R E S E A R C H
Open Access
New method for the existence and
uniqueness of solution of nonlinear parabolic
equation
Li Wei
1, Ravi P Agarwal
2,3and Patricia JY Wong
4**Correspondence:
4School of Electrical and Electronic
Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore, 639798, Singapore
Full list of author information is available at the end of the article
Abstract
There are two contributions in this paper. The first is that the abstract result for the existence of the unique solution of certain nonlinear parabolic equation is obtained by using the properties ofH-monotone operators, consequently, the proof is simplified compared to the corresponding discussions in the literature. The second is that the connections between resolvent ofH-monotone operators and solutions of nonlinear parabolic equations are shown, and this strengthens the importance ofH-monotone operators, which have already attracted the attention of mathematicians because of the connections with practical problems.
MSC: 47H05; 47H09
Keywords: H-monotone operator; resolvent; subdifferential; parabolic equation
1 Introduction and preliminaries 1.1 Introduction
Nonlinear boundary value problems involving the generalizedp-Laplacian operator arise from many physical phenomena, such as reaction-diffusion problems, petroleum extrac-tion, flow through porous media and non-Newtonian fluids, just to name a few. Thus, the study of such problems and their generalizations have attracted numerous attention in recent years. In particular, we would mention the books of Lieberman [, ] where in [] the theory of linear and quasilinear parabolic second-order partial differential equations is elaborated, with emphasis on the Cauchy-Dirichlet problem and the oblique deriva-tive problem in bounded space-time domains; while in [] a detailed qualitaderiva-tive analysis of second-order elliptic boundary value problems that involve oblique derivatives is pre-sented. A sample of other research work that contributes to the literature of parabolic and elliptic problems includes [–] listed chronologically as well as the references cited therein. For time-periodic case which is the concern of this paper, we refer the reader to [–].
In , Wei and Agarwal [] studied the following nonlinear elliptic boundary value problem involving the generalizedp-Laplacian:
–div[(C(x) +|∇u|)p– ∇u] +ε|u|q–u+g(x,u(x)) =f(x), a.e. in,
–ϑ, (C(x) +|∇u|)p– ∇u ∈β
x(u(x)), a.e. on,
(.)
where ≤C(x)∈Lp(),εis a non-negative constant andϑdenotes the exterior normal derivative of. It is shown that (.) has solutions inLs() under some conditions, where
N
N+<p≤s< +∞, ≤q< +∞ifp≥N, and ≤q≤
Np
N–p ifp<N, forN≥. We observe that the proof, which uses Theorem . (stated in Section .) as the main tool, is very com-plicated, since one needs to check that conditions (.) and (.) and the compactness of
A+Care satisfied.
In , Weiet al.[] extended the work on elliptic equation to the following nonlin-ear parabolic equation involving the generalizedp-Laplacian with mixed boundary con-ditions:
⎧ ⎪ ⎨ ⎪ ⎩
∂u
∂t –div[(C(x,t) +|∇u|
)p– ∇u] +ε|u|p–u=f(x,t), (x,t)∈×(,T),
–ϑ, (C(x,t) +|∇u|)p– ∇u ∈β(u) –h(x,t), (x,t)∈×(,T),
u(x, ) =u(x,T), a.e.x∈.
(.)
Some new technique has been used to tackle the existence of solutions of (.); specif-ically, the problem is divided into the following two auxiliary equations: (i) a parabolic equation with Dirichlet boundary conditions (.), and (ii) a parabolic equation with Neu-mann boundary value conditions (.):
⎧ ⎪ ⎨ ⎪ ⎩
∂u
∂t –div[(C(x,t) +|∇u|)
p–
∇u] +ε|u|p–u=f(x,t), (x,t)∈×(,T),
γu=w, (x,t)∈×(,T),
u(x, ) =u(x,T), a.e.x∈,
(.)
∂u
∂t –div[(C(x,t) +|∇u|
)p– ∇u] +ε|u|p–u=f(x,t), (x,t)∈×(,T),
–ϑ, (C(x,t) +|∇u|)p– ∇u ∈β(u) –h(x,t), (x,t)∈×(,T). (.)
By using Theorems . and . (stated in Section .), it is shown that (.) has a unique solution. By employing Theorem ., it is proved that (.) has a unique solution in
Lp(,T;W,p()), which implies that (.) has a unique solution inLp(,T;W,p()), where ≤p< +∞. However, we observe that the inequality (.) is not easy to check during the discussion.
