MARUO, KENJI

Osaka City University Repository

Maruo, K.
Osaka J. Math.
29 (1992), 455-470
ON THE EXISTENCE OF SOLUTIONS TO WAVE
INTEGRODIFFERENTIAL EQUATIONS WITH
SUBDIFFERENTIAL OPERATORS
Dedicated to Professor Hiroki Tanabe on his sixtieth birthday
KENJI MARUO
(Received December 25, 1989)
0. Introduction
In this paper we consider the following integrodifferential equation
d2u (t)+dψu(t)+dφu(t)+ Γ a(t—s) dφu(s)ds=> f(t, u(t))
Jo
(0.1)
in a real Hubert space H. Here ty and φ are lower semicontinuous proper con-
vex functions from H to [0, oo], and 9ψ and dφ are the subdifferentials of ^
and φ respectively. The functions a( ) and /( , ) are continuous from [0, Γ]
to (— oo, oo) and from [0, T] xH to H.
Our purpose here is to prove the existence of a global solution on [0, T~\ of
the initial value problem (0.1). In the case of a(t) = Q K. Maruo [3] proved
the existence of a solution to the above equation under some restrictions. More-
over, we showed that this class of equations contains vibrating string equations
with unilateral constraints which were deeply investigated by M. Schatzman
[4], A. Bamberger and M. Schatzman [1] and C. Citrini and L. Amerio in [5].
We will extend the result ol [3] to the equation containing a delay term
which corresponds to vibrating string with not only a unilateral constraint but
also a memory (see the example of section 4). In a general situation it seems
to be difficult to solve the above initial value problem (0.1). Hence we will seek
a solution which satisfies (0.1) in some generalized sense as in [3].
The outline of the present paper is as follows. In section 1 we list the
notations and state the assumptions and theorem. In section 2 we obtain an
energy estimate to Yosida approximate solutions of the initial value problem
(0.1). In section 3 we prove our theorem. In section 4 we show an example.
The auther would like to express his hearty gratitude to the referee for
456 K. MARUO
kind and helpful advices.
1. Assumptions, theorem and notations.
We list some notations which will be used throughout the paper. Let Xly
X2 and V be real Banach spaces and F* the dual space of V. We use the same
notation ( , ) as the inner product of H to denote the pairings between Xl9 V
and their corresponding duals. We denote the norm of a Banach space S by
I Is and use the usual notations 1 (^0, Γ; S), C([0, Γ]; S) etc. to denote variable
spaces of functions with values in S. By dφ
λ
 and φ
λ
( ) we denote Yosida ap-
proximations of dφ and φ( ) respectively, i.e dφ
λ
x=\~\x—J%x) and φ
λ
(χ)=
(2X)-11 x—JlxIH+φ(Jlx) where Jφ
λ
=(\+\dφ)-\ The notations d^/dt denote
the left and right derivatives of u in H.
Next we state the assumptions and theorem.
The Banach spaces F, Xly H and X2 hold the following properties.
(A-l) The following inclusion relations hold:
V C X! C H C X2 and X2 C {the dual space X£
where each inclusion mapping is continuous. Moreover, Xl is separable, the
imbedding mapping V~^Xl is compact, and V is reflexive and dense in H.
We introudce the assumptions of ψ ( ) (see [2]).
A-2) -v/τ( ) is a lower semicontinuous, convex function from Domain
D(^jr)=V to [0, oo ] and the subdifferential dψ* of -v^(-) is single valued and
bounded from V to V*. Moreover they satisfy the following conditions.
(1) The function ψ is coercive in the sense that
lim ψ(x)/1 x I
 v
= oo.
\*\r+™
(2) Suppose we are given a sequence of functions {u
n
} C PFU(0, Γ; //)
ΠΛoo(0, Γ F) such that
z/
Λ
-^w in C([0, Γ];fl),
MM—>w in the weak star topology of Z,oo(0, T; F).
Then a subsequence {w^} can be extracted so that d^u
nft-^d^u in
the weak star topology of Loo(0, T\ V*).
REMARK. In view of the coerciveness condition (1) ψ is lower semicon-
tinuous also in the topology of H.
