Acta Mathematica Scientia 2012,32B(6):2369–2376
http://actams.wipm.ac.cn
NOTE ON AN OPEN PROBLEM OF HIGHER ORDER
NONLINEAR EVOLUTION EQUATIONS∗
Tujin Kim
Institute of Mathematics, Academy of Sciences of DPR of Korea, Pyonyang, DPR of Korea
E-mail: [email protected]
Chang Qianshun (
)
Institute of Applied Mathematics, Academy of Mathematics and System Sciences,
Chinese Academy of Sciences, Beijing 100190, China
E-mail: [email protected]
)†
Xu Jing (
School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China
E-mail: [email protected]
Abstract In view of a new idea on initial conditions, an open problem of nonlinear
evolution equations with higher order, which was given by J. L. Lions, is solved. Effect of
our results is shown on an example.
Key words nonlinear evolution equation; Cauchy problem; higher order
2010 MR Subject Classification
1
35B37; 35K10; 93B05; 93C20
Introduction
It is well known that many mixed problems of equations of mathematical physics are
reduced to abstract Cauchy problems. A nonclassical equation of mathematical physics studied
in [2] and [3] can be transformed to the equation
u(n) + (An−1 u)(n−1) + · · · + (A1 u)′ + A0 u = f,
n ≥ 2,
(1.1)
where Ai are nonlinear operators from a Banach space into its dual space. Existence and
uniqueness of the initial value problem for (1.1) is open if Ai are essentially nonlinear (cf.
Problem 11.13, Ch. 2 in [4]).
In this paper, using a new idea on initial conditions, we give an answer to the open problem.
Apparently, the idea may seem artificial. However, it is useful since in application to partial
differential equations initial conditions may be given as usual. The results of this paper was
published in 1990 as a preprint of the Institute of Mathematics of the Academy of Sciences in
DPR of Korea, but seem to be known to only a few.
∗ Received
January 28, 2011. The first author’s research is supported by TWAS, UNESO and AMSS in
Chinese Academy. The research of the third author is partially supported by NSFC (11001239).
† Corresponding author: Xu Jing.
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The paper is organized as follows. In Section 2, the basic theorems of unique existence and
well posedness are proved (Theorems 1, 2). Then, in Section 3 we apply the abstract result to
a nonclassical partial differential equation.
2
Initial Value Problem
Let H be a real separable Hilbert space, V be a real separable reflexive Banach space
densely and continuously imbedded into H and V ∗ be its dual space. k · k, k · k∗ and | · | denote
norms of the spaces V , V ∗ and H, respectively. Also, h , iX means duality product between
X and X ∗ , h , iV is one between V and V ∗ , and ( , ) is inner product in H. We assume the
following,
Assumption 1 In equation (1.1), Ai , i = 0, 1, · · · , n − 1 are Volterra operators mapping
the space X = L2 (S; V ) ∩ L2 (S; H) ≡ L2 (S; V ) into X ∗ = L2 (S; V ∗ ), f ∈ L2 (S; V ∗ ) and
derivatives are in the sense of D∗ (S; V ∗ ) (cf. Definition 1.9, Ch. 4 in [1]), where S = [0, T ].
