R E S E A R C H
Open Access
The existence of a solution to a class of
degenerate parabolic variational inequalities
Yudong Sun
1*and Yimin Shi
2*Correspondence:
1School of Science, Guizhou Minzu
Uniwersity, Guiyang, Guizhou 550025, China
Full list of author information is available at the end of the article
Abstract
In this paper, we study the degenerate parabolic variational inequality problem in a bounded domain. First, the weak solutions of the variational inequality are defined. Second, the existence of the solutions in the weak sense are proved by using the penalty method and the reduction method.
MSC: 35B40; 35K35
Keywords: parabolic variational inequality; weak solution; penalty method; existence
1 Introduction
In this article, we consider the initial-boundary problem of the following parabolic varia-tional inequality:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
ut–udiv(|∇u|p–∇u) –γ|∇u|p≤, inT,
[ut–udiv(|∇u|p–∇u) –γ|∇u|p]·(u–u(x)) = , inT,
u(x,t)≤u(x), inT,
u(x, ) =u(x), in,
u(x,t) = , on∂×(,T),
()
whereT=×(,T),⊂RNis a bounded domain with appropriately smooth boundary
∂,p≥,γ > , andu(x) satisfies
≤u∈C(¯)∩W,p(). ()
Readers can refer to [] and [] for the motivation and references about the study of problem (). The linear parabolic variational inequality problem
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∂ ∂τV–
σ
∂
∂xV– (r–σ)∂∂xV+rV≥, inT,
V≥g(x), inT, (∂
∂τV– σ
∂
∂xV– (r–σ)∂∂xV+rV)(V–g(x)) = , inT,
V(t,x) = , on∂T,
V(x, ) =g(x), in,
()
is similar to (). The existence of solutions to problem () was studied in a series of pa-pers (see [] and [] and references therein). Here,randσ are positive constant. In [],
the authors discussed a general case in which the linear parabolic operator with constant coefficients can be replaced by a quasi-linear one with integro-differential terms. Later, the authors in [] extended the corresponding conclusions to theRd-values case in which the existence and uniqueness of solution to parabolic variational inequalities with integro-differential terms were proved by using the penalty method and the reduction method.
However, to the best of our knowledge, the existence of solutions to the variational in-equality problem with the degenerate parabolic operators has not been studied. The pur-pose of this paper is to fill this gap.
In the spirit of [] and [], we introduce the following maximal monotone graph:
G(λ) =
, λ> ,
[, +∞), λ= . ()
In addition, we define a function class for the solution as follows:
B=u∈L∞(T)∩Lp
,T;W,p() . ()
Based on the above basic knowledge, we define the weak solution of problem () below.
Definition A pair (u,ξ)∈B×L∞(T) is called a weak solution of problem (), if
(a) u(x,t)≤u(x),
(b) u(x, ) =u(x),
(c) ξ∈G(u–u),
(d) ∀ϕ∈C∞(T),
T
–uϕt+u|∇u|p–∇u∇ϕ+ ( –γ)|∇u|pϕ dxdt+
ξ φdxdt= , ()
(e) limt→∞|u
μ(x,t) –uμ
(x)|dx= holds for someμ> .
In Section , we prove that forp≥,γ ∈(, ), problem () admits a weak solution in the sense of Definition . We end the introduction by showing the following lemma which is used to prove our main results (see []).
Lemma Letθ≥and A(η) = (η+θ)P– η.Then
A(η) –Aη ·η–η≥Cη–ηp, ∀η,η∈R, ()
where C is a positive constant only depending on p.
2 The existence of weak solutions
This section is devoted to the proof of the existence of solutions to problem (). We prove the following theorem.
Theorem Let p≥andγ ∈(, ).Under the assumption(),problem()admits a weak solution u with ∂∂utμ∈L(T)where
μ= γp (p– )+
To prove Theorem , let us consider the approximation problem ⎧
⎪ ⎨ ⎪ ⎩
(uε)t–uεdiv(|∇uε|p–∇uε) –γ|∇uε|p–βε(uε–uε) = , inT,
uε(x, ) =uε(x) =u(x) +ε, on,
uε(x,t) = , on∂×(,T),
()
whereβε(·) is the penalty function satisfying
<ε≤, βε(x)∈C(R), βε(x)≤, βε() = –,
βε(x)≥, βε(x)≤, lim ε→βε(x) =
, x> , –∞, x< .
()
Definition A nonnegative functionuεis called a weak solution of problem (), if
(a) uε∈L∞(T)∩Lp(,T;W,p()),
(b)
T(uεϕt+uε|∇uε|
p–∇uε∇ϕ+ ( –γ)|∇uε|pϕ–β
ε(uε–uε)ϕ) dxdt= hold, for
any∀ϕ∈C∞(T),
(c) limt→∞|uμ
ε(x,t) –u μ
ε(x)|dx= holds for someμ> .