Motivated by the work of Kawohlet al.[, , , ], Serrinet al.[, , , ] as well as Weiet al.[, ], in this paper we shall consider the following parabolic problem:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
∂u
∂t –div[α(|∇u|p)|∇u|p–∇u] +λ|u|r–u
+λ|u|r–u+g(x,u,∂u∂t,ε∇u) =f(x,t), (x,t)∈×(,T),
–ϑ,α(|∇u|p)|∇u|p–∇u ∈β
x(u(x,t)), (x,t)∈×(,T),
u(x, ) =u(x,T), x∈,
(.)
whereα:R+∪{} →R+is a continuous nonlinear mapping such thatptα(t) + (p– )α(t) > ,α(t)≤k, fort≥,limt→+∞α(t) =k> , herekandkare positive constants.
Let ϕ:×R→Rbe a given function such that, for each x∈, ϕx=ϕ(x,·) :R→
Ris a proper, convex and lower-semicontinuous function withϕx() = . Letβxbe the subdifferential ofϕx,i.e.,βx≡∂ϕx. Suppose that ∈βx() and for eacht∈R, the function
There are some major differences between parabolic problems (.) and (.): (i) The main part –div[α(|∇u|p)|∇u|p–∇u] in (.) includes the main part –div[(C(x,t) +
|∇u|)p– ∇u] in (.); (ii) the termg(x,u,∂u
∂t,ε∇u) is considered in (.) but not in (.); (iii)βx(u(x,t)) in (.) is different fromβ(u) –h(x,t) in (.).
The existence of the unique solution of (.) will be discussed inL(,T;L()), which
does not change whilepis varying from NN+ to +∞forN≥. Hence, the result is dif-ferent from that on (.) in []. Our main tool in this paper will be Theorem . (stated in Section .). Consequently, the proof of our result is different from and comparatively simplified with respect to that of [].
Actually, (.) is very general and it includes the following special cases. The related work can be found in [–] and the references cited therein.
Example . If we setα(t) = +t( +t)–,t≥, then it is obvious thatα:R+∪ {} →R+
is a continuous nonlinear mapping,α(t)≤ andlimt→+∞α(t) = . Moreover,
ptα(t) + (p– )α(t) = pt ( +t)
+ (p– ) + (p– )√ t +t > .
So, ifλ≡λ≡λ, then (.) becomes the following parabolic capillarity equation:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∂u
∂t –div[( + |∇u|p
√
+|∇u|p)|∇u|
p–∇u] +λ|u|r–u+λ|u|r–u +g(x,u,∂u∂t,ε∇u) =f(x,t), (x,t)∈×(,T), –ϑ, ( +√|∇u|p
+|∇u|p)|∇u|
p–∇u ∈β
x(u(x,t)), (x,t)∈×(,T),
u(x, ) =u(x,T), x∈.
(.)
Example . For <p≤, if we setα(t) = (C+tp)p– t –p
p ,t> , whereC≥, then it is
ob-vious thatα:R+→R+is a continuous nonlinear mapping,α(t)≤ andlim
t→+∞α(t) = . Moreover,
ptα(t) + (p– )α(t) =C+tp p
–tp– tC( –p) + (p– )tC+tp
=C+tp p
–tp– Ct+ (p– )tp+> .
If λ ≡ , then (.) becomes the following parabolic equation with generalized p
-Laplacian: ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
∂u
∂t –div[(C(x) +|∇u|
)p– ∇u] +λ
|u|r–u+g(x,u,∂u∂t,ε∇u)
=f(x,t), (x,t)∈×(,T), –ϑ, (C(x) +|∇u|)p– ∇u ∈β
x(u(x,t)), (x,t)∈×(,T),
u(x, ) =u(x,T), x∈.
(.)
Example . If, in (.),C(x)≡, then (.) becomes the following parabolicp-Laplacian equation:
⎧ ⎪ ⎨ ⎪ ⎩
∂u
∂t –pu+λ|u|r–u+g(x,u, ∂u
∂t,ε∇u) =f(x,t), (x,t)∈×(,T), –ϑ,|∇u|p–∇u ∈β
x(u(x,t)), (x,t)∈×(,T),
u(x, ) =u(x,T), x∈.
Example . Fors≤, if we setα(t) = ( +tp)st
m–p+
p ,t> , wherem≥,m+s+ =
p, then it is obvious thatα:R+→R+ is a continuous nonlinear mapping,α(t)≤ and
limt→+∞α(t) = . Moreover,
ptα(t) + (p– )α(t) =tm–pp+ +tp s
– m+ (p– )tp> .