In this case, identifying the space L2 (S; H) with its dual space, we get
X ֒→ L2 (S; H) ֒→ X ∗ ,
(2.1)
and the operator imbedding the space L2 (S; V ) into L2 (S; V ∗ ) by (2.1) is denoted with I.
Lemma 1 If u is a solution to equation (1.1), then
u(t) ∈ C(S; H),
u(i) + (An−1 u)(i−1) + · · · + An−i u ∈ C(S; V ∗ ),
i = 1, · · · , n − 1.
Proof By Assumption 1 from (1.1) we get
[u(n−1) + (An−1 u)(n−2) + · · · + (A1 u)]′ = f − A0 u ∈ L2 (S; V ∗ ).
By Lemmas 1.8 and 1.9, Ch. 4 in [1]
[u(n−1) + (An−1 u)(n−2) + · · · + (A1 u)](t) = cn−1 +
Z
t
(f − A0 u)dt,
0
where cn−1 ∈ V ∗ , and
[u(n−1) + (An−1 u)(n−2) + · · · + (A1 u)] ≡ f1 ∈ C(S; V ∗ ).
Then,
[u(n−2) + (An−1 u)(n−3) + · · · + (A2 u)]′ = f1 − A1 u ∈ L2 (S; V ∗ )
and so
[u(n−2) + (An−1 u)(n−3) + · · · + (A2 u)] ∈ C(S; V ∗ ).
In this way we have
u(i) + (An−1 u)(i−1) + · · · + An−i u ∈ C(S; V ∗ ),
i = 1, · · · , n − 1.
Since u′ + An−1 u ∈ C(S; V ∗ ), by Assumption 1 we have u′ ∈ L2 (S; V ∗ ). Then, taking u ∈
L2 (S; V ) into account, it follows that u ∈ C(S; H).
2
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Considering the result above, we will study the initial value problem of equation (1.1) with
the following initial conditions
u(0) = u0 ∈ H,
(u(i) + (An−1 u)(i−1) + · · · + An−i u)t=0 = ui ∈ V ∗ ,
i = 1, · · · , n − 1.
(2.2)
Theorem 1 If An−1 + ωI ∈ (X 7→ X ∗ ) with ω ∈ R is a radially continuous and strongly
monotone operator and Ai ∈ Lip (X 7→ X ∗ ), i = 0, · · · , n−2, then there exists a unique function
u ∈ L2 (S; V ) satisfying (1.1), (2.2), and for fixed ui , i = 1, · · · , n − 1, mapping u0 → u(t)
is continuous from H into C(S; H). Moreover, if ui ∈ H, i = 0, · · · , n − 1, then mapping
(u0 , · · · , un−1 ) → u is continuous from H n into C(S; H).
Proof It is easy to prove that problem (1.1), (2.2) is equivalent to a problem
u′ + An−1 u +
n−1
X
R∗k An−1−k u = R∗n−1 f +
R∗k−1 uk ,
k=1
k=1
u(0) = u0 ,
n−1
X
u0 ∈ H,
(2.3)
where R∗ ∈ (X ∗ 7→ X ∗ ) is defined by
(R∗ v)(t) =
Z
t
v(s)ds
(in V ∗ ).
0
Therefore, we consider problem (2.3). Since An−1 + ωI ∈ (X 7→ X ∗ ) is strongly monotone, we
have
h(An−1 + ωI)u − (An−1 + ωI)v, u − v)iX ≥ mku − vk2X
∃m > 0, ∀u, v ∈ X.
Thus, using the way of proof similar to one of Lemma 1.1 of Ch. 6 in [1], we have
Z t
{h((An−1 + ωI)u)(s) − ((An−1 + ωI)v)(s), u(s) − v(s)iV − mku(s) − v(s)k2 }ds ≥ 0,
0
from which and Lemma 1.6 of Ch. 6 in [1], we have
Z t
e−2λs h(An−1 u)(s) − (An−1 v)(s), u(s) − v(s)iV ds
0
≥
Z
t
e−2λs (mku(s) − v(s)k2 − ω|u(s) − v(s)|2 )ds,
(2.4)
0
where λ > 0.
On the other band, if B ∈ Lip(X 7→ X ∗ ) with Lipschitz constant not greater than M , then
Z t
k−1
k(R∗k Bu − R∗k Bv)(t)k∗ = (R
(Bu
−
Bv))(s)ds
∗
0
∗
Z t Z s
k−2
≤
ds
(R∗ (Bu − Bv))(ξ)dξ 0
1
≤
(k − 1)!
≤
≤
tk−1
(k − 1)!
M tk−1
(k − 1)!
0
∗
Z
t
k(Bu − Bv)(r)k∗ (t − r)k−1 dr
0
Z
t
k(Bu − Bv)(r)k∗ dr
0
Z
0
t
k(u − v)(r)kdr.
(2.5)
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The inequality above and the condition such that Ai ∈ Lip (X 7→ X ∗ ), i = 1, · · · , n − 2
with Lipschitz constant not greater than M imply
n−1
2
X k
k
(R
A
u
−
R
A
v)(t)
∗ n−1−k
∗ n−1−k
∗
k=1
