MARUO, KENJI

Maruo, Kenji

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Existence of solutions of some nonlinear wave equations

Osaka City University Repository

Maruo, K.
Osaka J. Math.
22 (1985), 21-30
EXISTENCE OF SOLUTIONS
OF SOME NONLINEAR WAVE EQUATIONS
KENJI MARUO
(Received June 6, 1984)
0. Introduction and Theorem
Let if be a real Hubert space and A be a positive self adjoint operator
in H. Let φ be a lower semi continuous proper convex function from H
to (—00,00] and 9φ be the subdifferential of φ. Then we shall consider
the following equation
(0.1)
-
air
-
at
on [0, T]
where T1 is a positive number.
The above equation was studied in Schatzman [3], [4], [5] and Maruo
[2]. In this paper we prove the existence of a solution of the problem (0.1)
under certain assumptions which are somewhat weaker than those of Schatz-
man [5] and Maruo [2].
In [5] Schatzman showed the existence and uniqueness of a solution of
the following nonlinear wave equation
(0.2)
do?~J^ ' ' di2 Qx2
in the sense of distributions in [0, 1] X [0, Γ],
u(x, t)^r(x), u(xy 0) = u0(x) y for #e[0, 1],
» a.e. in [0,1],
dt
X#, ί) =
w(0, t) = w(l, ί) = 0 for te [0, Γ],
where r is a continuous given function such that r(0)<0, r(l)<0 and r(x)
dx
^0 (in the distribution sense). Set K= {/eL2(0, 1) /(*) ^
tion (0.2) is rewritten as the following equation in L2(Q, 1)
The equa-
22 K. MARUO
d2
d?
(0.3)
u+Au+dIK
f
«) = L
o if u<=κ
where l(u) = \ andloo if
A = — — (Dirichlet problem).
doc
We will show that we can apply our main theorem to this equation if only r
is a continuous function satisfying r(0)<0, r(l)<0 to deduce the existence
of a solution of (0.3). The solution of (0.3), however, does not always satisfy
the locally energy conserving condition (see [5]). Hence we cannot get the
uniqueness of a solution.
Let Ω be a domain in Rn with smooth boundary and consider the case
φ(u)=\ \u(x)\p+1dx, />>!. Then the equation (0.1) represents a nonlinearJΩ
Klein-Gordon equation. It will be shown that if (n+2)>p(n— 2), the result
of this paper can be applied to (0.1) in this case. Note that when n=3 this
inequality is satisfied for/>=3.
Now we state notations which will be used throughout this paper. The
inner product of H is denoted by ( , •). When S is a Banach space, its norm
is denoted by \ \
s
. We denote by V the domain of Al/2 endowed with the
graph norm. By 3φ
λ
 and φ
λ
 we denote the Yosida approximations of 8φ and
φ respectively (i.e. dφ
λ
x=\~1(l — J^~lx and φι(x)=(2\)~l\x— J
λ
x \H~\- φ(J
λ
x)
where /A=(l+λ8φ)-1).
Next we shall introduce the assumptions.
Let X L and X2 be real Banach spaces.
ASSUMPTION 1. The following inclusion relations hold:
V C Xj_ C H C X2 and Xl c {the dual space of X2}
where each inclusion mapping is continuous. Moreover X1 is separable and
the inclusion mapping from V to X1 is compact. H is dense in X2.
ASSUMPTION 2. There exists #e V such that
(Qφ
λ
x, x—z) ^Ci I dφ
λ
x I
 X2— c2
for Λ?e Vy I x I γ^R and | φ(x) \ ^ R where cλ and c2 are positive constants depend-
ing only on R and z.
ASSUMPTION 3. The continuous function/ from [0, T]yH to H satisfies
SOME NONLINEAR WAVE EQUATIONS 23
for any *e[0, T] and x,
dt
where h is a function belonging to Lι(0, T).
ASSUMPTION 4. The closure of D(φ) Π V in H is equal to the closure of
D(φ) in H.
Clearly V and X1 are dense in H. By assumption H is dense in X2. We
use the same notation ( , ) as the inner product of H to denote the pairings
between F, X19 X2 and their corresponding duals.
Now we define the solution of (0.1).
