Electronic Journal of Differential Equations, Vol. 2003(2003), No. 80, pp. 1–7.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu (login: ftp)
SOLUTIONS FOR A HYPERBOLIC SYSTEM WITH BOUNDARY
DIFFERENTIAL INCLUSION AND NONLINEAR
SECOND-ORDER BOUNDARY DAMPING
JONG YEOUL PARK & SUN HYE PARK
Abstract. In this paper we study the existence of generalized solutions for
a hyperbolic system with a discontinuous multi-valued term and nonlinear
second-order damping terms on the boundary.
1. Introduction
The main purpose of this paper is to investigate the initial boundary value problem for a hyperbolic system with differential inclusion on the boundary
u00 − ∆u0 − M (k∇uk2 )∆u = f
in (x, t) ∈ Q = Ω × (0, T ),
0
u(x, 0) = u (x, 0) = 0
u=0
in x ∈ Ω,
on Σ0 = Γ0 × (0, T ),
0
(1.1)
∂u
∂u
+ M (k∇uk2 )
+ K(u)u00 + |u0 |ρ u0 + Ξ = 0 on Σ1 = Γ1 × (0, T ),
∂ν
∂ν
Ξ(x, t) ∈ ϕ(u0 (x, t)) a.e. (x, t) ∈ Σ1 = Γ1 × (0, T ),
where Ω is a bounded open set of Rn (n ≥ 3) with sufficiently smooth boundary
Γ = ∂Ω such that Γ = Γ0 ∪ Γ1 , Γ¯0 ∩ Γ¯1 = ∅ and Γ0 , Γ1 have positive measures, ρ ∈
(1, ∞), M (s) is a C 1 class function such that M (s) > m0 > 0 for some constant m0 ,
Pn
2
K(s) is a continuously differentiable positive function, ∆u = i=1 ∂∂xu2 , k∇uk2 =
i
Pn R ∂u 2
i=1 Ω | ∂xi | dx, ν is the outward unit normal vector on Γ, ϕ is a discontinuous
and nonlinear set valued mapping and T is a positive real number. The precise
hypothesis on the above system will be given in the next section.
The background of these problems is in physics, especially in solid mechanics,
where non-monotone and multi-valued constitutive laws lead to differential inclusion. For a brief account of the works on such variational inequalities we refer
the reader to [3,4,5]. Doronin et al. [1] investigated the existence of generalized
2000 Mathematics Subject Classification. 35L85, 35B40.
Key words and phrases. Differential inclusion, boundary damping, generalized solution.
c
2003
Southwest Texas State University.
Submitted April 9, 2003. Published August 5, 2003.
Supported by grant R01-2002-000-00491-0 from the Korea Science and Engineering Foundation.
1
2
JONG YEOUL PARK & SUN HYE PARK
EJDE–2003/80
solutions for the hyperbolic equation of the form
u00 − ∆u = f
in (x, t) ∈ Q,
∂u
+ K(u)u00 + |u0 |ρ u0 = 0 on Σ1 = Γ1 × (0, T ),
∂ν
u = 0 on Σ0 = Γ0 × (0, T ),
u(x, 0) = u0 (x, 0) = 0
on Ω.
Motivated the results of [1], in this paper we study the existence of solutions of
the variational inequalities (1.1). It is important to observe that as far as we are
concerned it has never been considered differential inclusion acting on the boundary
in the literature. The plan of this paper is as follows. In section 2, the assumptions
and the main results are given. In section 3, the existence of a solution to problem
(1.1) is proved.
2. Assumptions and Main results
Throughout this paper we denote
Z
1
H1 (Ω) = {u ∈ H (Ω) : u = 0 on Γ0 }, (u, v) =
u(x)v(x)dx,
Ω
Z
Z
(u, v)Γ1 =
u(x)v(x)dΓ, kukp,Γ1 = (
|u(x)|p dΓ)1/p .
Γ1
Γ1
For simplicity, we denote k · kL2 (Ω) and k · k2,Γ1 by k · k and k · kΓ1 , respectively. We
formulate the following assumptions:
(A1) K(s) is a continuous real function satisfying the conditions
0 < K0 ≤ K(s) ≤ K1 (1 + |s|ρ ),
0
|K (s)|
ρ
ρ−1
≤ K2 (1 + K(s)),
(2.1)
(2.2)
for some K0 , K1 , K2 > 0.