≤ (n − 1)
n−1
X
k=1
M tk−1
(k − 1)!
2
·t·
Z
t
k(u − v)(r)k2 dr.
(2.6)
0
Using (2.6) and the following inequality
Z t
Z s
e−2λs
ku(r) − v(r)k2 drds
0
0
Z
Z t
e−2λt t
1
ku(r) − v(r)k2 dr +
e−2λs ku(s) − v(s)k2 ds
=−
2λ 0
2λ 0
Z t
1
≤
e−2λs ku(s) − v(s)k2 ds,
2λ 0
(2.7)
we get
Z
t
e
−2λs
0
≥−
≥−
n−1
X
k
k
(R∗ An−1−k u − R∗ An−1−k v)(s), u(s) − v(s) ds
V
k=1
m
2
Z
t
m
2
Z
t
e−2λs ku(s) − v(s)k2 ds −
0
e−2λs ku(s) − v(s)k2 ds −
0
1
2m
Z
0
n−1
2
t
X k
k
ds
A
u
−
R
A
v)(s)
e−2λs (R
∗ n−1−k
∗ n−1−k
∗
k=1
2
k−1
(n − 1)M t
t
4mλ
(k − 1)!
2 Z
t
e−2λs ku(s) − v(s)k2 ds. (2.8)
0
Thus, denoting
A ≡ An−1 +
n−1
X
R∗k An−1−k ,
k=1
2 )
n−1 T M 2 (n − 1) X T k−1
λ = max ω,
,
m2
(k − 1)!
(
k=1
and using (2.4) and (2.8), we have that
Z t
e−2λs h(Au − Av)(s) + λ(u(s) − v(s)), u(s) − v(s)iV ds
0
Z
m t −2λs
≥
e
ku(s) − v(s)k2 ds.
4 0
Also, it is easy to see that R∗n−1 f +
(2.9)
n−1
P
k=1
(2.10)
R∗k−1 uk ∈ X ∗ .
Therefore, we can see that for (2.3) all conditions of Theorem 1.3 of Ch. 6 in [1] are
satisfied. Hence, the first assert was proved.
Now, we are to prove the second one. Denote by u and v, respectively, the solutions
corresponding to the initial values (u0 , u1 , · · · , un−1 ) and (v0 , v1 , · · · , vn−1 ) ∈ H n . Then,
(u′ − v ′ )(s) + (An−1 u − An−1 v)(s) +
n−1
X
(R∗k An−1−k u − R∗k An−1−k v)(s)
k=1
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T. Kim et al: NOTE ON AN OPEN PROBLEM OF HIGHER ORDER NONLINEAR
=
n−1
X
(R∗n−1 uk − R∗n−1 vk )(s) in
a.a. s ∈ S.
(2.11)
k=1
Taking λ as in (2.9), from (2.11) we have that
t
Z
he−λs (u′ (s) − v ′ (s), e−λs (u(s) − v(s))iV ds
Z t
+
e−2λs h(An−1 u − An−1 v)(s), (u − v)(s)iV ds
0
0
+
Z
t
e
0
=
Z
0
−2λs
n−1
X
k
k
(R∗ An−1−k u − R∗ An−1−k v)(s), u(s) − v(s) ds
V
k=1
n−1
t
X
e−2λs
(R∗k−1 uk − R∗k−1 vk )(s), u(s) − v(s) ds.
(2.12)
V
k=1
Let uλ (t) = e−λt u(t), then e−λs u′ (s) = u′λ (s) + λe−λs u(s).
Thus, using Remark 1.22 of Ch. 4 in [1], we have
Z
t
he−λs (u′ (s) − v ′ (s)), e−λs (u(s) − v(s))iV ds
0
Z t
Z t
′
′
=
huλ (s) − vλ (s), uλ (s) − vλ (s)iV + λ
e−2λs (u(s) − v(s), u(s) − v(s))ds
0
0
Z t
1
2
2
= [|uλ (t) − vλ (t)| − |uλ (0) − vλ (0)| ] + λ
e−2λs (u(s) − v(s), u(s) − v(s))ds. (2.13)
2
0
Combining (2.12) with (2.10) and (2.13), we get
Z
1
1
m t −2λs
2
2
|uλ (t) − vλ (t)| ≤ |uλ (0) − vλ (0)| −
e
ku(s) − v(s)k2 ds
2
2
4 0
n−1
2
Z
1 t −2λs X k−1
k−1
+
e
(R∗ uk − R∗ vk )(s) ds
2 0
k=1
Z
1 t −2λs
+
e
|u(s) − v(s)|2 ds,
2 0
(2.14)
where the fact that R∗k−1 uk , R∗k−1 vk ∈ H for uk , vk ∈ H is used.
On the other hand, using deduction similar to the one of (2.5), we obtain
(n−1 )
n−1
2
2
k−2
X
X k−1
t
k−1
2
2
(R∗ uk − R∗ vk )(s) ≤ (n − 1)
· t|uk − vk | + |u1 − v1 | .
(k − 2)!
k=1
k=2
The inequality above and (2.14) imply
|e−λt (u(t) − v(t))|2
1 − e−2λt
≤ |u0 − v0 | +
(n − 1)
2λ
Z t
+
e−2λs |u(s) − v(s)|2 ds.
2
0
(n−1 )
2
X
tk−2
2
2
· t|uk − vk | + |u1 − v1 |
(k − 2)!
k=2
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Therefore, by Gronwall inequality
|e−λt (u(t) − v(t))|2 ≤
n−1
X
Ck |uk − vk |2 · et , Ck > 0,
k=0
and
sup |u(t) − v(t)| ≤ K
n−1
X
Ck |uk − vk |2
k=0
1/2
,
