Electronic Journal of Qualitative Theory of Differential Equations 2004, No. 14, 1-14;http://www.math.u-szeged.hu/ejqtde/
On the local integrability and boundedness of
solutions to quasilinear parabolic systems
∗
Tiziana Giorgi
†Mike O’Leary
September 6, 2004
Abstract
We introduce a structure condition of parabolic type, which allows for the gen-eralization to quasilinear parabolic systems of the known results of integrability, and boundedness of local solutions to singular and degenerate quasilinear parabolic equations.
1
Introduction
In this note, we investigate under which conditions it is possible to extend to systems the results of local integrability and local boundedness known to hold for solutions to a general class of degenerate and singular quasilinear parabolic equations. In particular, we show that the results presented by DiBenedetto in [1, Chp. VIII] are true for a larger class of problems, by providing conditions under which one can recover for weak solutions of quasilinear parabolic systems the work contained in [5, 6]. Fundamental to our approach is a new condition for the parabolicity of systems, which can be viewed as the extension of an analogous notion for parabolic equations, introduced in [1, Lemma 1.1 pg 19].
Generalizations of the results in [1, Chp. VIII] to initial-boundary value problems for systems have been proven in [7].
We study systems of the general form:
∂ ∂tui−
∂
∂xjAij(x, t, u,∇u) =Bi(x, t, u,∇u) (1)
fori= 1,2, ..., n, and(x, t)∈ΩT ≡Ω×(0, T)withΩ⊆RN; where we assumeAij
andBi to be measurable functions inΩ×(0, T)×Rn×RN n, herei = 1,2, ..., n;
j= 1,2, ..., N.
∗1991 Mathematics Subject Classifications: 35K40, 35K655.
Key words and phrases: Singular and degenerate quasilinear parabolic systems, local integrability, local
boundedness
†Partially supported by the National Science Foundation-funded ADVANCE Institutional Transformation
By a weak solution of (1), we mean a functionu = (u1, u2, . . . , un)withu ∈
L∞,loc(0, T;L2,loc(Ω))∩Lp,loc(0, T;Wp,loc1 (Ω))for somep >1, which verifies
Z Z
ΩT
−ui
∂φi
∂t +Aij(x, t, u,∇u) ∂φi ∂xj
dxdt=
Z Z
ΩT
Bi(x, t, u,∇u)φidxdt (2)
for allφ= (φ1, φ2, . . . , φn)∈C0∞(ΩT;Rn).
To the system (1), we add the following classical structure conditions (see [1, Chp. VIII]). For a.e.(x, t)∈ΩT, everyu∈Rn, andv∈RN n, we assume that
(H1)
N X
j=1
n X
i=1
Aij(x, t, u, v)vij ≥C0|v|p−C3|u|δ−φ0(x, t);
(H2) |Aij(x, t, u, v)| ≤C1|v|p−1+C4|u|δ(1− 1
p)+φ
1(x, t);
(H3) |Bi(x, t, u, v)| ≤C2|v|p(1− 1
δ)+C5|u|δ−1+φ2(x, t),
for C0 > 0,C1, C2, . . . , C5 ≥ 0, with δ s.t. 1< p≤δ <
N+ 2
N
p≡m, and
whereφ0, φ1, φ2are non-negative functions which satisfy
(H4) φ0∈L1,loc(ΩT),φ1∈L p p−1,loc
(ΩT), andφ2∈L m
m−1,loc(ΩT)
.
Finally, we introduce and assume the parabolicity condition
(H5)
n X
i,k=1
N X
j=1
Aij(x, t, u, v)uiukvkj ≥0.
The main result of our work is the complete recovery for systems of the form (1) of Theorem 1 in [5]:
Theorem 1 Letube a weak solution of (1), and suppose that the structure conditions (H1)-(H5) hold true, together with the following additional hypotheses:
(H6) φ0∈Lµ,loc(ΩT),φ1, φ2∈Ls,loc(ΩT), whereµ >1ands >
(N+ 2)p
(N+ 2)p−N;
(H7) u∈Lr,loc(ΩT), withr >1andN(p−2) +rp >0.