Next we state the assumptions of φ.
A-3) There exists #e V scuh that, for any
SOLUTIONS TO WAVE INTEGRODIFFERENTIAL EQUATIONS 457
where C1 and C2 are positive constants independent of x and λ.
The function /(ί, x) from [0, T]xH to H satisfies the following conditions.
A-4)
(1) For each x^H /(•, x) is continuous in H in [0, T].
(2) The following inequalities hold :
\f(t,x)-f(t,y)\ff<C\x-y\ffy
\f(ΐ,x)\H<C{l+\x\H}
for any x,y^H and any £e[0, T] where C is a constant independent
of x, y and t.
Let k(t) be the solution of the following integral equation
(1.1) k(t) = a+(t)-{ a-(s)k(t-s)dsy
where a+(t)=Max{a(t), 0} and a~(t)=Min{a(t), 0}. As is easliy seen the solu-
tion k(t) exists, is unique and nonnegative.
A-5) The function a(t) is real valued and belongs to Cl([Q, T]).
Furthermore, we assume the following condition either A-6) or A-7).
A-6) The function #(•) belongs to C2([0, T]) and the following inequali-
ties hold:
Max Γ {k(t-s) (S a+(ξ)dξ-a-(s)}ds<l and Γ k(ήds<l .
o^/^r Jo Jo Jo
In addition to A-3) we assume that
A-7) For any positive £ there exists a constant C
ε
 such that
\(Qψx, oc-y)\<te^(y)+Cs(^(x)+l) for any x,
REMARK. If Γ | a(s) \ ds< 1 and a( ) e C2([0, T]) then the assumption A-6)
Jo
is satisfied. Indeed, integrating both sides of (1.1) over [0, T] and noting that
a
+(t)= I a(t) I +a~(t) we have
(Tk(s)ds^(T \a(s)\ds+(Ta-(s)ds-(Ta-(s)ds (* k(s)ds ,
Jo Jo Jo Jo Jo
which implies
[Tk(s)ds<l.
Jo
Therefore
('{A(f-j) (' a+(ξ)dξ-a-(s)}ds<(T \a(ξ)\dξ<\ .
Jo Jo Jo
458 K. MARUO
With regard to the type of the initial value problem (0.1) we consider solu-
tions in the following sense.
DEFINITION. We say that a function u e C([0, T] X^ Π Wl(Q, T\ H) is the
solution of the initial value problem (0.1) if the following conditions are satisfied:
1) φ(u(t))+ I u(t) I
 v
 is bounded in [0, T].
2) There exists a linear functional F on C([0, T] X
λ
) such that
*X0-tt)£ f Tφ(v(s))ds- [Tφ(u(s))ds
Jo Jo
for any 0<=C([0, 71]; ^ ) and
=
 Γ A,)f ^ ,))Λ+ fJo tfί rfί J
αr
for any *€=C([0, Γ]; ^ ) 0^(0, Γ; F) Π W\(0, Γ; //).
3) The initial conditions are satisfied in the following sense
,
dt
where 7^ is the closure of the effective domain of φ, I
κ
 is the indicator
function of K and dl
κ
 is the subdifferantial of I
κ
.
Now we state our theorem.
Theorem. Let the initial values a and b be given so that
a£ΞVΓ(D(φ) and b^H .
Then under the assumptions A-l), A-2), A-3), A-4) and A-6) or A-l), A-2), A-3),
A-4), (A-5) and A-7) we have at least one solution to the initial value problem (0.1).
2. Approximate solutions.
To begin with we prove some lemmas concerning the properties of the
sub differential 9ψ. Throughout this paper we assume the conditions stated in
our Theorem.
Lemma 1. Let g be a continuous mapping from C([0, T] H) to L2(0, Γ; H)
such that the following inequality holds :
SOLUTIONS TO WAVE INTEGRODIFFERENTIAL EQUATIONS 459
for any v,w<= C([0, Γ] H) and t <Ξ [0, T].