DEFINITION. We say that a function weC([0, T]\ XJ Π WL(Q, Γ; H) is a
solution of the equation (0.1) when it satisfies the following requirements;
1) For any t e [0, T\ u(f) <Ξ Z)(φ) Γi F.
2) There exist weak right and left derivatives u(t)&H for any /e [0, T].
dt
Moreover for any *e[0, T]
+2 \T(f(s, «(,)), -|-«(Jo αί
(with necessary modifications at 0 and T).
3) There exists a linear functional F on C([0, T]; A^) such that
Jo JO
for any ι;eC([0, Γ]; XJ and
Γ / / /(/(ί,U
Jo
—Γ v * ds
o
A
 tt(ι), ^ - «(,)) Λ+
as as I 
for any v e C([0, T\ )^ Π ^ (0, Γ; F) Π PFi(0, Γ; H).
4) The initial conditions are satisfied in the following sense
Λ
where Iζ, is the closure of the domain of φ, I
κ
 is the indicator function of
24 K. MARUO
KQ and QIκQ is the subdifferential of !KO-
We state the theorem.
Theorem. Let a and b be given elements satisfying
Then under the assumptions 1, 2, 3 and 4 we have at least one solution of (0.1).
To prove the above theorem we consider the following approximate equa-
tions for λ>0
dt2(0.4)
In the next section using a method similar to that of [2] we shall investigate
the convergence of the solutions of the approximate equations (0.4). In section
2 we prove the theorem. In section 3 we show some examples.
1. Convergence of approximate solutions
In this section under the assumptions 1, 2, 3 and 4 we shall study the con-
vergence of the solutions of (0.4). In what follows let initial values a and b
belong to Vf]D(φ) and H respectively.
First we show some properties of the approximate solutions.
Lemma 1. For any λ>0 we have solutions of the problem (0.4) such that
if>eC([0, Γ]; H) nL-(0, Γ; V) Π WL(0, Γ; H) Π Wl(0, Γ; V*)
where V* is the dual space of V.
Proof. See p. 289 Barbu [1].
Lemma 2. We hold the following equality and inequality
at
a\2
γ
+2φ,(a)+2 (/(,, ι^ (,)), - u
λ
(s)) ds
as
2)
SOME NONLINEAR WAVE EQUATIONS 25
where C1 is a constant depending only on h and T.
Proof. See Lemma 2.2 in [2].
Lemma 3. There exists a constant independent of λ such that
T
I x2 ds^ Constant.
Proof. In the inequality of assumption 2 we put x=u
λ
(t). From Lemma
2 the constants c
ϊ
 and c2 are independent of λ. Replacing dφ^u by — (#"+
Au
λ
—f( , w
λ
)), integrating over [0, Γ], using the integration by parts and
noting 2) of Lemma 2 we get the conclusion of the Lemma.
Lemma 4. We have a continuous function u from [0, T] to H such that
a subsequence {u
λ
.} of the sequence {u
λ
} converges uniformly to u in H as λy— *0.
Proof. In view of 2) of Lemma 2 |w
λ
(0lv ig uniformly bounded. Hence
from the assumption 1 we know that {u
λ
(t)} is a relatively compact subset of
H for any ίe [0, T]. From 2) of Lemma 2 {u
λ
} is uniformly continuous. Thus
using Ascoli-Arzela's theorem this lemma is proved.
For simplicity we denote this subsequence by {u
λ
} .
Lemma 5. There exists a subsequence {λ;} of {\} such that {uλj} con-
verges to u in C([0, T]; XJ, \ - u
λ
.\ converges to - u in weak^-L^Q, Γ; H)
I dt ) dt
and {u
λ
.(t)} weakly converges to u(t) in V for any ίe[0, T]. Hence we know that
and
u(ί) eZ)(φ) Π V for any t e [0, T] .
Proof. From 2) of Lemma 2 it is easy to prove that some subsequence
I - u
λj\ converges to - u in weak*-Loo(0, T; H). Since {uλj(t)} is bounded
in V it follows from Lemma 4 that {u
λj(t)} weakly converges to u(f) in V. Next
we assume that there exists a sequence {ί, }Γ-ι such that
(urn ^=ίoo ,t+~K*t )-<*~)U^δ0>0 and f f e[O f Γ]
for i= 1,2, - .