Ifs, µ > (Np+p), thenu∈L∞,loc(ΩT);
ifs=µ= (Np+p), thenu∈Lq,loc(ΩT)for anyq <∞;
ifs, µ < (Np+p), thenu∈Lq,loc(ΩT)for anyq < q∗, where
q∗= min
s(N p+p−N)
sN−(s−1)(N+p),
µ(N p+ 2p)
µN−(µ−1)(N+p)
Remark We would like to point out that the parabolicity condition (H5) is a quite
natural one to consider. In fact, for the case of a single equation it reduces to the condition
Aj(x, t, u, v)u2vj ≥0,
which, foru 6= 0, is equivalent to the weak parabolicity condition presented in [1, Lemma 1.1, p.19].
Further, in the simple case where
∂ui ∂t −
∂ ∂xj
ajm(x, t, u,∇u)∂ui
∂xm
=Bi(x, t, u,∇u);
our requirement is satisfied if the matrixajm(x, t, u,∇u)is for example positive defi-nite. Indeed, since for the above system one has the identity
n X
i,k=1
N X
j=1
Aij(x, t, u, v)uiukvkj =
n X
i,k=1
N X
j,m=1
ajm(x, t, u, v) (uivim) (ukvkj),
(H5) can be rewritten as
n X
i,k=1
N X
j=1
Aij(x, t, u, v)uiukvkj =
N X
j,m=1
ajm(x, t, u, v)wmwj ≥0,
where we setwh=P lulvlh.
Finally, we note that (H5) is not so restrictive that the equation must have one of these simple forms. For example, consider the perturbation
Aij(x, t, u, v) =ajm(x, t, u, v)vim+αij(x, t, u, v)
where the matrixajmis positive definite. Define
λ(x, t, u, v) = min
|w|=1ajm(x, t, u, v)wjwm>0;
this exists and is obtained because
w7→ajm(x, t, u, v)wjwm
is positive and continuous for each(x, t, u, v)on the compact set{w∈RN :|w|= 1}.
Then for any vectorw∈RN, w6=0
ajm(x, t, u, v)wjwm=ajmwj |w|
wm |w||w|
2≥
λ|w|2.
Condition (H5) will be verified if the perturbationαij satisfies the smallness con-dition
N X
j=1
n X
i=1
|αij(x, t, u, v)ui| ≤λ(x, t, u, v)
N X
j=1
n X
i=1
Indeed, we have
We follow the approach of [1, 5, 6] and start with the derivation, presented in tion 2, of a local energy estimates for weak solutions to (1). We then outline, in Sec-tion 3 and SecSec-tion 4 how the methods in [5] can be applied to obtain local integrability and boundedness.
We also remark that the techniques presented can be modified to handle doubly degenerate problems, whereAij(x, t, u, v)vij ≥ Φ(|u|)|v|p−C
3|u|δ −φo(x, t)for
someΦ, following the same lines as the proof in [6].
2
Energy Estimates for
u
2.1
Notation & Preliminaries
Let(x0, t0) ∈ ΩT, without loss of generality we can assume(x0, t0) = (0,0). For kernel in space-time, and for a given functionf we use the notationfη ≡ Jη∗f to represent its convolution withJη.
verifies
0≤f′(s)≤
0 s < κ,
1
ǫ κ < s <2κ,
1
s κ s >2κ,
(4)
provided0< ǫ <12.
We are now ready to start the derivation of our energy estimate. Fixη >0,κ >0
and consider the test function{ui,η(x, t)f(|uη(x, t)|)ζkp(x, t)}η.Because this is aC0∞
function forηsufficiently small, we can substitute it into the definition of weak solution to obtain
Z Z
ΩT
−ui∂
∂t{ui,ηf(|uη|)ζ p
k}η dx dt
+
Z Z
ΩT
Aij(x, t, u,∇u) ∂
∂xj {ui,ηf(|uη|)ζ p
k}η dx dt
=
Z Z
ΩT
Bi(x, t, u,∇u){ui,ηf(|uη|)ζkp}η dx dt.
(5)
For convenience of notation, we rewrite (5) in compact form asI1+I2=I3, and
discuss each of these terms in turn.
2.2
Estimate of
I
1We begin by using the symmetry of the mollifying kernel, and integration by parts to rewriteI1as
I1=−
Z Z
ΩT
ui,η∂
∂t{ui,ηf(|uη|)ζ p k} dx dt
=
Z Z
Qτ R
∂ ∂tui,η
ui,ηf(|uη|)ζkpdx dt.