t ∈ [0, T ].
Thus, the second assert was proved.
2
Specially, in the case that n = 2, we have the following
Theorem 2 Let n = 2. Suppose that A1 + ωI ∈ (X → X ∗ ) with some real number
ω is a radially continuous, monotone and coercive operator, A0 ∈ (X → X ∗ ) is a Volterra
operator defined by (A0 u)(t) = Ā0 u(t), and Ā0 ∈ (V 7→ V ∗ ) is a bounded, strongly monotone
and self-adjoint operator. Then, the conclusions of Theorem 1 still are valid.
Proof In this case, problem (2.13) is written as follows:
u′ + A1 u + R∗ A0 u = R∗ f + u1 ,
(2.15)
u(0) = u0 .
Define R ∈ (X → X) by
(Ru)(t) =
Z
t
u(s)ds
(in V ).
0
The operator Ā0 is closed in V ∗ . Using this fact, we can prove that
(R∗ A0 u)(t) = Ā0 (Ru)(t),
(2.16)
which implies that for u ∈ L2 (0, T ; V )
h(R∗ A0 u)(t), u(t)iV = hĀ0 (Ru)(t), (Ru)′ (t)iV =
1 d
hĀ0 (Ru)(t), (Ru)(t)iV
2 dt
at a.a. t ∈ [0, T ]. This shows that derivative of the function hĀ0 (Ru)(t), (Ru)(t)iV is integrable,
and so hĀ0 (Ru)(t), (Ru)(t)iV is absolutely continuous. Thus,
Z t
1
h(R∗ A0 u)(s), u(s)iV ds = hĀ0 (Ru)(t), (Ru)(t)iV ≥ 0.
2
0
Therefore, by Theorem 1.14 of Ch. 4 in [1]
Z T
hR∗ A0 u, uiX =
h(R∗ A0 u)(t), u(t)iV dt ≥ 0,
∀u ∈ L2 (0, T ; V ).
(2.17)
0
Also, since R∗ ∈ Lip (X ∗ 7→ X ∗ ) and A0 ∈ Lip (X 7→ X ∗ ), we have
R∗ A0 ∈ Lip (X 7→ X ∗ ).
(2.18)
Let us consider the operator A = A1 + R∗ A0 . From the condition of theorem, (2.17)
and (2.18), it follows that A + ωI ∈ (X 7→ X ∗ ) is monotone and coercive. It is obvious that
R∗ f + u1 ∈ X ∗ .
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T. Kim et al: NOTE ON AN OPEN PROBLEM OF HIGHER ORDER NONLINEAR
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Consequently, using the Corollary of Theorem 1.3 of Ch. 6 in [1], we come to the asserted
first conclusion.
Proof of the second part is similar to one of Theorem 1.
2
Remark 1 In Theorem 2 the condition for A0 is stronger than one in Theorem 1, wile
the condition for A1 is weaker than one in Theorem 1.
3
Application
In [3] a mixed problem with homogeneous Dirichlet boundary condition and initial problem
of a nonclassical equation of mathematical physics
utt + ut −
n
n
X
X
∂2
∂
Fi (x, t, ux ) −
Ai (x, t, ux ) = f (x, t)
∂x
∂t
∂x
i
i
i=0
i=1
(3.1)
were studied by compact method, where u is a scalar defined in Rn . According to [3], for (3.1)
study of the case that n > 1 is very difficult than the case that n = 1 because monotone method
is impossible to be applied and compact method must be used and for this a priori estimate of
u in Sobolev space with an order higher enough must be obtained (cf. the comments of Ch. 4,
p.199 in [3]).
Now, applying Theorem 1, we can prove existence and uniqueness of solution for a system
of partial differential equations more general than ones in [3]. This shows effect of our monotone
method for above equation.
1
Let Ω be a bounded n-dimensional domain of class C 0,1 , u = (u1 , · · · , ul ), ω ≡ ( ∂u
∂x1 , · · · ,
∂ul
∂ul
∂u1
∂
∂xn , · · · , ∂x1 , · · · , ∂xn , u1 , · · · , ul ), ∂x0 A ≡ A.
We consider the mixed problem
 2
n
X
X
n
n
∂ u
∂ X
∂
∂
∂