Since \u(ti)\v is bounded there exists a subsequence {u(ίίy)} which converges
26 K. MARUO
to some element w of X
λ
. On the other hand u is continuous in H. Hence
w=u(t00). The above results contradict (1.1). Thus u is continuous in Xλ.
Combining that for any £^[0, T] {u
λj(t)} is a relatively compact set in Xlt
that {u
λj} is uniformly convergent to u in H and that u is continuous in X1
we can prove that {u
λj} is uniformly convergent to u in Xλ.
For simplicity we denote the subsequence {λy} by {λ} .
J t dφλUλ(t) dt by pλ(t) for any fe[0, Γ]. Then pλ belongs to
°
Lemma 6. There exists a subsequence -{λy}c-{λ} such that for
andt(Ξ[Q, T], {(p
λ
/0> «)> converges.
Proof. From Lemma 3 for any a^Xi we know that the total variation
of the function {(p
λ
(0> α)} on IP> *^] *s uniformly bounded in λ. Noting that
Xl is separable and using Kelly's choice theorem and the diagonal method we
have a subsequence {λy} such that
lim (p
λ
/£)> #) exists for any
-
For simplicity we denote {λy} by
Put
o
for »eC([0, ϊ1]; JCJ and ίe[0, Γ] .
Lemma 7. For each ίe[0, 71] ίAere eΛ wίί a linear continuous functional
Ft on C([0, T]; X,) such that
lim F
λ
» = F,(β) /or any
λ-ί O
Proof. Combining Lemmas 3 and 6 and approximating ϋ by step func-
tions we can prove this lemma.
Lemma 8. For any v e C([0, T] -ΪΊ) ίA^ ^  exist
right lim Ft(v) and left lim Ft(v)t-+tQ ί->/0
where t0^[Q, T] (with necessary modifications at 0 and T).
Proof. From Lemma 7 it follows
\Ft(v)-Fs(s)\^Km Γ |9φλ
λ-> 0 Js
Combining this inequality with Lemma 3 we see that Ft(v) is of bounded varia-
tion. Thus the lemma is proved.
SOME NONLINEAR WAVE EQUATIONS 27
We put
lim Ft(v) = F(v) .t+τ
Lemma 9. For any ίe[0, T] there exist weak right and left derivatives
d±
- u(t) in H (with necessary modifications at 0 and T).
dt
Proof. Let v be an arbitrary element of C\[Q, Γ]; JSΓ1)ΠCf([0, T]; V).
Forming the inner product of (0.4) and v and integrating by parts we get
(1.2) (A «
λ
 (ί), *(*)) = (*, «
Jo
From Lemmas 4, 5 and 7, the right side of the above equality converges for
any fe[0, Γ] as λ-*Ό. Since < - w
λ
(ί)> is uniformly bounded in H in view
I dt J
of Lemma 2, it follows that < - u
λ
(t) \ converges weakly in H for any /e[0, T],{ dt )
Put
Y0 = {ίe[0, Γ]; weak lim -~
Put z;(ί)= VQ^V in (1.2). We know that the total variation on Y0 of the right
side of (1.2) is uniformly bounded for λ. Hence the total variation on Y0 of
( - u(t), v0] is bounded. Thus using that V is dense in H we have the ex-\ dt /
istence of
weak left lim - u(ΐ) and weak right lim - u(t) .
/er0,/->/0 dt tGY0,t+tQ dt
Therefore this lemma is proved.
Lemma 10. Let »eC([0, Γ]; X^ and v(t)<=ΞD(φ) for a.e ίe[0, T}.
Then it follows
ίϊm Rm Γ (φ
λ
K*))-φ
λ
K(*))) Λ^ Γ (Φ(^
f-+T λ->0 JO JO
Proof. From Lemma 4 and 5 the sequence {J
λ
u
λ
(t)} converges to u(t).
Since φ is lower semi continuous it follows
lim
 λ
28 K. MARUO
From Theorem 2.2 in [1] (p. 57) we have
φ,(v(t))^φ(v(t)) and lim φ
λ
(v(t)) = φ(v(t)).
λ+O
= Γ (/k «w)- ^ "W, «-««) *- Γ (4- *W' 4- uJo Jo \ as ds
Combining the above two results and Fatou's lemma we can prove our asser-
tion.
