Electronic Journal of Differential Equations, Vol. 2002(2002), No. 91, pp. 1–22.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu (login: ftp)
Existence of global solutions to a quasilinear wave
equation with general nonlinear damping ∗
Mohammed Aassila & Abbes Benaissa
Abstract
In this paper we prove the existence of a global solution and study
its decay for the solutions to a quasilinear wave equation with a general
nonlinear dissipative term by constructing a stable set in H 2 ∩ H01 .
1
Introduction
We consider the problem
u00 − Φ(k∇x uk22 )∆x u + g(u0 ) + f (u) = 0 in Ω × [0, +∞[,
u = 0 on Γ × [0, +∞[,
u(x, 0) = u0 (x), u0 (x, 0) = u1 (x) on Ω,
(1.1)
where Ω is a bounded domain in Rn with a smooth boundary ∂Ω = Γ, Φ(s) is
a C 1 - class function on [0, +∞[ satisfying Φ(s) ≥ m0 > 0 for s ≥ 0 with m0
constant.
For the problem (1.1), when Φ(s) ≡ 1 and g(x) = δx (δ > 0), Ikehata and
Suzuki [11] investigated the dynamics, they have shown that for sufficiently small
initial data (u0 , u1 ), the trajectory (u(t), u0 (t)) tends to (0, 0) in H01 (Ω) × L2 (Ω)
as t → +∞. When g(x) = δ|x|m−1 x (m ≥ 1) and f (y) = −β|y|p−1 y (β > 0, p ≥
1), Georgiev and Todorova [6] have shown that if the damping term dominates
over the source, then a global solution exists for any initial data. Quite recently,
Ikehata [8] proved that a global solution exists with no relation between p and
m, and Todorova [27] proved that the energy decay rate is E(t) ≤ (1+t)−2/(m−1)
for t ≥ 0, she used a general method on the energy decay introduced by Nakao
[19]. Unfortunately this method does not seem to be applicable to the case of
more general functions g.
Aassila [2] proved the existence of a global decaying H 2 solution when g(x)
has not necessarily a polynomial growth near zero and a source term of the form
β|y|p−1 y, but with small parameter β. The decay rate of the global solution
∗ Mathematics Subject Classifications: 35B40, 35L70, 35B37.
Key words: Quasilinear wave equation, global existence, asymptotic behavior,
nonlinear dissipative term, multiplier method.
c
2002
Southwest Texas State University.
Submitted July 02, 2002. Published October 26, 2002.
1
2
Existence of global solutions
EJDE–2002/91
depends on the polynomial growth near zero of g(x) as it was proved in [3, 27,
15].
When Φ(s) is not a constant function, g(x) ≡ 0 and f (y) ≡ 0 the equation is
often called the wave equation of Kirchhoff type. This equation was introduced
to study the nonlinear vibrations of an elastic strings by Kirchhoff [14], and the
existence of global solutions was investigated by many authors [25, 13, 7]. In
[9], the authors discussed the existence of a global decaying solution in the case
(γ+2)
Φ(s) = m0 + s 2 , γ ≥ 0, g(v) = |v|r v, 0 ≤ r ≤ 2/(n − 2) (0 ≤ r ≤ ∞ if
n = 1, 2), f (u) = −|u|α u, 0 < α ≤ 4/(n − 2) (0 < α < ∞ if n = 1, 2) by use of a
stable set method due to Sattinger [26]. But, then, the method in [9] cannot be
applied to the case α > 4/(n − 2), which is caused by the construction of stable
set in H01 . Quite recently, in [10] (see also [1]) Ikehata, Matsuyama and Nakao
have constructed a stable set in H01 ∩ H 2 to obtain a global decaying solution to
the initial boundary value problem for quasilinear visco-elastic wave equations.
Our purpose in this paper is to give a global solvability in the class H01 ∩ H 2
and energy decay estimates of the solutions to problem (1.1) for a general nonlinear damping g. We use some new techniques introduced in [2] to derive a
decay rate of the solution. So we use the argument combining the method in
[2] with the concept of stable set in H01 ∩ H 2 . We also use some ideas from
[17] introduced in the study of the decay rates of solutions to the wave equation
utt − ∆u + g(ut ) = 0 in Ω × R+ .