Then there exists a solution ι*eL«(0, Γ; F) ΓΊ ϊΓi(0, Γ; H) Π PFi(0, T; F*) o/
the following equation
(2.1) dt
2
«(0) =
 β
,
on[0,
du
Moreover the solution satisfies the following energy inequality
2-ud*.
(2.2) dt
 x
'
J 0
/or ««J * e(0, Γ).
Proof. We consider the following approximate equation to the inital
value problem (2.1), for any μ,>0,
(2.3) dt
2
«μ(0) = a,
, = *(βμ) on[0,T]xV*,
dt
Wμ(0) = b .
Here ψ> is the Yosida approximation of -ψ considered as a convex function on
H which is lower semicontinuous also in the topology of H (Remark after A-2)).
Taking the inner products of both sides of (2.3) with (d/dt)uμ.(t) and integrating
the resultant equality over [0, t], we have
d
2->|^«μ(*)l*at
for
Using GronwalΓs lemma and the assumptions of the lemma we see that the
functions | (d/dt)uμ,(t) \
 H and ^rμ.(uμ.(£)) are uniformly bounded on [0, T]. Then
using (1) in A-2) we see that | J^Uμ.(t) \
 v
 are uniformly bounded on [0, T]. From
A-l) we know that {J^Uμ,(t)}μ. is relatively compact in H for each fixed t. Com-
bining the uniform boundedness of | (d/dl)uμ,(t) \
 H and the above result and using
Ascoli-Arezela's theorem we obtain that there exists a subsequence of {
such that
«μy(ί) = κ(*) in C([0,T];H).
Moreover, since both functions \(d/dt)uμ,(t)\H and \J$Uμ(t)\v are uniformly
460 K. MARUO
bounded it follows that u(t)SΞWl (0, T; H) nL~ (0, Γ; V). Noting the equa-
tion (2.3), (2) in A-2) and the above resultants we know that u belongs to
Wl(Q, T\ V*). Thus we complete the proof.
We consider the Yosida approximate equations of (0.1) with φ
λ
 in place
of φ:
(2.4)
We set
d2
-
at
Γ1 a( —s)dφ
λ
u
λ
(s)ds = f( , «
λ
) ,
Jo
«
λ
(0) = «, (^0) = fc.
- a(t-s)Qφ>μ(ήds+f(t, u(t)) .
O
From the assumption A-4) and the Lipschitz continuity of dφ
λ
 it follows that
the mapping £ satisfies the hypothesis of Lemma 1. Hence we have the fol-
lowing lemma.
Lemma 2. For each λ>0 there exists a solution of the equation (2.4)
in V*. Moreover the following energy inequality holds:
(2.5)
dt
Ί*ll
-('(Sa(S-ξ)(dφ,(uλ(ξ)),^-Jo Jo ds
foranyt<=(0, T).
Next we show that the functions | (d±/dt)u
λ
(t) \
 Hy φ^(uλ(ί)) and ψ(uλ(t)) are
uniformly bounded in t and λ.
For a while we assume the assumtion A-6).
We set
10 ® = Γ a±(t-s)(dφ
λ
u
λ
(s)y uλ(t)-u(s))ds ,Jo
where (ά)(t)=da(t)/dt.
Lemma 3. There exists a constant M such that
SOLUTIONS TO WAVE INTEGRODIFFERENTIAL EQUATIONS 461
(2.6)
Jo Jo
where k(i) is ihe function in the assumption A-6). Furthermore
(2.7) w-(ί)> Γ a-(t-s) {φ
λ
(u
λ
(t))-φ
λ
(u
λ
(ή)}ds .
Jo
Proof. Inductively we define functions h
n
(t) as follows:
(2.8) ktf) = a+(t) , h
n+1(t) = £ -a-(s)ha(t-s)ds
where n— 1, 2, 3, . Then we have
(2.9)
where M=Max | a(s) \ .
Oζs^T
From (2.8) we know
Σ h
Λ
(t) = a+(t)+ -a~(s) Σ h
n
(t-s)ds .
In view of the uniqueness of the solution of the integral equation (1.1) we have
Λ(0=ΣΓ-ιAι (0 Moreover from (2.8) we see that the functions k(t) and (d\
dί)k(t—s) are uniformly bounded in 0^ί<Γ.