(A2) b : R → R is a locally bounded function satisfying
|b(s)| ≤ µ1 (1 + |s|), ∀s ∈ R,
(2.3)
for some µ1 > 0.
The multi-valued function ϕ : R → R is obtained by filling in jumps of a function
b : R → R by means of the functions b , b , b, b : R → R as follows:
b (t) = ess inf |s−t|≤ b(s),
b(t) = lim+ b (t),
→0
b (t) = ess sup |s−t|≤ b(s),
b(t) = lim+ b (t),
→0
ϕ(t) = [b(t), b(t)].
We shall use the regularization of b defined by
Z ∞
m
b (t) = m
b(t − τ )ρ(mτ )dτ,
−∞
where ρ ∈
C0∞ ((−1, 1)), ρ
≥ 0 and
R1
−1
ρ(τ )dτ = 1.
Remark 2.1. It is easy to show that bm is continuous for all m ∈ N and that b ,
b , b, b, bm satisfy condition (A2) with a possibly different constant when b satisfies
(A2).
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HYPERBOLIC SYSTEMS WITH BOUNDARY DIFFERENTIAL INCLUSIONS 3
Definition A function u(x, t) such that
u ∈ L∞ (0, T ; H1 (Ω)),
u0 ∈ L2 (0, T ; H1 (Ω)) ∩ L∞ (0, T ; Lρ+2 (Γ1 )),
u00 ∈ L2 (0, T ; L2 (Ω) ∩ L2 (Γ1 )),
u(x, 0) = u0 (x, 0) = 0
is a generalized solution to (1.1) if there exists Ξ ∈ L2 (0, T ; L2 (Γ1 )) and for any
functions v ∈ W = H1 (Ω) ∩ Lρ+2 (Γ1 ) and ψ ∈ C 1 (0, T ) with ψ(T ) = 0 the relations
hold:
Z Tn
(u00 , v) + (∇u0 , ∇v) + M (k∇uk2 )(∇u, ∇v)
0
Z
o
+ (|u | u − K (u)(u ) + Ξ, v)Γ1 ψ(t)dt −
0 ρ 0
0
T
0 2
(K(u)u0 , v)Γ1 ψ 0 (t)dt
(2.4)
0
Z
=
T
(f, v)ψ(t)dt,
0
Ξ(x, t) ∈ ϕ(u0 (x, t)) a.e. (x, t) ∈ Σ1 .
(2.5)
Now we are in position to state our existence result.
Theorem 2.2. Assume that (A1) and (A2) hold and f ∈ L2 (0, T ; L2 (Ω)). Then,
for all T > 0 there exists a generalized solution to the problem (1.1).
3. Proof of main theorem
In this section we are going to show the existence of solution for problem (1.1)
using the Faedo-Galerkin’s approximation. For this end we represent by {wj }j≥1
a basis in W = H1 (Ω) ∩ Lρ+2 (Γ1 ).PLet Wm = span{w1 , w2 , . . . , wm }. Next we
m
define the approximations um (t) = j=1 gjm (t)wj , where gjm (t) are solutions to
the Cauchy problem
(u00m , wj ) + (∇u0m , ∇wj ) + M (k∇um k2 )(∇um , ∇wj )
+(K(um )u00m + |u0m |ρ u0m + bm (u0m ), wj )Γ1 = (f, wj ),
um (0) = u0m (0) = 0.
(3.1)
(3.2)
By the same argument as in [1], the approximate system (3.1) and (3.2) has solutions
um (t) in [0, tm ). The extension of these solutions to the whole interval [0, T ] is a
consequence of the priori estimate which we are going to prove below.
