Volume 2009, Article ID 873526, 15 pagesdoi:10.1155/2009/873526 Research Article Almost Periodic Viscosity Solutions of Nonlinear Parabolic Equations Shilin Zhang and Daxiong Piao School
Trang 1Volume 2009, Article ID 873526, 15 pages
doi:10.1155/2009/873526
Research Article
Almost Periodic Viscosity Solutions of Nonlinear Parabolic Equations
Shilin Zhang and Daxiong Piao
School of Mathematical Sciences, Ocean University of China, Qingdao 266071, China
Correspondence should be addressed to Daxiong Piao,dxpiao@ouc.edu.cn
Received 26 March 2009; Accepted 9 June 2009
Recommended by Zhitao Zhang
We generalize the comparison result 2007 on Hamilton-Jacobi equations to nonlinear parabolic equations, then by using Perron’s method to study the existence and uniqueness of time almost periodic viscosity solutions of nonlinear parabolic equations under usual hypotheses
Copyrightq 2009 S Zhang and D Piao This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
1 Introduction
In this paper we will study the time almost periodic viscosity solutions of nonlinear parabolic equations of the form
∂ t u Hx, u, Du, D2u
ft, x, t ∈ Ω × R,
u x, t 0, x, t ∈ ∂Ω × R,
1.1
whereΩ ∈ RN is a bounded open subset and ∂Ω is its boundary Here H : R N× R × RN×
SN → R and SN denotes the set of symmetric N × N matrices equipped with its usual
orderi.e., for X, Y ∈ SN, we say that X ≤ Y if and only if p t Xp ≤ p t Y p, ∀p ∈ R N;
Du and D2u denote the gradient and Hessian matrix, respectively, of the function u w.r.t
the argument x f is almost periodic in t Most notations and notions of this paper relevant
to viscosity solutions are borrowed from the celebrated paper of Crandall et al.1 Bostan and Namah 2 have studied the time periodic and almost periodic viscosity solutions of first-order Hamilton-Jacobi equations Nunziante considered the existence and uniqueness
of viscosity solutions of parabolic equations with discontinuous time dependence in3,4, but the time almost periodic viscosity solutions of parabolic equations have not been studied yet as far as we know We are going to use Perron’s Method to study the existence of time almost periodic viscosity solutions of1.1 Perron’s Method was introduced by Ishii 5 in
Trang 2the proof of existence of viscosity solutions of first-order Hamilton-Jacobi equations, Crandall
et al had applications of Perron’s Method to second-order partial differential equations in 1 except to parabolic case
To study the existence and uniqueness of viscosity solutions of1.1, we will use some results on the Cauchy-Dirichlet problem of the form
∂ t u Hx, t, u, Du, D2u
0, in Ω × 0, T,
u x, t 0, for x ∈ ∂Ω, 0 ≤ t < T,
u x, 0 u0x, for x ∈ Ω,
1.2
where u0x ∈ CΩ is given Crandall et al studied the comparison result of the
Cauchy-Dirichlet problem in1, and it follows the maximum principle of Crandall and Ishii 6 This paper is structured as follows InSection 2, we present the definition and some properties of almost periodic functions InSection 3, first we list some hypotheses and some results that will be used for existence and uniqueness of viscosity solutions, here we give an improvement of comparison result in paper2 to fit for second-order parabolic equations; then we prove the uniqueness and existence of time almost periodic viscosity solutions In the end, we concentrate on the asymptotic behavior of time almost periodic solutions for large frequencies
2 Almost Periodic Functions
In this section we recall the definition and some fundamental properties of almost periodic functions For more details on the theory of almost periodic functions and its application one can refer to Corduneanu7 or Fink 8
Proposition 2.1 Let f : R → R be a continuous function The following conditions are equivalent:
i ∀ε > 0, ∃lε > 0 such that ∀a ∈ R, ∃τ ∈ a, a lε satisfying
f t τ − ft< ε, ∀t ∈ R; 2.1
ii ∀ε > 0, there is a trigonometric polynomial T ε t Σ n
k1{a k · cosλ k t b k · sinλ k t}
where a k , b k , λ k ∈ R, 1 ≤ k ≤ n such that |ft − T ε t| < ε, ∀t ∈ R;
iii for all real sequence h nn there is a subsequence h n kk such that f· h n kk converges uniformly on R.