For any natural number n we set
(2.10) w
n
(t) = Γ h
n
(t-s)(dφ,u
λ
(s)y uλ(t)-uλ(s))ds,Jo
/ϊ(ί) =
fl(t) =
and
/SW = ('*•(*-*)(- ('^-^(β^λUλU), U
λ
(μ)-U
λ
(s))dμds.Jo Jo
From (2.4) we see
462 K. MARUO
d2 Γ
, tt
λ
)— 1 a+( -μ)dφ
λ
U
λ
(μ)dμ
Jo
— \ <*~( —μ)d<PλU
λ
(μ)dμ.Jo
Substituting this in (2.10), noting (2.8) and using Fubini's theorem we get
V>n(t] = Σ /?(*)+ Γ ^ (t-S^
i — l Jo
In view of (2.9) we see
Thus it follows that
Ί(f) = Σ Σ /?
Set L
ί
(ί)=Σ/?(0>^1>2>3>4>5> and ^(0=ΣA (0 Solving the above in-
tegral equation we get the following equality
(2.11)
 Wl(ί) = L(t)+
where 3?(ί) is a positive continuous function in 0<s^t<T. With the aid of
an integration by parts we get
Ll(t) = k(t)(b,»k(t)-a)
+ (' (dlds)k(t-s)((dlds)u
λ
(ή,
Jo
-['*(«-*) I (rf/ΛXW I !A.
Jo
Noting that
we obtain
(2.12) I L
λ
(t) I < C - *λ « Λ - « λ HJo
Using the assumption A-4) and Schwarz's inequality we see
(2.13) \L3(t)\<C (' (\uλ(t)-uλ(s)\2H+ \uλ(s)\2a+l)ds .Jo
SOLUTIONS TO WAVE INTEGRODIFFERENTIAL EQUATIONS 463
The definition of the subdifferential yeilds
(2.14) L2(t)> Γ k(t-s)ψ(uλ(s))ds- Γ k(t-sJo Jo
(2.15) Lt(t) <, Γ Γ k(t-ήa+(s-Jo Jo
(2.16) L5(t)> - Γ Γ k(t-S)a-(s-μ)φ(U,(μ))dμdsJo Jo
+ 1 \ k(t—s)a-(s—μ)φ(u
λ
(s))dμds.
Jo Jo
Combining (2.11), (2.12), (2.13), (2.14), (2.15), (2.16) and the assumption A-6)
we obtain (2.6). The equlity (2.7) is a direct consequence of the difinition of
the subdifferential.
Noting Lemma 3 and using the argument of the proof of Lemma 3 we
can establish the following lemma, where k(t) is the solution of
k(t) = -(ά)-(f)+ Γ -a-(s)k(t-s)ds .
Jo
Lemma 4. There exists a constant M such that
W+ (t) > -M Γ { I 4- "λ(*) I HJo as
-Γ ~k(t-s)d
JO
Moreover
Proposition 5. Under the assumptions A-l), A-2), A-3), A-4) and A-6) the
functions \(dldt)u
λ
(t)\ffj φ\(u^(t)) and ψ(uλ(t)) are uniformly bounded in λ and t.
Proof. Using Fubini's theorem and the integration by parts we see
o J o
464 K. MARUO
Combining the inequality (2.5), Lemma 3, Lemma 4 and the above equality and
using the assumption A-6) and GronwalΓs inequality we complete the proof.
Proposition 6. Under the assumptions of Proposition 5 there exists a cons-
tant M independent of \ and t such that
Proof. We set
y(t) = Γ (dφ
λ
(u
λ
(s)), u
λ
(s)-z)dsJo
where z is the element in the assumption A-3).
In view of (2.4) we get
y® = o as
Mtf), «
λ
W-«
λ
(f )) dξds
dξds
x(f )-*) dξds
= /l+/,+/S+/4+/5
Using the integration by parts, the definition of the usbdifferential and the as-
sumption A-4) we get
/,<Ξ Γ {|4- «χ(*) I l+l+^(MJo rfί
From the difinition of the subdifferential it follows
J ί W^0
Lemma 3 and Proposition 5 we see /2^ Constant.