0
Step 1 : A priori estimate. Multiplying (3.1) by gjm
(t) and summing from
j = 1 to j = m, we get
Z
o
1 dn 0
kum (t)k2 + M̄ (k∇um (t)k2 ) +
K(um (t))(u0m (t))2 dΓ
2 dt
Γ1
+ (bm (u0m (t)), u0m (t))Γ1 + k∇u0m (t)k2 + ku0m (t)kρ+2
ρ+2,Γ1
Z
1
0
0
3
−
K (um (t))(um (t)) dΓ
2 Γ1
= (f (t), u0m (t)),
(3.3)
4
JONG YEOUL PARK & SUN HYE PARK
where M̄ (s) =
EJDE–2003/80
Rs
M (r)dr. By the condition (A2), we have
Z
2
kbm (u0m (t))k2Γ1 =
bm (u0m (x, t)) dΓ
ZΓ1
≤
c1 (1 + |u0m (x, t)|)2 dΓ
Γ1
Z
≤ 2c1
(1 + |u0m (x, t)|2 )dΓ = c2 + 2c1 ku0m (t)k2Γ1 ,
0
(3.4)
Γ1
where c1 , c2 are positive constants(dependent on the geometry of Γ but independent
of m). In what follows ci (i ≥ 3) denote generic constants independent of m.
Inequality (3.4) and Hȯlder’s inequality imply that
Z
t m 0
(b (um (s)), u0m (s))Γ1 ds
0
Z t
Z t
≤(
kbm (u0m (s))k2Γ1 ds)1/2 (
ku0m (s)k2Γ1 ds)1/2
0
0
Z t
Z t
1/2
1/2
≤
(c2 + 2c1 ku0m (s)k2Γ1 )ds
ku0m (s)k2Γ1 ds
0
0
Z t
0
2
≤ c3 (1 +
kum (s)kΓ1 ds).
0
Note that, by Young’s inequality,
Z
Z t
1
K 0 (um (s))(u0m (s))3 dΓ}ds
{ku0m (s)kρ+2
−
ρ+2,Γ1
2 Γ1
0
Z tZ
ρ
≥
|u0m (s)|2 {|u0m (s)|ρ − |u0m (s)|ρ − C()|K 0 (um (s))| ρ−1 }dΓds,
0
(3.5)
Γ1
where is an arbitrary positive number. Therefore, integrating (3.3) over (0, t) and
taking = 12 in (3.6), from (2.2), (3.5) and (3.6) we obtain
Z
1 0
2
2
kum (t)k + M̄ (k∇um (t)k ) +
K(um (t))(u0m (t))2 dΓ
2
Γ1
Z t
Z t
1
+
k∇u0m (s)k2 ds +
ku0m (s)kρ+2
ρ+2,Γ1 ds
2 0
0
(3.6)
Z t
Z t
Z t
≤ c3 (1 +
ku0m (s)k2Γ1 ds) +
kf (s)k2 ds +
ku0m (s)k2 ds
0
0
0
Z tZ
+ c4
|u0m (s)|2 (1 + K(um (s)))dΓds.
0
Γ1
On the other hand, note that K(u) ≥ C0 (1 + K(u)) where 2C0 = min{1, K0 }.
Thus, letting
Z
1
0
2
2
Em (t) = (kum (t)k + M̄ (k∇um (t)k ) + C0
(1 + K(um (t)))|u0m (t)|2 dΓ),
2
Γ1
Rt
from (3.7) we have Em (t) ≤ c5 (1 + 0 Em (s)ds). Thus, by Gronwall’s lemma, we
conclude that
Em (t) ≤ c6 , ∀t ∈ [0, T ].
(3.7)
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HYPERBOLIC SYSTEMS WITH BOUNDARY DIFFERENTIAL INCLUSIONS 5
This inequality and (3.7) imply that for all t ∈ (0, T )
Z t
Z
k∇u0m (s)k2 ds ≤ c7 ,
|u0m (t)|2 dΓ ≤ c8 .
0
(3.8)
Γ1
By imbedding theorem, from (3.9) we have
Z t
ku0m (s)k2 ds ≤ c9 .
(3.9)
0
Furthermore, from (3.4) and (3.9) we obtain
Z t
kbm (u0m (s))k2Γ1 ds ≤ c10 .
(3.10)
0
Since M̄ (k∇um (t)k2 ) ≥ m0 k∇um (t)k2 , by (3.8)
k∇um (t)k2 ≤ c11 .