We then notice that summing over the indexiimplies
X
i ui,η∂
∂tui,η=
1 2
X
i ∂ ∂t(ui,η)
2
= 1 2
∂ ∂t|uη|
2=|
uη|∂
∂t|uη|, (6)
and we derive
I1=
Z Z
Qτ R
|uη|∂|uη|
∂t f(|uη|)ζ p k dx dt.
If we now letk→ ∞, thanks to the uniform convergence ofζk →ζ, and the smooth-ness of the mollified functions we obtain
lim
k→∞I1=
Z Z
Qτ R
|uη|∂|uη|
Proceeding in a standard fashion, we rewrite the integral on the right hand side as
whereγ2is a constant that depends onσ, Randp. (Note that the above limits are zero
2.3
Estimate of
I
2We start as in Section 2.2, and use the symmetry of the mollifying kernel to rewriteI2:
I2=
Z Z
Qτ R
Aij,η(x, t, u,∇u) ∂
∂xj{ui,ηf(|uη|)ζ p
k} dx dt.
We then take the limit fork→ ∞, and by the smoothness of the mollified functions we obtain
lim
k→∞I2=
Z Z
Qτ R
Aij,η(x, t, u,∇u) ∂
∂xj {ui,ηf(|uη|)ζ
p} dx dt. (9)
As done while deriving the estimate for I1, we would like to consider the limit
forη ↓ 0as well. To do so, we notice that the structure condition (H2) implies the inequality
Z Z
Qτ R
|Aij(x, t, u,∇u)|p−p1 dx dt≤γ Z Z
Qτ R
h
|∇u|p+|u|δ+φ
p p−1
1
i dx dt.
From which, we have thatAij(x, t, u,∇u)∈ L p p−1(
Qτ
R), sinceδ < mand since by
the classical embedding theorems for parabolic spaces we know
u∈L∞,loc(0, T;L2,loc(Ω))∩Lp,loc(0, T;Wp,loc1 (Ω))֒→Lm,loc(ΩT). (10)
Therefore, we obtain Aij,η(x, t, u,∇u)−→η↓0 Aij(x, t, u,∇u)inL p p−1(
QτR).
On the other hand,
∂
∂xj{ui,ηf(|uη|)ζ
p}= ∂ui,η
∂xj f(|uη|)ζ p
+ui,ηf′(|uη|)∂|uη|
∂xj ζ p+
pui,ηf(|uη|)ζp−1 ∂ζ ∂xj;
hence fromui,η→uiand∇ui,η→ ∇uialmost everywhere [3, Appendix C, Theorem 6] we conclude that
∂
∂xj{ui,ηf(|uη|)ζ
p} → ∂
∂xj{uif(|u|)ζ
p} a.e.
If next we use our estimates forfandf′, we have the upper bound
∂
∂xj{ui,ηf(|uη|)ζ p}
p
≤
|∇ui,η|+ 2κ1
ǫ|∇uη|+|uη|
1
κ|uη||∇uη|+C|uη| p
≤C{|∇uη|p+|uη|p},
which, applying a slight generalization of Lebesgue’s Dominated Convergence Theo-rem [4,§1.8], gives
∂
∂xj{ui,ηf(|uη|)ζ
p}−→η↓0 ∂
∂xj{uif(|u|)ζ p} in
We then have that equation (9) yields
The first integral above can be estimated with the help of (H1) as follows:
Z Z
To handle the second integral, we use the parabolicity assumption (H5), and the
equal-ity∂|u|
For the last integral, we need (H2) to derive
Z Z
Finally, we combine (11), (12), (13), and (14) so to obtain the inequality:
2.4
Estimate of
I
3Once again, our first step is to rewriteI3in the form
I3=
Z Z
Qτ R
Bi,η(x, t, u,∇u){ui,ηf(|uη|)ζkp} dx dt,
and to consider the limit fork→ ∞:
lim
k→∞I3=
Z Z
Qτ R
Bi,η(x, t, u,∇u){ui,ηf(|uη|)ζp} dx dt.
To justify taking the limit forη ↓ 0 in this case, we proceed by noticing that (H3) implies
|Bi(x, t, u,∇u)|mm−1 ≤C2|∇u|p(1−
1
δ)( m
m−1) +C5|u|m
δ−1
m−1 +φ
m m−1
2 (x, t).