According to the standard theory for parabolic equations [], problem () admits a weak solution
uε∈L∞(T)∩Lp
,T;W,p(T) ()
in the sense of Definition , which satisfies
uμε t∈L(T), ∇uε∈Lp(T). () Further, it follows by the comparison principle and the maximum principle [, ] that
ε≤uε≤uε≤ |u|∞+ε, uε≤uε forε≤ε. ()
Moreover, from () and (), we assert that there exists a subsequenceε(still denoted by
ε) such that
uε→u∈Lp
,T;W,p(T) asε→, ()
uε≥u≥ for anyε> . ()
Lemma Assume Qε
c={(x,t)∈T;uε≥c,c> },Qc={(x,t)∈T;u≥c,c> }such that,
asε→,
Qεc
|∇uε–∇u|pdxdt→,
Qc
|∇uε–∇u|pdxdt→. ()
Proof Choosingϕ=upε–(uε–ε–u) as the test function in
T
Then it is easy to see that
T ∂uε
∂t u
p– ε
uε–ε–u dxdt
= –
T
|∇uε|p–∇uε∇upε–
uε–ε–u dxdt
+γ
T
upε–|∇uε|p
uε–ε–u dxdt
+
T
upε–uε–ε–u βε(uε–uε) dxdt
= –
T
upε–|∇uε|p–∇uε∇
uε–u dxdt
+
T
upε–
uε–ε–u βε(uε–uε) dxdt
+ (γ–p+ )
T
upε–|∇uε|p
uε–ε–u dxdt. ()
From () and the definition ofβε, we derive
T
upε–|∇uε|p
uε–ε–u dxdt≥–ε
T
upε–|∇uε|pdxdt, ()
T
upε–uε–ε–u βε(uε–uε) dxdt
≤|u|∞+ε p–
T
uε–ε–u dxdt. ()
Observing thatγ –p+ < , and combing (), (), and (), we have
T ∂uε
∂t u
p– ε
uε–ε–u dxdt ≤––p
T
∇uεp–∇uε∇uε–u dxdt+pε
T
upε–|∇uε|pdxdt
+|u|∞+ε p–
T
uε–ε–u dxdt. ()
Note thatε≤uε≤ |u|∞+ε. Thus, it follows by the trigonometrical inequality and the Hölder inequality that
T ∂uε
∂t u
p– ε
uε–ε–u dxdt
+
T
∇u ε
p–
∇uε–∇u
p–
∇u∇uε–u dxdt
≤––p
T
∇up–∇u∇uε–u dxdt+pε
T
upε–|∇uε|pdxdt
+|u|∞+ε p–
T
uε–ε–u dxdt
≤–p
T
∇updxdt
p–
p
T
∇
uε–u
p dxdt
p
+|u|+ε p–με
T
|∇uε|pdxdt
+|u|∞+ε p–
T
uε–ε–u dxdt
. ()
Since∂∂tuμ
ε ∈L(T), using the Hölder inequality, one derives
T
∂upε–
∂t
dxdt
= (p– )
T
upε–
∂uε
∂t
dxdt
≤(p– )|u|∞+ε p–
T
∂uε
∂t
dxdt
≤(p– )|u|∞+ε p–|T|
T
∂uε
∂t
dxdt<∞. ()
From the above equation and (), we may conclude that, asε→,
T ∂uε
∂t u
p– ε
uε–ε–u dxdt
≤
p–
T
∂upε–
∂t
uε–ε–u dxdt
≤
p– T
∂upε– ∂t
dxdt·
T
u
ε–ε–u
dxdt
≤
p– ε
| T| T
∂upε– ∂t
dxdt→, ()
|u|∞+ε p–
T
uε–ε–u dxdt
→. ()
Substituting () and () into () yields
lim sup ε→
T
∇u ε
p–
∇uε–∇u
p–
∇u∇uε–u dxdt≤. ()
This and Lemma lead to
lim ε→
T
∇uε–∇u
p
dxdt= . ()
Hence, in view of
we derive
p
T
upε|∇uε–∇u|pdxdt ≤p
T
∇uε–∇u
p
dxdt+ p
T
|∇u|p|uε–u|pdxdt→ (ε→), ()
p
T
up|∇uε–∇u|pdxdt ≤p
T
∇uε–∇u
p
dxdt+ p
T
|∇uε|p|uε–u|pdxdt→ (ε→). ()
Note thatQε
c⊂T,Qc⊂T. Then it is easy to see that asε→,
cp
Qεc
|∇uε–∇u|pdxdt≤
T
upε|∇u
ε–∇u|pdxdt→, ()
cp
Qc
|∇uε–∇u|pdxdt≤
T
up|∇uε–∇u|pdxdt→. ()
Thus, the lemma is proved.