So, ifλ≡, then (.) becomes the following parabolic curvature equation:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
∂u
∂t –div[( +|∇u|
)s|∇u|m–∇u] +λ
|u|r–u+g(x,u,∂u∂t,ε∇u)
=f(x,t), (x,t)∈×(,T), –ϑ, ( +|∇u|)s|∇u|m–∇u ∈βx(u(x,t)), (x,t)∈×(,T),
u(x, ) =u(x,T), x∈.
(.)
1.2 Preliminaries
LetXbe a real Banach space with a strictly convex dual spaceX∗. We shall use (·,·) to denote the generalized duality pairing betweenXandX∗. We shall use “→” and “w-lim” to denote strong and weak convergence, respectively. Let “X→Y” denote the spaceX
embedded continuously in spaceY. For any subsetGofX, we denote byintGits interior andGits closure, respectively. For two subsetsG andG inX, ifG=G andintG=
intG, then we sayG isalmost equalto G, which is denoted byGG. A mapping
T :X→X∗ is said to behemi-continuousonX[] ifw-limt→T(x+ty) =Txfor any
x,y∈X.
A function is called aproper convex functiononX [] ifis defined from Xto (–∞, +∞], not identically +∞, such that(( –λ)x+λy)≤( –λ)(x) +λ(y), whenever
x,y∈Xand ≤λ≤.
A function : X → (–∞, +∞] is said to be lower-semicontinuous on X [] if
lim infy→x(y)≥(x), for anyx∈X.
Given a proper convex functiononXand a pointx∈X, we denote by∂(x) the set of allx∗∈X∗such that(x)≤(y) + (x–y,x∗), for everyy∈X. Such elementx∗is called thesubgradientofatx, and∂(x) is called thesubdifferentialofatx[].
LetJrdenote theduality mappingfromXinto X ∗
, which is defined by
Jr(x) =
f ∈X∗: (x,f) =xr,f=xr–, ∀x∈X,
wherer> is a constant. We useJto denote the usualnormalized duality mapping. It is known that, in general,Jr(x) =xr–J(x), for allx= . SinceX∗ is strictly convex,Jis a single-valued mapping [].
A multi-valued mappingA:X→Xis said to beaccretive[, ] if (v–v,Jr(u–u))≥
, for anyui∈D(A) andvi∈Aui,i= , . The accretive mappingAis said to bem-accretive ifR(I+λA) =Xfor someλ> .
A multi-valued operatorB:X→X∗is said to bemonotone[] if its graphG(B) is a monotone subset ofX×X∗in the sense that (u–u,w–w)≥, for any [ui,wi]∈G(B),
i= , . Further,Bis calledstrictly monotoneif (u–u,w–w)≥ and the equality holds
if and only ifu=u. The monotone operatorBis said to bemaximal monotoneifG(B)
coercive[] iflimn→+∞(xn,x∗n)/xn= +∞for all [xn,x∗n]∈G(B) such thatlimn→+∞xn= +∞.
LetB:X→X∗be a maximal monotone operator such that [, ]∈G(B), then the equa-tionJ(ut–u) +tBut has a unique solutionut∈D(B) for everyu∈Xandt> . The
resolvent JB
t and theYosida approximation BtofBare defined by []
JtBu=ut,
Btu= –
tJ(ut–u),
for everyu∈Xandt> . (Hence, [JtBu,Btu]∈G(B).)
Definition .([]) LetCbe a closed convex subset ofXand letA:C→X∗be a multi-valued mapping. ThenAis said to be apseudo-monotone operatorprovided that
(i) for eachx∈C, the imageAxis a non-empty closed and convex subset ofX∗; (ii) if{xn}is a sequence inCconverging weakly tox∈Cand iffn∈Axnis such that
lim supn→∞(xn–x,fn)≤, then to each elementy∈C, there corresponds an
f(y)∈Axwith the property that(x–y,f(y))≤lim infn→∞(xn–x,fn);
(iii) for each finite-dimensional subspaceKofX, the operatorAis continuous from
C∩KtoX∗in the weak topology.
Definition .([, ]) Let Hbe a Hilbert space. Let H:H→Hbe a single-valued mapping andA:H→Hbe a multi-valued mapping. We say thatAisH-monotoneifA
is monotone andR(H+λA)(H) =H, for everyλ> .
Lemma .([]) If A:X→X∗is a everywhere defined,monotone,and hemi-continuous
mapping,then A is maximal monotone.If,moreover,A is coercive,then R(A) =X∗.