Which yieldsBi,η −→ Bi inL m m−1(
QτR), in view of the embedding (10), and the
relationsδ < m,p 1−1
δ
m
m−1
=p11−−11/m/δ< p. Moreover, since we know that
ui,ηf(|uη|)ζp −→uif(|u|)ζp for a.e.(x, t), and |ui,ηf(|uη|)ζp|m≤C|uη|m;
we can apply the same generalization of Lebesgue’s Dominated Convergence Theorem to see that
ui,ηf(|uη|)ζp−→uif(|u|)ζp inLm(QτR).
Thus,
lim
η↓0klim→∞I3=
Z Z
Qτ R
Bi(x, t, u,∇u)uif(|u|)ζpdx dt, (16)
and we can use (H3) once more to conclude
lim
η↓0klim→∞I3≤C2
Z Z
Qτ R
|∇u|p(1−1δ)|u|f(|u|)ζpdx dt
+C5
Z Z
Qτ R
|u|δf(|u|)ζpdx dt+
Z Z
Qτ R
φ2(x, t)|u|f(|u|)ζpdx dt. (17)
2.5
The Energy Estimate
To derive our energy estimate (presented in Proposition 3 below), we use the interme-diate result stated as Lemma 2. This is a direct consequence of equations (8), (15) and (17): starting from (5), one can use the bounds given in the previous sections, and
then apply Young’s inequality to treat the terms involving|∇u|p−1,|∇u|p(1−1
δ), and
Lemma 2 Letp > 1, letf be defined by (3), and letu ∈L∞,loc(0, T;L2,loc(Ω))∩ Lp,loc(0, T;Wp,loc1 (Ω))be a weak solution of (1). If the assumptions (H1)-(H5) are verified, then for anyQτ
R(x0, t0) =BR(x0)×(t0−Rp, τ)⊂⊂ΩT we have
To extract useful information from Lemma 2, we need to substitute our choice of
Moreover, since
f(|u|) = (|u| −κ)+ (|u| −κ)++ǫ
↑χ[|u|> κ] asǫ↓0,
we can use the Monotone Convergence Theorem to pass to the limit asǫ ↓ 0 in the remaining terms of (18), and gather the bound
1 2
Z
BR
(|u| −κ)2+ζpdx
t=τ
+
Z Z
Qτ R
|∇u|pχ[|u|> κ]ζpdx dt
≤γ Z Z
Qτ R
|u|δχ[|u|> κ]dx dt+
Z Z
Qτ R
|u|pχ[|u|> κ]|∇ζ|pdx dt
+
Z Z
Qτ R
|u|2χ[|u|> κ]ζtdx dt+
Z Z
Qτ R
φ0(x, t)χ[|u|> κ]dx dt
+
Z Z
Qτ R
φ1(x, t)|u|χ[|u|> κ]|∇ζ|dx dt+
Z Z
Qτ R
φ2(x, t)|u|χ[|u|> κ]dx dt
!
.
In turn, the above inequality leads to the classical local energy estimate stated in Propo-sition 3 below, if one takes in account the relation∇|u|
p
≤ |∇u|p.
Proposition 3 (Local Energy Estimate) Under the hypotheses (H1)-(H5), ifuis a weak solution of (1) then forQR(x0, t0) = BR(x0)×(t0−Rp, t0)⊂⊂ ΩT,0 < σ <1,
andκ >0
ess sup
−Rp<τ <0
Z
BσR
(|u| −κ)2+dx
t=τ
+
Z Z
QσR
|∇(|u| −κ)+|pζpdx dt
≤γ Z Z
QR
|u|δχ[|u|> κ]dx dt+ 1
(1−σ)pRp Z Z
QR
|u|pχ[|u|> κ]dx dt
+ 1
(1−σ)pRp Z Z
QR
|u|2χ[|u|> κ]dx dt+Z Z
QR
φ0(x, t)χ[|u|> κ]dx dt
+ 1
(1−σ)R Z Z
QR
φ1(x, t)|u|χ[|u|> κ]dx dt
+
Z Z
QR
φ2(x, t)|u|χ[|u|> κ]dx dt
.
(20)
for some constantγ=γ(C0, C1, C2, C3, C4, C5, p).