Lemma The solution of()satisfies
T
|∇uε|pu–αε dxdt≤C, ()
where C is independent ofε,α∈[, –γ).
Proof Multiply () byu–α
ε and integrate both sides of the equation overT. After integrat-ing by parts, we obtain
T ∂uε
∂t u
–α ε dxdt
=
T
u–αε div|∇uε|p–∇uε
+γu–αε |∇uε|p+uε–αβε(uε–uε) dxdt
= T
dt
∂
u–αε |∇uε|p– ∂uε
∂ν
dx– ( –α–γ)
T
u–αε |∇uε|pdxdt
+
T
u–αε βε(uε–uε) dxdt, ()
whereνdenotes the outward normal to∂×(,T). Further, putting together () and () implies
T
u–αε βε(uε–uε) dxdt≤. ()
Sinceuε≥ε, we have ∂uε
This leads to T
∂
u–αε |∇uε|p– ∂uε
∂ν
dxdt≤. ()
Now, we drop the non-positive terms () and () in () to get
T ∂uε
∂t u
–α
ε dxdt≤–( –α–γ)
T
u–αε |∇uε|pdxdt. ()
Clearly, using integration by parts, we derive
T ∂uε
∂t u
–α ε dxdt=
–α
u–αε (x,T) –u–αε (x, ) dx. ()
This and () lead to
T
|∇uε|pu–αε dxdt≤
( –α–γ)( –α)
u–αε (x, ) dx≤C, ()
whereC> depending onα,γ,, and|u|. Hence, the proof is completed. Lemma Asε→,we have
T
|∇uε|p–|∇u|pdxdt→, ()
T
uε|∇uε|p–∇uε–u|∇u|p∇udxdt→, ()
βε(uε–uε)→ξ∈G(u–u). ()
Proof Letχηandχη(ε)be the characteristic functions of{(x,t)∈T;u(x,t) <η}and{(x,t)∈
T;uε(x,t) <η}, respectively. Sinceuε→u, asε→,χη≤χη(ε), we have
T
|∇uε|p–|∇u|pdxdt
≤
T
|∇u|pχη–|∇uε|pχη(ε)dxdt+
T
|∇u|p( –χη) –|∇uε|p
–χη(ε) dxdt
≤
T
|∇uε|pχη(ε)dxdt+
T
|∇u|pχηdxdt
+
T
|∇u|pχη(ε)–χη dxdt+
T
|∇uε|p–|∇u|p –χη(ε) dxdt
=H+H+H+H. ()
Takingα= ( –γ)/ in Lemma one obtains
H=
T
|∇uε|pu
α ε
uα ε
χη(ε)dxdt
≤ηα
T
|∇uε|pu–αε χη(ε)dxdt
≤ηα
T
|∇uε|pu–αε dxdt≤Cη α→
Applying Lemma , (), and the fact thatχη≤χη(ε)implies
H≤
T
χη(ε)|∇u|pdxdt
≤
T
χη(ε)|∇uε|pdxdt+
T
χη(ε)|∇uε–∇u|pdxdt→ (η→), ()
H→ (η→). ()
For fixedη> ,χ(ε)
η →χη(ε→) a.e. inT, so
H→ (η→). ()
Putting together (), (), (), and (), we have
T
|∇uε|p–|∇u|pdxdt→ (η→). ()
Thus () holds.