Using the integration by parts we see
I,= -\[a-(t-ξ)y(ξ)dξds.
Then we get
SOLUTIONS TO WAVE INTEGRODIFFERENTIAL EQUATIONS 465
y(t)<,Const— Γ a'(t— s)y(s)ds .Jo
Combining the assumption A-3) and Proposition 5 \ve know that ( y(t)+N) is
a positive function on [0, T] where N is some large positive number. Then
using GronwalΓs lemma and the above inequality we get
onst.
Using a similar method to the proof of lemma 3 of [3] and combining the above
inequality and Proposition 5 we complete the proof.
Next we assume the assumption A-5) and A-7).
We define w
n
(t, ξ) by
*. 0 = (' "(*-ζ)-u^ds and(2.18)
w
n+1(t, ζ) = —a(s—ζ)wn(t, s)ds inductively .
Lemma 7. We have the following inequalities
where ,4= Max KOI , L=Max|— a(t)\ and
0<(<Γ dt
Proof. With the aid of the integation by parts we see
The remaing part can be established by induction.
REMARK. From Lemma 2 we know w
n
(t, )eLTO(0, T; V) for each t, n=
1,2,3,-.
We set, forw=l, 2, 3, •••,
,2(0 = - Γ (».(ί, r),Jo and
Lemma 8. We get the following equality
466 K. MARUO
Γ Γ «(ί-r) (9<z>rf), ^-u
κ
(s))dζds = Σ F
n
(t)
Jo Jo as »=l
F.(t)= {/..ι(ί)+Λ.1(ί)+/..J(<)}
Proof. Using the equation (2.5) and Fubini's theorem we get
Γ (w
a
(t, ξ),
Jo
Noting that
S
 a(sΓ
J o J o
Jo
and Lemma 7 we can prove this lemma.
Lemma 9. jΓAe^ ixώfc a constant C independent of λ and t such that
/or »==!, 2, 3, — , a Aβr^ ^  wί (At)"ll(— 1)1 = 1.
Proof. In view of Lemma 7 and the assumption A-4) we find
(2.19) |
On the other hand there exist a constant K independet of λ and t such that
(2.20)
The desired result on/
n 3(£) follows from (2.19) and (2.20). Using the integra-
tion by parts and noting w^t, t)=0 we see
I /ι,ι W I < I »ι(ί, 0) U I b \H+ Γ I α(0) I I -f«λ W I πdsJo t/ί
Noting Lemma 7, (2.20) and choosing a constant M so large that M>((LT+
A)T+(K+AT)\b\H) we get the required inequality for /^(t). Noting the
following equalities
SOLUTIONS TO WAVE INTEGRODIFFERENTIAL EQUATIONS 467
and lemma 7 we have
I ^«.(ί,*)lι
where n>2.
On the other hand it follows
o ' ~cΓ
W
"^ ^'
a
' ~d~U^' ^  '
Then using Lemma 7, the above two inequalities and (2.20) we know
n-2)\{AT\b\
a
+(A+LT) (' |-f^(*)lj
Jo as
*l(n-2) ! {1+ | (^*) I id*}Jo rfί
where M is a positive large number independent of X, ί and n. Our required
inequalities for f
nl(t) are obtained.
Lemma 10. For any £>0, fair* ^Λ?ώ^ a constant Kz independent of n
and t such that
Jo
Proof. From (2.18) we see that the functions w
n
(t, ξ) are equal to
where d§
n
=dξ
n
-1dξn-2~ dξ1 and w=l, 2, •••.
From (2.18) we have the following equality
_!)- Γ έ
J * w — I
where ά=(d/dt)a(t).
Using the above two lemmas and the assumption A-7) and noting
we obtain the following inequalities
468 K. MARUO
Γ - Γ (
J
«
 J
*»-l
Then it follows
Y(ιt- 1) ! -
i) (fcί) -γ(n-l)!
Γ ψ(«
λJo
Therefore the proof of the lemma is complete.
