Applied Mathematics E-Notes, 2(2002), 98-101 c
Available free at mirror sites of http://www.math.nthu.edu.tw/∼amen/
ISSN 1607-2510
Existence of Absorbing Set for a Nonlinear Wave
Equation ∗
Ayfer Kurt†
Received 6 February 2002
Abstract
We prove the existence of an absorbing set for a Cauchy problem involving a
nonlinear wave equation.
(1)
Let u0 ∈ W2 (0, l), u1 and h are given elements in L2 (0, l), α, γ and c be positive
numbers and f (.) ∈ C 1 (R) such that
] s
F(s) =
f (η)dη ≥ −c,
(1)
0
and
f (s)s − F(s) ≥ −c
(1)
W2 (0, l)
for all s ∈ R, where
following Cauchy problem:
(2)
= {u : u, u ∈ L2 (0, l), u(0) = u(l) = 0}. We consider the
utt − uxx + αutx + γut + f (u) = h(x), x ∈ (0, l) , t ∈ R+
(3)
u (x, 0) = u0 (x), ut (x, 0) = u1 (x) , x ∈ (0, l)
(4)
u (0, t) = u (l, t) = 0, t ∈ R .
(5)
+
(1)
A continuous semigroup of operators S (t) from X = W2 (0, l) × L2 (0, l) into itself can
be defined and satisfies the semigroup condition S (t + s) = S (t) S (s) . We will prove
the existence of an absorbing set in X by defining a Lyapunov like functional φ (u, ut ).
The methods used are inspired from the results of Ladyzhenskaya [3, 4] and used in
Eden and Kalantarov [1, 2]. In the following we shall use the following notations:
u
2
]
=
l
u2 (x)dx,
0
and
(u, v) =
]
l
u(x)v(x)dx.
0
∗ Mathematics
† Cag
Subject Classifications: 35L05, 35L70, 35B40, 35B41.
University, Adana Mersin karayolu uzeri, 33800 Yenice-ersin, Turkey
98
A. Kurt
99
We also use the Wirtinger inequality
]
0
l
u2 (x)dx ≤ λ
]
l
u2x (x)dx
(6)
0
where λ = l2 /π2 .
THEOREM 1. Let u0 , u1 , h, α, γ, c be given. Suppose f ∈ C 1 (R) satisfies the
conditions (1) and (2). Suppose further that the problem (3)-(5) has a unique weak
solution. Then the semigroup S(t), t > 0, defined by S(t)(u0 , u1 ) = (u(t), u1 (t))
generated by the problem (3)-(5) is bounded and dissipative.
PROOF. Suppose u(x, t) is a weak solution of (3)-(5). Let δ be a positive number.
Multiplying the equation (3) by ut + δu and integrating over (0, l) we get
0 =
1
d 1
2
2
( ut +
ux + (F(u), 1) − (h, u) + δ (u, ut )
dt 2
2
δγ
2
2
2
+
u ) + γ ut − αδ (ux , ut ) + δ ux
2
−δ ut 2 + δ(f (u), u) − δ(h, u).
(7)
We consider the functional
φ (u, ut ) =
1
ut
2
2
+
1
ux
2
2
+ (F(u), 1) − (h, u) + δ (u, ut ) +
δγ
u
2
2
.
(8)
From (7) we write
d
φ (u, ut ) = −γ ut
dt
2
2
+ αδ (ux , ut ) − δ ux
+ δ ut
2
− δ(f (u), u) + δ(h, u).
(9)
Let η be a positive number such that η < δ. Then from (7) we write
d
φ (u, ut ) + ηφ (u, ut )
dt
η
η
δηγ
2
2
2
=
ut +
ux + η(F(u), 1) − η(h, u) + δη (u, ut ) +
u
2
2
2
−γ ut 2 + αδ (ux , ut ) − δ ux 2 + δ ut 2 − δ(f (u), u) + δ(h, u).
(10)
By using (6) we obtain
δη |(u, ut )| ≤
δη
ut
2
(δ − η) |(h, u)| ≤
η
ux
2
2
2
+
+
and
αδ |(ux , ut )| ≤
γ
ut
2
2
+
δηl2
ux
2π2
2
,
l2 (δ − η)2
h
2ηπ 2
δ 2 α2
ux
2γ
2
.
(11)
2
,
(12)
(13)
100
Nonlinear Wave Equation
With the help of the inequalities (11), (12) and (13) we obtain from (10) that
d
φ (u, ut ) + ηφ (u, ut )
dt
γ
δη η
≤
δ− +
+
ut
2
2
2
δηl2 δηγl2
δ 2 α2
+ η+
+
+
− δ ux
2π 2
2π 2
2γ
l2 (δ − η)2
2
−δ((f (u), u) − (F(u), 1)) − (δ − η) (F(u), 1) +
h .
2ηπ 2
2
Thus by choosing
δ < min
η < min
qγ γ r
,
,
4 α2
γπ2
2γ
, 2 2
γ + 4 2π α + γ 2 l2 + γl2
2
(14)
,
and writing
(δ − η) (F(u), 1) ≥ −c (δ − η) l,
δ((f (u), u) − (F(u), 1)) ≥ −cδl,
we get from (14) that
d
φ (u, ut ) + ηφ (u, ut ) ≤ c1
dt
where
c1 =
It follows from (15) that
and
l2 (δ − η)2
h
2ηπ2
2
(15)
+ cl (2δ − η) .
d ηt
e φ (u, ut ) ≤ eηt c1
dt
(16)
φ (u (., t) , ut (., t)) ≤ φ (u (·, 0) , ut (·, 0)) e−ηt +
c1
.
η
(17)
By using (8) and (1) and writing inequalities similar to (11) and (12), we may obtain
1 δγl2
δl2
δ
φ (u (., t) , ut (., t)) ≥
−
−
−
ux 2
2
2π 2
2π 2 2
1 δ
l2
2
2
.
(18)
+
h
−
ut − cl +
2 2
2δπ 2
If we choose
δ < min 1,
π2
2
l γ + l2 + π 2
,
then we get from (18) that
φ (u (., t) , ut (., t)) ≥ a0
ux
2
+ ut
2
− c2
A. Kurt
101
where
1
δl2
a0 = min 1 − 2 (γ + 1) − δ, 1 − δ
2
π
and
c2 =
l2
h
2δπ 2
2
+ cl.
If we use this result in (17) we obtain
ux
2
+ ut
2
e−ηt
1
≤
φ (u (·, 0) , ut (·, 0)) +
a0
a0
c1
+ c2 .
η
√
Then for R = a10 cη1 + c2 , B = {{u, v} ∈ X : {u, v} X ≤ 2R} is the absorbing
set for the semigroup S(t) in X. Thus the semigroup S(t), t > 0, is bounded and
dissipative.
References
[1] A. Eden and V. K. Kalantarov, Finite dimensional attractors for a class of semilinear
wave equations, Turk. J. Math., 20(1996), 425—450.
[2] A. Eden and V. K. Kalantarov, On the discrete squeezing property for semilinear
wave equations, Turk. J. Math., 22(1998), 335—341.
[3] O. A. Ladyzhenskaya, The finite-dimensionality of bounded invariant sets for the
Navier-Stokes system and other dissipative systems (Russian), Boundary value
problems of mathematical physics and related questions in the theory of functions,
14, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 115(1982),
137—155, 308.
[4] O. A. Ladyzhenskaya, Finding minimal global attractors for the Navier-Stokes equations and other partial differential equations, Upseki Mat. Nauk., 42(6)(1987), 25—
60.