Lemma 11. The function u satisfies the initial conditions in the sense stated
in 4) of Definition.
Proof. For any v^D(φ) Π V from (1.2) it follows
ds
The left side of the above tends to ( — ιι(0)— ή, v— a] as t^O. From
\ at J
ιιeL«(0, Γ; F) Π ϊΓi(0, Γ; //) we have
lim /! = 0 and lim I2 = 0 ./•ί O ί^ O
On the other hand from Lemmas 5 and 7 it follows
Ft(v-u) = lim ^(ϋ-ifO .
λ->Ό
Hence arguing as in the proof of Lemma 10
lim F^-ttJ^Πm lim (' (φ
λ
(^)-φλKW)) ds = 0 .
/-^O i +0 λ->0 Jo
Thus
(4- «(0)— *»»—«) ^ ° for any\ αί /
Therefore using the assumption 4 we obtain
2. The proof of Theorem
Combining the definition of the sub differential and Lemma 5, 7 and 10
we have the first half of 3) in Definition of the solution. From Lemmas 4, 5, 7
SOME NONLINEAR WAVE EQUATIONS 29
(1.2), 1) and the second half of 3) follow. Combining 2) of Lemmas 2, 5 and
9 we have 2) in Definition. From Lemma 11 we know 4). Thus the proof
of the theorem is complete.
3. Examples
EXAMPLE 1. Let Ω be a bounded domain in R" with a smooth boundary.
Set
H = L2(Ω), X, = L,+1(ίl), X2 =
A = — Δ (Dirichlet problem) and
where p>l.
Then we know dφu=(p+l)\u\p~1 u.
Putting w
λ
 = (ί+\Qφ)-lf we have 8φ
λ
(/) = 9φ(w
λ
), I «>*(*) I ^  I /(*) I and
w
λ(Λ) * f(χ) ^0. Hence it follows that
and
1 9ΦΛ/ U =
Then we have
Thus the assumption 2 is satisfied.
If (n-^2)>p(n—2) using Sobolev's lemma we know that the assumption 1 is
satisfied. Since A= — Δ (Dirichlet problem) it is easy to show the assumption 4.
EXAMPLE 2. Put H=L2(Q, 1), -X^CflΌ, 1]), -XΓ2=Lι(0, 1) and A=--^~dor
(Dirichlet problem).
Let r be a continuous function on [0, 1] such that r(0)<0 and r(l)<0. Set
K=
Let φ=I
κ
 which is the indicator function of K. From Sobolev's lemma the
assumptions 1 and 4 follow. We choose a function fleC^fO, 1]) such that 0(0)
=0(1)=0 and θ(x)-r(x)^8Q>0 for any *e[0, 1],
Since
30 K. MARUO
_ (0 if f(x)^r(x)
\\~\f(x)-r(x)) if f(x)<r(x),
andf(x)<r(x) implies
we have
Hence the assumption 2 holds with z=θ> c1=S0 and c2=0.
EXAMPLE 3. Let K be a closed convex set in H with inner points and
X1=H=X2. Let A be a positive self adjoint operator in H and V be Domain
(Al/2) endowed with the graph norm of A1/2. If an inclution mapping from
V to H is compact it follows that the assumption 1 holds. From Lemma 2.3
in [2] we have the assumption 2.
Biblography
[1] V. Barbu: Nonlinear semigroup and differential equations in Banach space,
NoordhofT International, 1976
[2] K. Maruo: On certain nonlinear differential equations of second order in time, to
appear
[3] M. Schatzman: Sur une classe de problemes hyperboliques nonlineaies, C.R. Acad.
Sci Paris 277 (1973), A 671-A 674.
[4] M. Schatzman: A class of nonlinear differential equations of second order in time,
Nonlinear Anal., Theory, Meth. Appl. 2 (1978), 335-373.
[5] M. Schatzman: A hyperbolic problem of scond order with unilateral constraints',
the vibrating string with a concave obstacle, J. Math. Anal. Appl. 73 (1980), 138-191.
Department of Mathematics
Himeji Institute of Technology
Shosha 2167, Himeji 671-22
Japan


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Existence of solutions of some nonlinear wave equations

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