−
(∗)
F
(x,
t,
ω)
−
(∗)
A
(x,
t,
ω)
=
(∗)i
fi (x, t),

i
i
i
i

 ∂t2
∂t i=0
∂xi
∂x
∂x
i
i
i=0
i=0
(3.2)

u(x, 0) = u0 (x) ∈ H01 (Ω)l , ut (x, 0) = u1 (x) ∈ H −1 (Ω)l ,




u|∂Ω = 0,
where
(∗)i =

 −1,
i = 0,
 1,
i 6= 0.
Denote ω for ui = u0i , i = 1 − l, by ω 0 , where u0i are components of u0 (x).
Definition 1 (cf. Definition 4.1, Ch. 4 in [3]) The vector function u ∈ L2 (S; H01 (Ω)l ) ∩
C(S; L2 (Ω)l ) such that ut ∈ L2 (S; H −1 (Ω)l ), u(x, t)|t=0 = u0 (x) and
Z
hut (s), v(s)iV ds +
S
+
S
Z X
n Z
S
=
Z
S
Z X
n
i=0
1
0
s
i=0
Fi (x, τ, ω(τ )), vxi (τ ) dτ
Ai (x, τ, ω(τ ))dτ, vxi (s) ds
hu (x), v(s)iV ds +
Z X
n
S
i=0
Fi (x, 0, ω ), vxi (s) ds
0
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Z X
n Z
S
i=0
0
s
fi (s, τ )dτ, vxi (s) ds,
Vol.32 Ser.B
∀v ∈ L2 (S, H01 (Ω)l )
is called a generalized solution to problem (3.2), where V = H01 (Ω)l , V ∗ = H −1 (Ω)l and ( , ) is
inner-product of L2 (Ω)l .
We will use following notation





 

F1i
ξ1
η1
ζ1





 

∂Fji
 . 
 . 
 . 
 . 
Fi =  ..  , ξ =  ..  , η =  ..  , ζ =  ..  , Fjik (x, t, ξ, η) =
,






 
∂ξk
ηl
ζl×n
Fli
ξl×n


F111 F112 · · · F11(l×n)


 ···
··· ···
··· 




F

 1n1 F1n2 · · · F1n(l×n) 


F(x, t, ξ, η) = 
··· ···
··· 
 ···
.


 Fl11 Fl12 · · · Fl1(l×n) 




 ···

·
·
·
·
·
·
·
·
·


Fln1 Fln2 · · · Fln(l×n)
Theorem 3 (cf. Theorems 4.2, 4.4 in [3]) Suppose that
1) Fji (x, t, ξ, η) is continuously differentiable with respect to ξ and
(F(x, t, ξ, η)ζ, ζ) ≥ mkζk2Rl×n ,
m > 0, ∀ζ ∈ Rl×n
at a.a.(x, t);
2) Arbitrary component a(x, t, ξ, η) of Fi and Ai is measurable with respect to (x, t) at
every (ξ, η) and continuous with respect to ξ and η at a.a. (x, t), and
X
l×n
l
X
¯ η̄)| ≤ K
|a(x, t, ξ, η) − a(x, t, ξ,
|ξi − ξ̄i +
|ηi − η̄i | , a(x, t, 0, 0) ∈ L2 (Ω × [0, T ]);
i=1
i=1
3) Every component of fi (x, t) belongs to L2 (Ω × [0, T ]).
Then, there exists a unique generalized solution to problem (3.2).
Acknowledgments The authors would like to thank to Prof. J. L. Lions and Yanpin
Lin for their comments for our original preprints. We also thank Prof. G. O. H. Katona for his
attention to our original preprints.
References
[1] Gajewski H, Grëger K, Zacharias K. Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Mathematische Nachrichten Academic-Verlag, 1975, 67(22): 4–4
[2] Larikin N A, Novikov V A, Yanenko N N. Equations of nonclassical types and their applications in
continuum mechanics//Current Problems in Numerical and Applied Mathematics (Novosibirsk, 1981).
Novosibirsk: Nauka, Sibirsk Otdel, 1983: 22–27 (Russian)
[3] Larikin N A, Novikov V A, Yanenko N N. Nonlinear Equations of Variable Type. Novosibirsk: Nauka,
1983 (Russian)
[4] Lions J L. Quelques Méthodes de Résolution des Problemes aux Limites non Linéaires. Paris: Dunod,
1969, 76
[5] Zhao Y D, Li K T. Existence of global attractors for a nonlinear evolution equation in Sobolev space H k .
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