Lemma . ([]) If:X→(–∞, +∞]is a proper convex and lower-semicontinuous
function,then∂is maximal monotone from X to X∗.
Lemma . ([]) If A and A are two maximal monotone operators in X such that
(intD(A))∩D(A)=∅,then A+Ais maximal monotone.
Theorem .([]) Let X be a real Banach space with a strictly convex dual space X∗.Let
J:X→X∗ be a duality mapping on X and there exists a functionη:X→[, +∞)such
that for all u,v∈X,
Ju–Jv ≤η(u–v). (.)
Let A,C:X→Xbe accretive mappings such that
(i) either bothAandCsatisfy condition(.),orD(A)⊂D(C)andCsatisfies
condition(.):
foru∈D(A)andv∈Au,there exists a constantC(a,f)such that
v–f,J(u–a)≥C(a,f). (.)
LetC:X→Xbe a bounded continuous mapping such that, for anyy∈X, there is a
constantC(y) satisfying (C(u+y),Ju)≥–C(y) for anyu∈X. Then the following results
hold:
(a) [R(A) +R(C)]⊂R(A+C+C);
(b) int[R(A) +R(C)]⊂intR(A+C+C).
Theorem .([]) Let T:X→X∗be a bounded and pseudo-monotone operator,K be
a closed and convex subset of X.Suppose thatis a lower-semicontinuous and convex
function defined on K which is not always+∞such that(v)∈(–∞, +∞]for all v∈K.
Suppose there exists v∈K such that(v) < +∞and
(v–v,Tv) +(v)
v → ∞, asv → ∞,v∈K.
Then there exists u∈K such that
(u–v,Tu)≤(v) –(u), ∀v∈K.
Theorem .([]) Let X be a real reflexive Banach space with both X and its dual X∗
being convex spaces.Let S:D(S)⊂X→X∗be a linear maximal monotone operator and
T:X→X∗be a pseudo-monotone and coercive operator.Then,for each f ∈X∗,there exists an u∈D(S)such that,in the weak sense,Su+Tu=f.
Theorem .([]) Let X be a real reflexive Banach space with both X and its dual X∗
being strictly convex.Let J be the normalized duality mapping.Let A and B be two maximal monotone operators in X.Suppose there exist≤k< and C,C> such that
a,J–(Btv)
≥–kBtv–CBtv–C (.)
for v∈D(A),a∈Av and t> ,where Bt is the Yosida approximation of B.Then R(A) +
R(B)R(A+B).
Theorem .([]) Let A:H→Hbe a maximal monotone operator and H:H→Hbe
a bounded,coercive,hemi-contiunuous,and monotone mapping.Then A is H-monotone.
2 Main results
In this paper, unless otherwise stated, we shall assume that
N≥, N
N+ <p< +∞, ≤ri≤min
p,p, i= , ,
p+
p = ,
r
+
r = ,
r
+
r = .
In (.),is a bounded conical domain of a Euclidean spaceRNwith its boundary∈C
[],T is a positive constant,λ,λandεare non-negative constants, andϑ denotes the
Suppose thatg:×RN+→Ris a given function satisfying the following conditions:
(a) Carathéodory’s conditions
x→g(x,r)is measurable on, for allr∈RN+;
r→g(x,r)is continuous onRN+, for almost allx∈.
(b) Growth condition
g(x,s, . . . ,sN+)≤h(x) +k|s|min{p/p
,} ,
where(s,s, . . . ,sN+)∈RN+,h(x)∈L()∩Lp
()andkis a positive constant.
(c) Monotone conditiongis monotone with respect tor,i.e.,
g(x,s, . . . ,sN+) –g(x,t, . . . ,tN+)
(s–t)≥
for allx∈and(s, . . . ,sN+), (t, . . . ,tN+)∈RN+.
(d) Coercive condition
g(x,s, . . . ,sN+)s≥ks,
wherekis a fixed positive constant.
Now, we present our discussion in the sequel.
Lemma .([]) Let Xdenote the closed subspace of all constant functions in W,p().
Let X be the quotient space W,p()/X
.For u∈W,p(),define the mapping P:W,p()→
Xby
Pu=
meas()
u dx.
Then there is a constant k> such that for all u∈W,p(),
u–Pup≤k∇u(Lp())N.