(3.11)
00
Next, multiplying (3.1) by gjm
(t) and summing from j = 1 to j = m, we have
1 d
d
k∇u0m (t)k2 + M (k∇um (t)k2 ) (∇um (t), ∇u0m (t))
2 dt
dt
− M (k∇um (t)k2 )k∇u0m (t)k2 + (bm (u0m (t)), u00m (t))Γ1
Z
1 d 0
kum (t)kρ+2
+
K(um (t))(u00m (t))2 dΓr +
ρ+2,Γ1
ρ
+
2
dt
Γ1
ku00m (t)k2 +
(3.12)
= (f (t), u00m (t)).
Integrating this inequality over (0, t) and using (2.1) and Young’s inequality, we
obtain
Z t
Z t
1
1
ku00m (s)k2Γ1 ds +
ku0m (t)kρ+2
ku00m (s)k2 ds + k∇u0m (t)k2 + K0
ρ+2,Γ1
2
ρ
+
2
0
0
Z t
≤
M (k∇um (s)k2 )k∇u0m (s)k2 ds − M (k∇um (t)k2 )(∇um (t), ∇u0m (t))
0
Z t
0
2
0
2
+ 2M (k∇um (t)k )(∇um (t), ∇um (t)) + ku00m (s)k2Γ1 ds
(3.13)
0
Z t
Z t
Z t
+ C()
kbm (u0m (s))k2Γ1 ds + ku00m (s)k2 ds + C()
kf (s)k2 ds,
0
0
0
u0m (0)
where we have used um (0) =
= 0. Since is arbitrary and M (s) is a C 1
function, from (3.8),(3.9),(3.11) and (3.12), we conclude that
Z t
Z t
ku00m (s)k2 ds + k∇u0m (t)k2 +
ku00m (s)k2Γ1 ds + ku0m (t)kρ+2
(3.14)
ρ+2,Γ1 ≤ c12 .
0
0
From (3.8)-(3.12), and (3.15), taking into consideration that uΓ0 = 0, we obtain
(um ) is bounded in L∞ (0, T ; H1 (Ω)),
(u0m ) is bounded in L∞ (0, T ; H1 (Ω)) ∩ L∞ (0, T ; Lρ+2 (Γ1 )),
(u00m ) is bounded in L2 (0, T ; L2 (Ω) ∩ L2 (Γ1 )),
(bm (u0m )) is bounded in L2 (0, T ; L2 (Γ1 )).
(3.15)
6
JONG YEOUL PARK & SUN HYE PARK
EJDE–2003/80
Step 2 : Passage to the limit. Multiplying (3.1) by ψ ∈ C 1 (0, T ) with ψ(T ) = 0
and integrating over (0, T ), we obtain
Z Tn
(u00m (t), wj ) + (∇u0m (t), ∇wj ) + M (k∇um (t)k2 )(∇um (t), ∇wj )
0
o
+ (bm (u0m (t)), wj )Γ1 + (|u0m (t)|ρ u0m (t) − K 0 (um (t))(u0m (t))2 , wj )Γ1 ψ(t)dt
Z T
(3.16)
−
(K(um (t))u0m (t), wj )Γ1 ψ 0 (t)dt
0
Z
=
T
(f (t), wj )ψ(t)dt.
0
From (3.16), we have subsequences (in the sequel we denote subsequences by the
same symbols as original sequences) such that
weakly star in L∞ (0, T ; H1 (Ω)),
um → u
u0m
→u
0
∞
∞
ρ+2
weakly star in L (0, T ; H1 (Ω)) ∩ L (0, T ; L
u00m
→u
b
m
00
(u0m )
2
2
(3.17)
(Γ1 )),
2
weakly in L (0, T ; L (Ω) ∩ L (Γ1 )),
→Ξ
2
2
weakly in L (0, T ; L (Γ1 )).
(3.18)
(3.19)
(3.20)
From (3.18)–(3.21), considering that the imbedding H1 (Ω) ,→ L2 (Γ) is continuous
and compact and using Aubin compactness theorem [2], we have
ρ+2
> 1,
ρ+1
→ u0 a.e. on Σ1 .
|u0m |ρ u0m , K(um )u0m , K 0 (um )(u0m )2 ∈ Lq (Σ1 ),
um → u a.e. on Σ1
and u0m
q=
(3.21)
(3.22)
Therefore,
|u0m |ρ u0m → |u0 |ρ u0 ,
K(um )u0m → K(u)u0 ,
K 0 (um )(u0m )2 → K 0 (u)(u0 )2
a.e. on Σ1 .
