Zieme ,“Weakly Diffe en iable Func ions,” Sp inge -Ve lag,
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Depa amen d’An`alisi Ma em`a ica
Uni e si a de Val`encia
D . Moline 50
46100 Bu jasso
Val`encia
SPAIN
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