Lemma . Define the mapping B:Lp(,T;W,p())→Lp(,T; (W,p())∗)by
(w,Bu) = T
α|∇u|p|∇u|p–∇u,∇w
dx dt+λ
T
|u|r–uw dx dt
+λ
T
|u|r–uw dx dt
for any u,w∈Lp(,T;W,p()).Then B is strictly monotone,pseudo-monotone,and
coer-cive.
(Here,·,·and| · |denote the Euclidean inner-product and Euclidean norm inRN,
Proof Step.Bis everywhere defined. Foru,w∈Lp(,T;W,p()), we find
(w,Bu)≤ T
k|∇u|p–|∇w|dx dt+λ
T
|u|r–|w|dx dt
+λ
T
|u|r–|w|dx dt
≤kup/p
Lp(,T;W,p())wLp(,T;W,p())+λwLr(,T;Lr())ur/r
Lr(,T;Lr())
+λwLr(,T;Lr())u
r/r
Lr(,T;Lr()).
SinceW,p()→Lp()→Lr() andW,p()→Lp()→Lr(), forv∈W,p(), we havevLr()≤kvW,p(),vLr()≤kvW,p(), wherekandkare positive
con-stants. Hence,
(w,Bu)≤kup/p
Lp(,T;W,p())wLp(,T;W,p())
+λku
r/r
Lp(,T;W,p())wLp(,T;W,p())
+λku
r/r
Lp(,T;W,p())wLp(,T;W,p()),
which implies thatBis everywhere defined.
Step.Bis strictly monotone. Foru,v∈Lp(,T;W,p()), we have
(u–v,Bu–Bv)≥ T
α|∇u|p|∇u|p––α|∇v|p|∇v|p–|∇u|–|∇v|dx dt
+λ
T
|u|r––|v|r–|u|–|v|dx dt
+λ
T
|u|r––|v|r–|u|–|v|dx dt.
If we setf(s) =s–pα(s),s> , then in view of the assumption ofα, we have
f(s) =
–
p
α(s) +sα(s)
s–p> ,
which implies thatf is strictly monotone. Hence,Bis strictly monotone.
Step.Bis hemi-continuous.
It suffices to show that for anyu,v,w∈Lp(,T;W,p()) andt∈[, ], (w,B(u+tv) –
Bu)→ ast→. Since
w,B(u+tv) –Bu
≤
T
α|∇u+t∇v|p|∇u+t∇v|p–(∇u+t∇v)
+λ
T
|u+tv|r–(u+tv) –|u|r–u|w|dx dt
+λ
T
|u+tv|r–(u+tv) –|u|r–u|w|dx dt,
by Lebesque’s dominated convergence theorem and noting thatαis continuous, we find
lim
t→
w,B(u+tv) –Bu= .
Hence,Bis hemi-continuous.
Step.Bis coercive.
We shall first show that foru∈Lp(,T;W,p()),
uLp(,T;W,p())≤k
T
|∇u|pdx dt
p
+k, (.)
wherekandkare positive constants.
In fact, using Lemma ., we know that, foru∈Lp(,T;W,p()),
u–
meas()
u dx
Lp()≤
k
|∇u|pdx
p
.
Thus,
u–
meas()
u dx
p
W,p()
=u–
meas()
u dx
p
Lp()
+∇
u–
meas()
u dx
p
(Lp())N ≤kp+
|∇u|pdx.
Since
u–
meas()
u dx
W,p()
≥ uW,p()–
meas()
u dx
W,p()
,
we have
uW,p()≤
u–
meas()
u dx
W,p()
+ Const.
Therefore,
uLp(,T;W,p())≤
u–
meas()
u dx
Lp(,T;W,p())
+k
≤kp+ p
T
|∇u|pdx dt
p
+k.
If we setk= (kp+ )
Sincelimt→+∞α(t) =k> , there exists sufficiently largeK> such thatα(t) >l
when-evert>K. Now, foru∈Lp(,T;W,p()), letuLp(,T;W,p())→+∞. Using (.), we find
(u,Bu)
uLp(,T;W,p())
= T
α(|∇u|p)|∇u|pdx dt
uLp(,T;W,p())
+λ
T
|u|rdx dt
uLp(,T;W,p())
+λ
T
|u|rdx dt
uLp(,T;W,p())
>
uLp(,T;W,p())
l
T
|∇u|pdx dt+λ
T
|u|rdx dt
+λ
T
|u|rdx dt
>
uLp(,T;W,p())
l
T
|∇u|pdx dt→+∞.
This completes the proof.