Combining Lemmas 8, 9 and 10 and the inequality (2.5), choosing £ suffi-
cently small and using GronwalΓs lemma we get the following proposition.
Proposition 11. Under the assumptions A-l), A-2), A-3), A-4), A-7) the
functions \ — u
λ
(t) \
 H) φκ(uλ(t)) and ty(ux(t)) are uniformly bounded in λ and t.dt
Noting the above proposition and using a similar argment to the proof of
Proposition 6 we have the following lemma.
Proposition 12. Under the assumptions of Proposition 11 there exists a
constant M independent of λ and t such that
3. Proof of Theorem.
We set
= J (8?w)> «(*))* foro
From the definition of F
λ
 and Fubini's theorem we get the following lemma.
Lemma 13. We have the following equality
Γ \ζa(ζ~s) (dφ
λ
u
λ
(ή, v(ζ))dsdζ = FΛTa(ζ-.)v(ζ)dζ) .
Jo Jo J.
Combining Propositions 5, 6 and lemma 13 or Proposition 11, 12 and
Lemma 13 and using the argument of the proof of Theorem in [3] we obtain
our theorem.
SOLUTIONS TO WAVE INTEGRODIFFERENTIAL EQUATIONS 469
4. Example, (see the example in [3])
Put H=L2(Q, 1), -XΊ=C([O, 1]), X2=Ll(Q, 1) and 7= (^0, 1). Then from
Sobolev's imbedding theorem the assumption A-l) follows.
We consider the following symmetric sesqulinear form a(u, v) defined on
VxV.
1) a(u,u)>S\u\2
v
2) \a(u,v)\£K\u\
v
\v\
γ
for any uy v^V where δ and K are some positive constants. We put ψ>(u)=
a(u, u). Then it is easy to prove that ψ ( ) satisfies the assumption A-2). More-
over we know that it satisfies the assumption A-7).
Set
K= {/e£2(0, 1);/(*)>K*) ^Λ?e[0, 1]}
where r(Ξθ([0, 1]) and r(0), r(l)<l.
Let φ=I
κ
 be the indicator function of K. Then we show that the Yosida ap-
proximation of Qφ satisfies the assumption A-3). We choose a function 0€Ξ
C^fO, 1]) such that 0(0)=0(1)=0 and θ(x)-r(x)>σ>0 for any *e[0, T]. In
the assumption A-3) we define #, GI and cz as θ, σ and 0 respectively.
Since
/™ = l° ίf f
φ
^
J(X)
 \\-\f(X)-r(X)) if /
and/(Λ?)<r(Λ?) implies
we have
Therefore we have the assumption A-3).
Now we can consider the term dφu as a unilateral constraint and the integral
term in the equation (0.1) as a memory term. Then we can regard the initial
value problem (0.1) as the vibrating equations with a unilateral constraint and a
memory term.
References
[1] A. Bamberger and M. Schatzman: New results on the vibrating string with a con-
tinuous obstacle, SIAM J. Math. Anal. 14(1983), 560-595.
[2] V. Barbu: Nonlinear semigroup and differential equations in Banach spaces,
Noordhoff International, 1976.
470 K. MARUO
[3] K. Maruo: Existence of solutions of some nonlinear wave equations, Osaka J. Math.
22(1985), 21-30.
[4] M. Schatzman: A hyperbolic problem of second Order with Unilateral Constrains:
The Vibrating String with a Concave String with a Concave Obstacle, J. Math. Anal.
Appl. 73, (1980), 138-191.
[5] H. Cabannes and C. Citrini (editors): Vibrations with unilateral constraints,
EUROMECH 209, Como (Italy), 1986.
Department of Mathematical Sciences
Faculty of Engineering
Osaka University
Suita, Osaka 565,
Japan
Current Address
Department of Mathematics
Kobe University of
Mercantile Marine
Higashinada, Kobe
Japan


On the existence of solutions to wave integrodifferential equations with subdifferential operators (Dedicated to Professor Hiroki Tanabe on his sixtieth birthday)

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On the existence of solutions to wave integrodifferential equations with subdifferential operators (Dedicated to Professor Hiroki Tanabe on his sixtieth birthday)

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