3
Higher Integrability of
u
parabolic spaces [1, Chap. 1], and hypotheses (H6) for the functionsφ0, φ1, andφ2,
we have
Z Z
QσR
(|u| −κ)p(
N+2
N )
+ ζpdx dt
1+1p N
≤γ Z Z
QR
|u|δχ[|u|> κ]dx dt
+ γ
(1−σ)pRp Z Z
QR
|u|pχ[|u|> κ]dx dt
+ γ
(1−σ)pRp Z Z
QR
|u|2χ[|u|> κ]dx dt+γ||φ0||Lµ(QR)(meas[|u|> κ])
1−1
µ
+γ ||φ
1||Ls(QR)
(1−σ)R +||φ2||Ls(QR)
Z Z
QR
|u|χ[|u|> κ]dx dt.
(21)
Inequality (21) is the key link needed to obtain for our systems exactly the same higher integrability result proven in [5, Proposition 3] for single equations:
Proposition 4 Under the hypotheses and notation of Theorem 1, we have that
ifs, µ≥ (Np+p), thenu∈Lq,loc(ΩT)for anyq <∞;
ifs, µ < (Np+p), thenu∈Lq,loc(ΩT)for anyq < q∗.
Indeed, supposeu∈Lβ,loc. Then we can use (21) to see that
κ(NN+2)pmeas
QσR
[|u|>2κ]
1+1p/N
≤Cγ,R,σ,p||u||βLβ(QR) (
1 κ
β−δ
+
1 κ
β−p
+
1 κ
β−2)
+C||u||β(1−
1
µ)
Lβ(QR)
1
κ
β(1−1µ)
+C||u||β(1−
1
s)
Lβ(QR)
1
κ
β(1−1s)−1
.
Therefore, if
α(β) = N+ 2
N p+
1 + p
N
min
β−2, β−δ, β
1−1 s
−1, β
1− 1 µ
thenu∈Lweakα(β),loc. Thusu∈Lq,loc(QR)for allq < α(β), and we can iterate this
process starting fromβ0= max{2,
N+ 2
N p, r}to obtain the result. The details can be
found in [5].
4
Boundedness of
u
thatQρ⊂⊂ΩT. For each integern, we define
ρn=σρ+(1−σ) 2n ρ,
and setQn=Qρn. Next we fixκ >0to be chosen later, and set
κn=κ
1− 1
2n+1
.
ForNN+2p >2, we consider
Yn = 1
measQn Z Z
Qn
|u−κn|mdx dt,
while forNN+2p≤2, we take
Yn= 1
measQn Z Z
Qn
|u−κn|λdx dt,
forλsufficiently large. This is well defined thanks to the local integrability proven in Section 3. We then apply the local energy estimate (21)in a standard way to obtain an estimate of the form
Yn+1≤γ(B1nYn1+ǫ1+B2nYn1+ǫ2+Bn3Yn1+ǫ3),
for positive constantsγ, B1, B2, B3, ǫ1, ǫ2 andǫ3. As final step, we chooseκ
suffi-ciently large so to haveYn→0asn→ ∞which implies|u|< κinQσρ.
It should be clear from the above presentation how the crucial roles in the gener-alization of the results in [5] to system of the form (1) are played by the local energy estimate of Proposition 3, and by the fact that the techniques in [5] really depend just on|u|. In a similar fashion, it is an easy exercise to check that the same ingredients (Proposition 3 and replacement ofuby|u|) lead to the more general results of [6] .
References
[1] E. DiBenedetto, Degenerate parabolic equations, Springer-Verlag, New York, 1993.
[2] , Partial differential equations, Birkh¨auser, Boston, 1995.
[3] L. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998.
[4] E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, vol. 14, Amer-ican Mathematical Society, 1997.
[5] M. O’Leary, Integrability and boundedness of solutions to singular and degenerate
quasilinear parabolic equations, Differential Integral Equations 12 (1999), 435–
[6] , Integrability and boundedness for local solutions to doubly degenerate
quasilinear parabolic equations, Adv. Differential Equations 5 (2000), no. 10-12,
1465–1492.
[7] W. H. Zajaczkowski,L∞-Estimate for solutions of nonlinear parabolic systems, Singularities and differential equations (Warsaw, 1993), 491–501, Banach Center Publ., 33, Polish Acad. Sci., Warsaw, 1996.
TIZIANAGIORGI
Department of Mathematical Sciences, New Mexico State University Las Cruces, NM 88003 USA.
E-mail address: tgiorgi@nmsu.edu
MIKEO’LEARY
Mathematics Department, Towson University Towson, MD 21252 USA.
E-mail address: moleary@towson.edu