Lemma . The mapping:Lp(,T;W,p())→Rdefined by
(u) = T
ϕx
u|(x,t)
d(x)dt,
for any u∈Lp(,T;W,p()), is proper, convex, and lower-semicontinuous on Lp(,T;
W,p()).Moreover,the subdifferential∂ofis maximal monotone in view of Lemma..
Proof The proof is similar to that of Lemma . in [].
Lemma .([]) Define S:D(S)→Lp(,T; (W,p())∗)by
Su(x,t) =∂u
∂t,
where
D(S) =
u∈Lp,T;W,p()∂u
∂t ∈L
p,T;W,p()∗,u(x, ) =u(x,T)
.
The mapping S is linear maximal monotone.
Definition . Define a mappingA:L(,T;L())→L(,T;L())by
Au=w(x)∈L,T;L()|w(x)∈Bu+∂(u) +Su
foru∈D(A) ={u∈L(,T;L())|there exists aw(x)∈L(,T;L()) such thatw(x)∈
Bu+∂(u) +Su}.
Lemma . Define the mapping F:Lp(,T;W,p())→Lp(,T; (W,p())∗)by
(v,Fu) = T
g
x,u,∂u
∂t,ε∇u
v(x,t)dx dt
Proof Step. Foru(x,t)∈Lp(,T;W,p()),x→g(x,u,∂u
∂t,ε∇u) is measurable on. From the fact thatu(x,t),∂x∂u
i∈L
p(),i= , , . . . ,N, we see thatx→(u, ∂u ∂x, . . . ,
∂u ∂xN) is
measurable on. Combining with the fact thatgsatisfies Carathéodory’s conditions, we know thatx→g(x,u,∂u∂t,ε∇u) is measurable on.
Step.Fis everywhere defined. Foru,v∈Lp(,T;W,p()), we have
(v,Fu)≤ T
h(x)v(x,t)dx dt+k
T
u(x,t)p/pv(x,t)dx dt
≤T
ph(x)
Lp()+kup/p
Lp(,T;W,p())
vLp(,T;W,p()),
which implies thatFis everywhere defined.
This completes the proof.
Definition . Define the mappingH:L(,T;L())→L(,T;L()) by
Hu(x) =v(x)∈L,T;L()|v(x) =Fu(x)
foru∈D(H) ={u(x)∈L(,T;L())|there existsv(x)∈L(,T;L()) such thatv(x) =
Fu(x)}, whereFis the same as in Lemma ..
Lemma . The mapping H:L(,T;L())→L(,T;L())defined in Definition.
is bounded,coercive,hemi-continuous,and monotone.
Proof Step.His bounded.
From condition (b) ofg, we know that
HuL(,T;L())= T
g
x,u,∂u
∂t,ε∇u
dx dt ≤kuL(,T;L())+kh(x)L(),
wherekandkare positive constants. This implies thatHis bounded.
Step.His coercive.
From condition (d) ofg, we know that
(u,Hu) = T
g
x,u,∂u
∂t,ε∇u
u dx dt≥k
T
|u|dx dt
=kuL(,T;L())→+∞, asuL(,T;L())→+∞. Hence,His coercive.
Step.His hemi-continuous.
Sincegsatisfies condition (a), we have, for anyw(x,t)∈L(,T;L()),
w,H(u+tv) –Hu
=
g
x,u+tv,∂u
∂t +t ∂v
∂t,ε(∇u+t∇v)
–g
x,u,∂u
∂t,ε∇u
w dx dt→,
Step.His monotone.
In view of condition (c) ofg, we have
(u–v,Hu–Hv)
= T
g
x,u,∂u
∂t,ε∇u
–g
x,v,∂v
∂t,ε∇v
u(x,t) –v(x,t)dx dt≥,
which implies thatHis monotone.
This completes the proof.
Lemma . For all u,v∈Lp(,T;W,p()),we have
v,∂(u)= T
βx
u|(x,t)
v|(x,t)d(x)dt.
Moreover, ∈∂().
Proof The idea of the proof mainly comes from Proposition .(ii) in []. For
complete-ness, we give the outline of the proof as follows.
Define the mappingG:Lp(,T;Lp())→Lp(,T;Lp()) byGu=β
x(u), for anyu∈
Lp(,T;Lp()). Also, define the mappingK:Lp(,T;W,p())→Lp(,T;Lp()) byK(v) =
v|, for anyv∈Lp(,T;W,p()). ThenK∗GK=∂, whereis the same as in Lemma .. In fact, it is obvious that Gis continuous. Foru(x,t),v(x,t)∈Lp(,T;Lp()), we have (u–v,Gu–Gv) =T(βx(u) –βx(v))(u–v)d(x)dt≥, sinceβxis monotone. Thus,
Gis monotone. In view of Lemma .,G:Lp(,T;Lp())→Lp(,T;Lp()) is maximal monotone.