We conclude this section by stating our plan and giving some notations.
In section 2 we shall prepare some lemmas needed for our arguments. Section
3 is devoted to the proof of the global existence and decay estimates to the
problem (1.1). Section 4 is devoted to the proof of the global existence and
decay estimates to the problem (1.1) in the case α = 0, i.e., f (u) = −u. In
this case the smallness of |Ω| (the volume of Ω) will play an essential role in
our argument. In the last section we shall treat the case Φ ≡ 1, we prove only
the global decaying H01 solution, but we obtain more results than the case when
Φ 6≡ 1. The condition that β (k1 in our paper) is small is removed here, also we
extend some results obtained by Ono [24] and Martinez [17].
Throughout this paper the functions considered are all real valued. We omit
the space variable x of u(t, x), ut (t, x) and simply denote u(t, x), ut (t, x) by u(t),
u0 (t), respectively, when no confusion arises. Let l be a number with 2 ≤ l ≤ ∞.
We denote by k . kl the Ll norm over Ω. In particular, L2 norm is denoted k . k2 .
( . ) denotes the usual L2 inner product. We use familiar function spaces H01 ,
H 2.
2
Preliminaries
Let us state the precise hypotheses on Φ, g and f .
(H1) Φ is a C 1 -class function on R+ and satisfies
Φ(s) ≥ m0
and |Φ0 (s)| ≤ m1 sγ/2
for 0 ≤ s < ∞
(2.1)
EJDE–2002/91
Mohammed Aassila & Abbes Benaissa
3
for some constants m0 > 0, m1 ≥ 0, and γ ≥ 0.
(H2) g is a C 1 odd increasing function and
c2 |x| ≤ |g(x)| ≤ c3 |x|q
if |x| ≥ 1
with 1 ≤ q ≤
N +2
,
(N − 2)+
where c1 , c2 and c3 are positive constants.
(H3) f (.) is a C 1 (R) satisfying
|f (u)| ≤ k2 |u|α+1
and |f 0 (u)| ≤ k2 |u|α
for allu ∈ R
(2.2)
with some constant k2 > 0, and
0<α<
2
,
(N − 4)+
(2.3)
where (N − 4)+ = max{N − 4, 0}. A typical example of these functions
is f (u) = −|u|α u.
We first state three well known lemmas, and then we prove two other lemmas
that will be needed later.
Lemma 2.1 (Sobolev-Poincaré inequality) Let q be a number with 2 ≤ q <
+∞ (n = 1, 2) or 2 ≤ q ≤ 2n/(n − 2) (n ≥ 3), then there is a constant c∗ =
c(Ω, q) such that
kukq ≤ c∗ k∇uk2 for u ∈ H01 (Ω).
Lemma 2.2 (Gagliardo-Nirenberg) Let 1 ≤ r < q ≤ +∞ and p ≤ q. Then,
the inequality
kukW m,q ≤ CkukθW m,p kuk1−θ
r
u ∈ W m,p ∩ Lr
for
holds with some C > 0 and
θ=
k
1 1 m 1 1 −1
+ −
+ −
n r
q n
r
p
provided that 0 < θ ≤ 1 (we assume 0 < θ < 1 if q = +∞).
Lemma 2.3 ([15]) Let E : R+ → R+ be a non-increasing function and assume
that there are two constants p ≥ 1 and A > 0 such that
Z +∞
p+1
E 2 (t) dt ≤ AE(S), 0 ≤ S < +∞.
S
Then
−2
E(t) ≤ cE(0)(1 + t) p−1
−ωt
E(t) ≤ cE(0)e
∀t ≥ 0,
∀t ≥ 0,
if
if
p > 1,
p = 1,
where c and ω are positive constants independent of the initial energy E(0).