Definition 2.2 One says that a continuous function f is almost periodicif and only if f satisfies
one of the three conditions ofProposition 2.1
A number τ verifying2.1 is called ε almost period By usingProposition 2.1we get the following property of almost periodic functions
Proposition 2.3 Assume that f : R → R is almost periodic Then f is bounded uniformly
continuous function.
Trang 3Proposition 2.4 Assume that f : R → R is almost periodic Then 1/Ta T
a f tdt converges as
T → ∞ uniformly with respect to a ∈ R Moreover the limit does not depend on a and it is called
the average of f:
∃f : lim
T→ ∞
1
T
a T
a
f tdt, uniformly w.r.t a ∈ R. 2.2
Proposition 2.5 Assume that f : R → R is almost periodic and denote by F a primitive of f Then
F is almost periodic if and only if F is bounded.
For the goal of applications to the differential equations, Yoshizawa 9 extended almost periodic functions to so called uniformly almost periodic functions
Definition 2.6 9 One says that u : Ω × R → R is almost periodic in t uniformly with respect to x if u is continuous in t uniformly with respect to x and ∀ε > 0, ∃lε > 0 such that all interval of length lε contain a number τ which is ε almost periodic for ux, ·, ∀x ∈ Ω
|ux, t τ − ux, t| < ε, ∀x, t ∈ Ω × R. 2.3
3 Almost Periodic Viscosity Solutions
In this section we get some results for almost periodic viscosity solutions
We consider the following two equations to get some results used for the existence and uniqueness of almost periodic viscosity solutions That is, the Dirichlet problems of the form
∂ t u Hx, t, u, Du, D2u
0, in Ω × 0, T,
u x, t 0, for x ∈ ∂Ω, 0 ≤ t < T,
3.1
H
x, u, Du, D2u
0, in Ω,
in3.2 Ω is an arbitrary open subset of RN
In1, Crandall et al proved such a theorem
Theorem 3.1 see 1 Let O i be a locally compact subset ofRN i for i 1, , k,
O O1× · · · × Ok , 3.3
u i ∈ USCO i , and ϕ be twice continuously differentiable in a neighborhood of O Set
w x u1x1 · · · u k x k for x x1, , x k ∈ O, 3.4
Trang 4and suppose x x1, , x k ∈ O is a local maximum of w − ϕ relative to O Then for each ε > 0
there exists X i ∈ SN i such that
D x i ϕ x, X i
∈ J 2,Oiu i x i for i 1, , k, 3.5
and the block diagonal matrix with entries X i satisfies
−
1
ε A I≤
⎛
⎜
⎜
X1 · · · 0
0 · · · X k
⎞
⎟
⎟
⎠ ≤A A2, 3.6
where A D2ϕ x ∈ SN, N N1 · · · N k
Put k 2, O1 O2 Ω, u1 u, u2 −v, ϕx, y α/2|x − y|2, where α > 0, recall
that J 2,−Ω v −J 2,Ω −v, then, fromTheorem 3.1, at a local maximum x, y of ux − vy −
ϕ x, y, we have
D x ϕ x, y −D y ϕ x, y α x − y ,
A α
I −I
−I I
, A2 2αA, A 2α. 3.7
We conclude that for each ε > 0, there exists X, Y ∈ SN such that
α x − y , X
∈ J 2,Ω u x, α x − y , Y
∈ J 2,−Ω v y ,
− 1
0 I
≤
0 −Y
≤ α1 2εα
I −I
−I I
.
3.8
Choosing ε 1/α one can get
−3α
I 0
0 I
≤
0 −Y
≤ 3α
I −I
−I I
To prove the existence and uniqueness of viscosity solutions, let us see the following main hypotheses first
As in Crandall et al.1, we present a fundamental monotonicity condition of H, that
is,
H x, r, p, X
≤ H x, s, p, Y
whenever r ≤ s, Y ≤ X, 3.10
where r, s ∈ R, x ∈ Ω, p ∈ R N , X, Y ∈ SN Then we will say that H is proper.
Trang 5Assume there exists γ > 0 such that
γ r − s ≤ H x, r, p, X
− H x, s, p, X
, for r ≥ s, x, p, X
∈ Ω × RN × SN, 3.11
and there is a function ω : 0, ∞ → 0, ∞ that satisfies ω0 0 such that
H y, r, α x − y , Y
− H x, r, α x − y , X
≤ ωαx − y2x − y
whenever x, y ∈ Ω, r ∈ R, X, Y ∈ SN, and 3.9 holds.