Define:Lp(,T;Lp())→Rby(u) =Tϕx(u)d(x)dt. It is easy to see that is a proper, convex, and lower-semicontinuous function onLp(,T;Lp()), which implies that∂:Lp(,T;Lp())→Lp(,T;Lp()) is maximal monotone in view of Lemma .. Since
(u) –(v) = T
ϕx(u) –ϕx(v)
d(x)dt
≥
T
βx(v)(u–v)d(x)dt= (Gv,u–v)
for allu(x,t),v(x,t)∈Lp(,T;Lp()), we haveGv∈∂(v). SoG=∂.
Now, it is clear thatK∗GK:Lp(,T;W,p())→Lp(,T; (W,p())∗) is maximal mono-tone since bothKandGare continuous. Finally, for anyu,v∈Lp(,T;W,p()), we have
(v) –(u) =(Kv) –(Ku)
= T
ϕx
v|(x,t)
–ϕx
u|(x,t)
d(x)dt
≥
T
βx
u|(x,t)
v|(x,t) –u|(x,t)
d(x)dt
= (GKu,Kv–Ku) =K∗GKu,v–u.
It now follows that for allu,v∈Lp(,T;W,p()),
v,∂(u)= T
βx
u|(x,t)
v|(x,t)d(x)dt.
Moreover, ∈∂() since ∈βx(). This completes the proof.
Lemma . The mapping A:L(,T;L())→L(,T;L())defined in Definition.is
maximal monotone.
Proof Noting Lemmas .-., we can easily get the result thatAis monotone.
Next, we shall show thatR(I+A) =L(,T;L()), which ensures thatAis maximal
monotone.
Case.p≥. We defineF:Lp(,T;W,p())→Lp(,T; (W,p())∗) by
Fu=u, (v,Fu)Lp(,T;W,p())×Lp(,T;(W,p())∗)= (v,u)L(,T;L()),
where (·,·)L(,T;L())denotes the inner-product ofL(,T;L()). ThenFis everywhere defined, monotone and hemi-continuous, which implies thatFis maximal monotone in view of Lemma .. Combining with the facts of Lemmas ., .-., we haveR(B+∂+
S+F) =Lp(,T; (W,p())∗).
For f ∈ L(,T;L())⊂ Lp(,T; (W,p())∗), there exists u ∈ Lp(,T;W,p())⊂
L(,T;L()) such that
f =Bu+∂(u) +Su+Fu=Au+u,
which implies thatR(I+A) =L(,T;L()).
Case. N
N+ <p< , thenp≥. Similar to Lemma ., we defineB:L
p(,T;W,p())→
Lp(,T; (W,p())∗) by
(w,Bu) = T
α|∇u|p|∇u|p–∇u,∇w
dx dt+λ
T
|u|r–uw dx dt
+λ
T
|u|r–uw dx dt
for anyu,w∈Lp(,T;W,p()). ThenBis maximal monotone and coercive. Similar to Lemma ., define the mapping:Lp(,T;W,p())→Rby
(u) = T
ϕx
u|(x,t)
d(x)dt,
for any u∈Lp(,T;W,p()), then ∂ is maximal monotone. Similar to Lemma ., define S: D(S) ={u ∈Lp(,T;W,p())|∂u
∂t ∈ L
p(,T; (W,p())∗),u(x, ) =u(x,T)} →
Lp(,T; (W,p())∗) by Su(x,t) =∂u
ThenS is linear maximal monotone. Similar to Case , defineF:Lp(,T;W,p())→
Lp(,T; (W,p())∗) by
Fu=u, (v,Fu)Lp(,T;W,p())×Lp(,T;(W,p())∗)= (v,u)L(,T;L()),
then we have R(B+∂+S +F) =Lp(,T; (W,p())∗). So, for f ∈L(,T;L())⊂
Lp(,T; (W,p())∗), there existsu∈Lp(,T;W,p())⊂L(,T;L()) such that
f =Bu+∂(u) +Su+Fu=Au+u,
which implies thatR(I+A) =L(,T;L()).