(3.23)
Step 3 : (u, Ξ) is a solution of (1.1). Letting m tend to infinity in (3.17) and
using (3.18)-(3.24), we have
Z T
00
(u (t), wj ) + (∇u0 (t), ∇wj ) + M (k∇u(t)k2 )(∇u(t), ∇wj )
0
+ (Ξ(t), wj )Γ1 + (|u0 (t)|ρ u0 (t) − K 0 (u(t))(u0 (t))2 , wj )Γ1 ψ(t)dt
Z T
(3.24)
−
(K(u(t))u0 (t), wj )Γ1 ψ 0 (t)dt
0
Z
=
T
(f (t), wj )ψ(t)dt.
0
Since {wj } is dense in H1 (Ω) ∩ Lρ+2 (Γ), we conclude that (2.4) hold. It remains
to show that (2.5), i.e., Ξ(x, t) ∈ ϕ(u0 (x, t)) a.e. (x, t) ∈ Σ1 . By the Aubin-Lions
compactness Lemma[2], we get from (3.19)-(3.20) that
u0m → u0
strongly in L2 (0, T ; L2 (Γ1 )).
This implies u0m (x, t) → u0 (x, t) a.e. in Σ1 . Thus, for given η > 0, using the
theorems of Lusin and Egoroff, we can choose a subset ω ⊂ Σ1 such that meas(ω) <
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HYPERBOLIC SYSTEMS WITH BOUNDARY DIFFERENTIAL INCLUSIONS 7
η, u0 ∈ L∞ (Σ1 \ ω) and u0m → u0 uniformly on Σ1 \ ω. Thus, for each > 0, there
is an N > 2 such that
|u0m (x, t) − u0 (x, t)| < , ∀(x, t) ∈ Σ1 \ ω.
2
Then, if |u0m (x, t) − s| < 1/m, we have |u0 (x, t) − s| < for all m > N and
(x, t) ∈ Σ1 \ ω. Therefore,
b (u0 (x, t)) ≤ bm (u0m (x, t)) ≤ b (u0 (x, t)),
∀m > N, (x, t) ∈ Σ1 \ ω.
∞
Let φ ∈ L (Σ1 ), φ ≥ 0. Then
Z
Z
b (u0 (x, t))φ(x, t)dΓdt ≤
bm (u0m (x, t))φ(x, t)dΓdt
Σ1 \ω
Σ \ω
Z 1
b (u0 (x, t))φ(x, t)dΓdt.
≤
(3.25)
Σ1 \ω
Letting m approach ∞ in (3.26) and using (3.21), we obtain
Z
Z
0
Ξ(x, t)φ(x, t)dΓdt
b (u (x, t))φ(x, t)dΓdt ≤
Σ \ω
Σ1 \ω
Z 1
≤
b (u0 (x, t))φ(x, t)dΓdt.
(3.26)
Σ1 \ω
+
Letting → 0 in (3.27), we infer that
Ξ(x, t) ∈ ϕ(u0 (x, t))
a. e. in Σ1 \ ω,
+
and letting η → 0 we get
Ξ(x, t) ∈ ϕ(u0 (x, t))
a.e. in Σ1 .
This completes the proof.
References
[1] G. G. Doronin, N. A. Lar’kin and A. J. Souza; A hyperbolic problem with nonlinear secondorder boundary damping, Electronic J. Diff. Eqs. 1998(1998) No. 28, 1-10.
[2] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, DunodGauthier Villars, Paris (1969)
[3] M. Miettinen, A parabolic hemivariational inequality, Nonlinear Anal. 26(1996), 725-734
[4] M. Miettinen and P. D. Panagiotopoulos, On parabolic hemivariational inequalities and applications, Nonlinear Anal. 35(1999), 885-915
[5] J. Rauch, Discontinuous semilinear differential equations and multiple valued maps, Proc.
Amer. Math. Soc.64(1977), 277-282
Department of Mathematics, Busan National University,
30 Changjeon-dong, Keumjeong-ku, Busan, 609-735, South Korea
E-mail address, Jong Yeoul Park: jyepark@pusan.ac.kr