Next we prove (). It follows by the trigonometrical inequality that
T
uε|∇uε|p–∇uε–u|∇u|p–∇udxdt
≤
T
|uε–u||∇uε|p–dxdt+
T
u|∇uε|p–|∇uε–∇u|dxdt
+
T
u|∇u| ·|∇uε|p––|∇u|p–dxdt
=H+H+H. ()
Using the Hölder inequality and Lemma , we obtain
H≤C
T
|uε–u|pdxdt
p
→ asε→. ()
With the inequality|ar–br| ≤ |a–b|r(r∈[, ],a,b≥), the Hölder inequality, and (), we have
H =
T
u|∇u| ·|∇uε|p––|∇u|p–dxdt
=
T
u|∇u| ·|∇uε|p
p–
p –|∇u|p p–p dxdt
≤C
T
|∇u| ·|∇uε|p–|∇u|p
p–
p dxdt
≤C
T
|∇u| ·|∇uε|p–|∇u|pdxdt
p–
p
×
T
|∇u|pdxdt
p
Finally, we estimateH. Again by using trigonometrical inequality, we arrive at
H=
T
u|∇uε|p–· |∇u
ε–∇u|χηdxdt
+
T
u|∇uε|p–· |∇uε–∇u|( –χη) dxdt ≤η
T
|∇uε|p–· |∇uε–∇u|χρdxdt
+C
T
|∇uε|(p–)p/(p–)dxdt
p/(p–)
T
|∇uε–∇u|p( –χρ) dxdt
/p
=η
T
|∇uε|p–· |∇u
ε–∇u|χρdxdt
+C
T
|∇uε–∇u|p( –χρ) dxdt
/p
=H+H. ()
For allδ> andε∈(, ), letηbe small enough and use Lemma such that, asε→,
H≤
T
|∇uε|pdxdt
p–
p
T
|∇uε–∇u| p
χ
p ρ dxdt
p
≤ |T|
p
T
|∇uε|pdxdt
p–
p
T
|∇uε–∇u|pχρdxdt
p
→. ()
Clearly, for fixedη> , using Lemma , we have
H→ asε→. ()
Substituting () and () into (), we obtain
H→ asε→. ()
Hence, () is proved by putting together (), (), and (). Finally we prove (). Using () and the definition ofβε, we have
βε(uε–uε)→ξ asε→. ()
Now we proveξ∈G(u–u). According to the definition ofG(·), we only need to prove
that ifu(x,t) <u(x),ξ(x,t) = . In fact, ifu(x,t) <u(x), there exist a constantλ>
and aδneighborhoodBδ(x,t) such that, ifεis small enough, we have
uε(x,t)≤uε(x) –λ, ∀(x,t)∈Bδ(x,t). ()
Thus, ifεis small enough, we have
Furthermore, it follows byε→ that
ξ(x,t) = , ∀(x,t)∈Bδ(x,t). ()
Hence, () holds, and the proof of Lemma is completed.
With Lemma and (), it is easy to check thatusatisfies (c) and (d) in Definition . Moreover, applying (), it is clear that
u(x,t)≤u(x), inT, u(x, ) =u(x), in. ()
Thus (a) and (b) hold.
To show the existence of problem (), we only need to prove that (e) holds. Define
I=
uμε –u μ
εdx. ()
Applying the Hölder inequality twice, we obtain
I=
uμε(x,t) –u μ
ε(x)dx=
t
∂ ∂su
μ εdx
dx
≤√t
t
∂uμε ∂s
dtdx≤ ||√t
t
∂uμ ε ∂s
dtdx
≤√t||dxdt. ()
It follows by () that
uμε(x,t) –u μ
ε(x)dx≤C √
t, ()
whereCis independent ofε. Using () and lettingε→ yields
uμ(x,t) –uμ(x)dx≤C√t. ()
So
uμ(x,t) –uμ
(x)dx→ ast→. ()
Thus the proof of existence is completed.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally in this paper and they read and approved the final manuscript.
Author details
1School of Science, Guizhou Minzu Uniwersity, Guiyang, Guizhou 550025, China.2School of Science, Northwestern
Acknowledgements
This work was supported by National Nature Science Foundation of China (Grant Nos. 71171164, 11426176) and the Doctorate Foundation of Northwestern Polytechnical University (Grant No. CX201235). The authors are sincerely grateful to the referee and the Associate Editor handling the paper for their valuable comments.
Received: 21 November 2014 Accepted: 2 June 2015 References
1. Chen, X, Yi, F, Wang, L: American lookback option with fixed strike price 2-D parabolic variational inequality. J. Differ. Equ.251, 3063-3089 (2011)
2. Zhou, Y, Yi, F: A free boundary problem arising from pricing convertible bond. Appl. Anal.89, 307-323 (2010) 3. Chen, X, Yi, F: Parabolic variational inequality with parameter and gradient constraints. J. Math. Anal. Appl.385,
928-946 (2012)
4. Zhou, Y, Yi, F: A variational inequality arising from American installment call options pricing. J. Math. Anal. Appl.357, 54-68 (2009)
5. Sun, Y, Shi, Y, Gu, X: An integro-differential parabolic variational inequality arising from the valuation of double barrier American option. J. Syst. Sci. Complex.27, 276-288 (2014)
6. Sun, Y, Shi, Y, Wu, M: Second-order integro-differential parabolic variational inequalities arising from the valuation of American option. J. Inequal. Appl.2014, 8 (2014)
7. Zhou, W, Wu, Z: Some results on a class of degenerate parabolic equations not in divergence form. Nonlinear Anal. TMA60, 863-886 (2005)
8. Ladyzenskaja, OA, Solonnikov, VA, Ural’ceva, NN: Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs, vol. 23. Am. Math. Soc., Providence (1968)