3.12
Now we can easily prove the following result There is a similar result for first-order Hamilton-Jacobi equations in the book of Barles10
Lemma 3.2 Assume that H ∈ CΩ × 0, T × R × R N × SN and u ∈ CΩ × 0, T is a viscosity
subsolution (resp., supersolution) of ∂ t u Hx, t, u, Du, D2u 0, x, t ∈ Ω × 0, T Then u is a
viscosity subsolution (resp., supersolution) of ∂ t u Hx, t, u, Du, D2u 0, x, t ∈ Ω × 0, T.
Proof Since u ∈ CΩ × 0, T is a viscosity subsolution of ∂ t u Hx, t, u, Du, D2u 0, x, t ∈
Ω × 0, T, if ∀ϕ ∈ C2Ω × 0, T and local maximum x, t ∈ Ω × 0, T of u − ϕ, we have
∂ t ϕ
x, t H x, t, u x, t, Dϕ
x, t, D2ϕ
x, t≤ 0. 3.13 Now we prove that ifx0, T is a local maximum of u − ϕ in Ω × 0, T, then
∂ t ϕ x0, T Hx0, T, u x0, T , Dϕx0, T , D2ϕ x0, T≤ 0. 3.14 Suppose thatx0, T is a strict local maximum of u − ϕ in Ω × 0, T, we consider the function
ψ ε x, t ux, t − ϕx, t − εT − t−1 3.15
for small ε > 0 Then we know that the function ψ ε x, t has a local maximum point x ε , t ε
such that t ε < T and x ε , t ε → x0, T when ε → 0 So at the point x ε , t ε we deduce that
∂ t ϕ x ε , t ε ε
T − t ε2 Hx ε , t ε , u x ε , t ε , Dϕx ε , t ε , D2ϕ x ε , t ε≤ 0. 3.16
As the term ε/T − t ε2is positive, so we obtain
∂ t ϕ x ε , t ε Hx ε , t ε , u x ε , t ε , Dϕx ε , t ε , D2ϕ x ε , t ε≤ 0. 3.17
The results following upon letting ε → 0 This process can be easily applied to the viscosity
supersolution case
By time periodicity one gets the following
Trang 6Proposition 3.3 Assume that H ∈ CΩ × R × R × R N × SN and u ∈ CΩ × R are T periodic
such that u is a viscosity subsolution (resp., supersolution) of ∂ t u Hx, t, u, Du, D2u 0, x, t ∈
Ω × 0, T Then u is a viscosity subsolution (resp., supersolution) of ∂ t u Hx, t, u, Du, D2u
0, x, t ∈ Ω × R.
Crandall et al have proved the following two comparison results
Theorem 3.4 see 6 Let Ω be a bounded open subset of R N , F ∈ CΩ × R × R N × SN be
proper and satisfy 3.11, 3.12 Let u ∈ USCΩ (resp., v ∈ LSCΩ) be a subsolution (resp.,
supersolution) of F 0 in Ω and u ≤ v on ∂Ω Then u ≤ v in Ω.
Theorem 3.5 see 1 Let Ω ∈ R N be open and bounded Let H ∈ CΩ × 0, T × R × R N × SN
be continuous, proper, and satisfy3.12 for each fixed t ∈ 0, T, with the same function ω If u is a
subsolution of1.2 and v is a supersolution of 1.2, then u ≤ v on 0, T × Ω.
We generalize the comparison result in article 2 for first-order Hamilton-Jacobi equations, and get two theorems for second-order parabolic equations Let us see a proposition we will need in the proof of the comparison resultsee 1
Proposition 3.6 see 1 Let O be a subset of R M , Φ ∈ USCO, Ψ ∈ LSCO, Ψ ≥ 0, and
M α sup
for α > 0 Let −∞ < lim α→ ∞M α < ∞ and x α ∈ O be chosen so that
lim
α→ ∞M α − Φx α − αΨx α 0. 3.19
Then the following holds:
i lim
α→ ∞α Ψx α 0,
ii Ψ x 0, lim
α→ ∞M α Φ x sup
{Ψx0} Φx
whenever x ∈ O is a limit point of x α as α −→ ∞.
3.20
Remark 3.7 In Proposition 3.6, when M, O, x, Φx, Ψx are replaced by 2N, O ×
O, x, y, ux − vy, 1/2|x − y|2, respectively, we can get the following results:
i lim
α→ ∞αx α − y α2 0,
ii Ψ x 0, lim
α→ ∞M α u x − v x sup
O ux − vx
whenever x ∈ O is a limit point of x α as α −→ ∞.