Theorem . For f(x,t)∈L(,T;L()),the nonlinear parabolic equation(.) has a
unique solution u(x,t)in L(,T;L()),i.e.,
(a) ∂u
∂t –div[α(|∇u|
p)|∇u|p–∇u] +λ
|u|r–u+λ|u|r–u+g(x,u,∂u∂t,ε∇u) =f(x,t),a.e.
(x,t)∈×(,T);
(b) –ϑ,α(|∇u|p)|∇u|p–∇u ∈βx(u(x,t)),a.e.x∈×(,T);
(c) u(x, ) =u(x,T),x∈.
Proof We split our proof into two steps.
Step . There exists a unique u(x,t) which satisfies Hu+λAu =f, where f(x,t)∈
L(,T;L()) is a given function.
From Theorem ., Lemmas . and ., we know that A is H-monotone. Thus,
R(H+λA) =L(,T;L()). Then, forf(x,t)∈L(,T;L()) in (.), there existsu(x,t)∈
L(,T;L()) such thatHu(x,t) +λAu(x,t) =f(x,t). Next, we shall prove thatu(x,t) is
unique.
Suppose thatu(x,t) andv(x,t) satisfyHu+λAu=f andHv+λAv=f, respectively. Then ≤λ(u–v,Au–Av) = –(u–v,Hu–Hv)≤, which ensures that
= (u–v,Au–Av) = (u–v,Bu–Bv) +u–v,∂(u) –∂(v)+ (u–v,Su–Sv).
Using Lemmas ., ., and ., we have (u–v,Bu–Bv) = , which implies thatu(x,t) =
v(x,t), sinceBis strictly monotone.
Step. Ifu(x,t)∈L(,T;L()) satisfiesf =Hu+Au, thenu(x,t) is the solution of (.).
Since(u+ϕ) =(u) for anyϕ∈C∞(×(,T)), we have (ϕ,∂(u)) = . Then, for
ϕ∈C∞(×(,T)), we have
(ϕ,f –Hu) = (ϕ,Bu) +ϕ,∂(u)+ (ϕ,Su) = (ϕ,Bu) + (ϕ,Su).
So T
f –g
x,u,∂u
∂t,ε∇u
ϕdx dt
= T
α|∇u|p|∇u|p–∇u,∇ϕdx dt+λ
T
|u|r–uϕdx dt
+λ
T
|u|r–uϕdx dt+ T
= – T
div α|∇u|p|∇u|p–∇uϕdx dt+λ
T
|u|r–uϕdx dt
+λ
T
|u|r–uϕdx dt+ T
∂u ∂tϕdx dt,
which implies that the equation
∂u ∂t –div α
|∇u|p|∇u|p–∇u+λ|u|r–u
+λ|u|r–u+g
x,u,∂u
∂t,ε∇u
=f(x,t), a.e.x∈×(,T), (.)
is true.
By using (.) and Green’s formula, we have T
ϑ,α|∇u|p|∇u|p–∇uv|d(x)dt
= T
div α|∇u|p|∇u|p–∇uv dx dt+ T
α|∇u|p|∇u|p–∇u,∇vdx dt
=
v,∂u
∂t +λ|u|
r–u+λ
|u|r–u+g
x,u,∂u
∂t,ε∇u
–f
+v,Bu–λ|u|r–u–λ|u|r–u
= (v,Su+Bu+Hu–f) =v, –∂(u)
= – T
βx
u|(x)
v|(x)d(x)dt. (.)
Then
–ϑ,α|∇u|p|∇u|p–∇u∈βx
u(x,t), a.e. on×(,T). (.)
From the definition ofS, we can easily obtainu(x, ) =u(x,T) for allx∈. Combining with (.) and (.) we see thatuis the unique solution of (.).
This completes the proof.
Lemma . DefineB:Lp(,T;W,p())→Lp(,T; (W,p())∗)byBu≡Bu–f(x,t),for
u∈Lp(,T;W,p()).ThenB is maximal monotone.
Proof Similar to the proof of Lemma ., we know thatBis everywhere defined,
mono-tone, and hemi-continuous. It follows thatBis maximal monotone.
Definition . Define a mappingA:L(,T;L())→L(,T;L())by
Au=w(x)∈L,T;L()|w(x)∈Bu+∂(u) +Su
for u∈ D(A) = {u ∈L(,T;L())|there existsw(x)∈ L(,T;L()) such thatw(x) ∈
Bu+∂(u) +Su}.
Definition . LetHbe a Hilbert space andAbe aH-monotone operator. The resolvent operator ofA,RHA,λ:H→H, is defined by