3.21
Now we have the following
Trang 7Theorem 3.8 Let Ω ∈ R N be open and bounded Assume H ∈ CΩ × 0, T × R × R N × SN
be continuous, proper, and satisfy3.11, 3.12 for each fixed t ∈ 0, T Let u, v be bounded u.s.c.
subsolution of ∂ t u Hx, t, u, Du, D2u fx, t in Ω × 0, T, ux, t 0 for x ∈ ∂Ω and 0 ≤
t < T, respectively, l.s.c supersolution of ∂ t v Hx, t, v, Dv, D2v gx, t in Ω × 0, T, vx, t
0 for x ∈ ∂Ω and 0 ≤ t < T where f, g ∈ BUCΩ × 0, T.
lim
t ux, t − ux, 0 lim
t vx, t − vx, 0− 0, uniformly for x ∈ Ω,
u ·, 0 ∈ BUCΩ or v ·, 0 ∈ BUCΩ.
3.22
Then one has for all t ∈ 0, T
e γt u·, t − v·, t L∞Ω ≤ u·, 0 − v·, 0L∞Ω
t
0
e γs f·,s − g·,s L∞Ωds,
3.23
where γ γ R0, R0 maxu L∞Ω×0,T , v L∞Ω×0,T .
Proof Let us consider the function given by
w α x, y, t
ux, t − v y, t
− ϕ x, y, t
where ϕx, y, t α/2|x − y|2 φt, and φt ∈ C10, T As we know that u and
v are bounded semicontinuous in Ω × 0, T and Ω ∈ R N is open and bounded, we can find xt α , yt α ∈ Ω × Ω, for t α ∈ 0, T such that M α t α : supΩ×Ωux, t α − vy, t α −
ϕ x, y, t α u xt α , t α − v yt α , t α − ϕ xt α , yt α , t α , here without loss of generality, we can assume that M α t α 0 Since Ω × Ω × 0, T is compact, these maxima xt α , yt α , t α converge to a point of the form zt, zt, t from Remark 3.7 From Theorem 3.1 and its
following discussion, there exists X α , Y α ∈ SN such that
α xt α − yt α , X α
∈ J 2,Ω u xt α , t α , α xt α − yt α , Y α
∈ J 2,−Ω v yt α , t α
,
−3α
I 0
0 I
≤
X α 0
0 −Y α
≤ 3α
I −I
−I I
,
3.25
which implies X α ≤ Y α At the maximum point, from the definition of u being a subsolution
and v being a supersolution we arrive at the following:
∂ t α ϕ xt α , yt α , t α
H xt α , t α , u xt α , t α , α xt α − yt α , X α
− H yt α , t α , v yt α , t α
, α xt α − yt α , Y α
≤ f xt α , t α − g yt α , t α
, 3.26
Trang 8by the proper condition of H, we have
H yt α , t α , v yt α , t α
, α xt α − yt α , Y α
≤ H yt α , t α , v yt α , t α
, α xt α − yt α , X α
,
3.27
as we know that H satisfying3.12 then we deduce that
H xt α , t α , u xt α , t α , α xt α − yt α , X α
− H yt α , t α , v yt α , t α
, α xt α − yt α , X α
H xt α , t α , u xt α , t α , α xt α − yt α , X α
− H yt α , t α , u xt α , t α , α xt α − yt α , X α
H yt α , t α , u xt α , t α , α xt α − yt α , X α
− H yt α , t α , v yt α , t α
, α xt α − yt α , X α
≥ H yt α , t α , u xt α , t α , α xt α − yt α , X α
− H yt α , t α , v yt α , t α
, α xt α − yt α , X α
− ωα xt α − yt α2 xt α − yt α,
3.28
hence we get
∂ t α ϕ xt α , yt α , t α
H yt α , t α , u xt α , t α , α xt α − yt α , X α
− H yt α , t α , v yt α , t α
, α xt α − yt α , X α
− ωα xt α − yt α2 xt α − yt α
≤ ht α ,
3.29
where ht α f xt α , t α − g yt α , t α , ∀t α ∈ 0, T For any t α ∈ 0, T consider
u xt α , t α − v yt α , t α
H yt α , t α , u xt α , t α , α xt α − yt α , X α
− γu xt α , t α − H yt α , t α , v yt α , t α
, α xt α − yt α , X α
γv yt α , t α
,
3.30
Trang 9if u xt α , t α / v yt α , t α , and rt α 0 otherwise From hypothesis 3.11 we deduce that
H x, t, z, p, X − γ · z is nondecreasing with respect to z, then we have rt α ≥ 0 for all t α ∈
0, T Hence we have
H yt α , t α , u xt α , t α , α xt α − yt α , X α
− H yt α , t α , v yt α , t α
, α xt α − yt α , X α
γ rt α u xt α , t α − v yt α , t α
, ∀t α ∈ 0, T.
3.31
Notice that u xt α , t α − v yt α , t α ϕ xt α , yt α , t α , we get
∂ t α ϕ xt α , yt α , t α
γ rt α ϕ xt α , yt α , t α
− ωα xt α − yt α2 xt α − yt α
≤ ht α .
3.32
Replacing u xt α , t α − v yt α , t α by ϕ xt α , yt α , t α in the expression of rt α we
know that r· is integrable and denote by At α the function At α t α
0{γ rσ}dσ, t α ∈
0, T After integration one gets
ϕ t α ≤ e −Atα
ϕ0
t α
0
e A sα·h s α ωα xs α − ys α2 xs α − ys αds α,
3.33
t α ∈ 0, T Now taking u xt α , t α −v yt α , t α instead of ϕ xt α , yt α , t α for any t α ∈ 0, T and letting α → ∞ we can get
u zt, t − vzt, t ≤ e −At
u z0, 0 − vz0, 0
t
0
e A s · hsds
, t ∈ 0, T 3.34
Finally we deduce that for all t ∈ 0, T
e γt u·, t − v·, t L∞Ω ≤ u·, 0 − v·, 0L∞Ω
t
0
e γs f·,s − g·,s L∞Ωds.
3.35
Theorem 3.9 Let Ω ∈ R N be open and bounded Assume H ∈ CΩ × R × R × R N × SN be
continuous, proper, T periodic, and satisfy3.11, 3.12 Let u be a bounded time periodic viscosity
u.s.c subsolution of ∂ t u Hx, t, u, Du, D2u fx, t in Ω × R, ux, t 0 for x, t ∈ ∂Ω × R
Trang 10and v a bounded time periodic viscosity l.s.c supersolution of ∂ t v Hx, t, v, Dv, D2v gx, t in
Ω × R, vx, t 0 for x, t ∈ ∂Ω × R, where f, g ∈ BUCΩ × R Then one has
sup
x∈Ω
ux, t − vx, t ≤ sup
s ≤t
t
s
sup
x∈Ω
f x, σ − gx, σ dσ. 3.36
Proof As the proof ofTheorem 3.8, we get equation3.34
u zt, t − vzt, t ≤ e −At
u z0, 0 − vz0, 0
t
0
e A s · hsds
, t ∈ 0, T 3.37
We introduce that Fs −t
s h σdσ, s, t ∈ 0, T By integration by parts we have
t
0
e A s h sds
t
0
e A s F sds
t
0
h σdσ
t
0
e A s A s
t
s
h σdσ ds
≤
t
0
h σdσ e A t− 1sup
0≤s≤t
t
s
h σdσ.
3.38
We deduce that for all t ∈ 0, T we have
sup
x∈Ω
ux, t − vx, t ≤ e −γtsup
x∈Ω
ux, 0 − vx, 0
sup
0≤s≤t
t
s
sup
x∈Ω
f x, σ − gx, σ dσ.
3.39
Similar to the proof of Corollary 2.2 in paper2, we can reach the conclusion
In order to prove the existence of viscosity solution, we recall the the Perron’s method
as followssee 1,5 To discuss the method, we assume if u : O → −∞, ∞ where O ⊂ R N ,
then
u∗x lim sup
r↓0
u y
: y∈ O andy − x ≤ r,
u∗x lim inf
r↓0
u y
: y∈ O andy − x ≤ r. 3.40
Theorem 3.10 Perron’s method Let comparison hold for 3.2; that is, if w is a subsolution of
3.2 and v is a supersolution of 3.2, then w ≤ v Suppose also that there is a subsolution u and
... existence and uniqueness of viscosity solutions, let us see the following main hypotheses firstAs in Crandall et al.1, we present a fundamental monotonicity condition of H, that
is,...
viscosity subsolution (resp., supersolution) of ∂ t u Hx, t, u, Du, D2u 0, x, t ∈ Ω × 0, T.
Proof Since u ∈ CΩ × 0, T is a viscosity. .. second-order parabolic equations Let us see a proposition we will need in the proof of the comparison resultsee 1
Proposition 3.6 see 1 Let O be a subset of R M