4
Existence of global solutions
EJDE–2002/91
Lemma 2.4 ([17]) Let E : R+ → R+ be a non increasing function and φ :
R+ → R+ an increasing C 2 function such that
φ(0) = 0
φ(t) → +∞
and
as t → +∞.
Assume that there exist p ≥ 1 and A > 0 such that
Z
+∞
E(t)
p+1
2
(t)φ0 (t) dt ≤ AE(S).
0 ≤ S < +∞,
S
Then
E(t) ≤ cE(0)(1 + φ(t))−2/(p−1)
−ωφ(t)
E(t) ≤ cE(0)e
∀t ≥ 0,
∀t ≥ 0,
if
if
p > 1,
p = 1,
where c and ω are positive constants independent of the initial energy E(0).
Proof Let f : R+ → R+ be defined by f (x) := E(φ−1 (x)), (we remark that
φ−1 has a sense by the hypotheses assumed on φ). f is non-increasing, f (0) =
E(0) and if we set x := φ(t) we obtain
Z
φ(T )
f (x)
p+1
2
dx =
Z
=
Z
φ(S)
φ(T )
E(φ−1 (x))(p+1)/2 dx
φ(S)
T
E(t)
p+1
2
φ0 (t) dt
S
≤AE(S) = Af (φ(S))
0 ≤ S < T < +∞.
Setting s := φ(S) and letting T → +∞, we deduce that
Z
+∞
f (x)
p+1
2
dx ≤ Af (s)
0 ≤ s < +∞.
s
Thanks to Lemma 2.3, we deduce the desired results.
Remark 2.5 The use of a ‘weight function’ φ(t) to establish the decay rate of
solutions to hyperbolic PDE was successfully done by Aassila [3], Martinez [17],
and Mochizuki and Motai [18].
Lemma 2.6 ([17]) There exists a function φ : R+ → R increasing and such
that φ is concave and φ(t) → +∞ as t → +∞, φ0 (t) → 0 as t → +∞, and
Z
1
+∞
2
φ0 (t) g −1 (φ0 (t)) dt < +∞.
EJDE–2002/91
Mohammed Aassila & Abbes Benaissa
5
Proof. If such a function exists, we can assume that φ(1) = 1. Setting s :=
φ(t) we obtain
+∞
Z
0
φ (t) g
−1
1
0
(φ (t)) dt =
=
+∞
Z
2
g −1 (φ0 (φ−1 (s))) ds
1
+∞
Z
g −1
1
Let us define
ψ(t) := 1 +
Z
t
1
1
ds,
g(1/s)
2
1
ds.
(φ−1 )0 (s)
t ≥ 1.
Note that ψ is increasing, of class C 2 , and
ψ 0 (t) =
1
→ +∞ as
g(1/t)
t → +∞.
Hence ψ(t) → +∞ as t → +∞ and
Z
+∞
1
g
−1
+∞
Z
1 2
ds =
ψ 0 (s)
1
1
ds < +∞.
s2
Furthermore ψ 0 is non-decreasing, and hence ψ is convex. Let us verify that
ψ −1 is concave: from ψ(ψ −1 (s)) = s we have
(ψ
2
ψ 00 (ψ −1 (s)) (ψ −1 )0 (s)
ψ 00 ψ −1 (s)
) (s) = −
=−
3 ≤ 0.
ψ 0 (ψ −1 (s))
(ψ 0 (ψ −1 (s)))
−1 00
In conclusion, if we set φ(t) := ψ −1 (t) for all t ≥ 1, we see that φ verify all the
hypotheses of lemma 2.6.
First, we shall construct a stable set in H01 ∩ H 2 . For this, we define the
following functionals:
1
J(u) ≡
2
Z
k∇x uk22
Φ(s) ds +
u
Z Z
0
Ω
˜
J(u)
≡ Φ(k∇x uk22 )k∇x uk22 +
f (η) dη dx for u ∈ H01 ,
0
Z
f (u)u dx
for u ∈ H01
Ω
1
E(u, v) ≡ kvk22 + J(u)
2
for (u, v) ∈ H01 × L2 .
Lemma 2.7 Let 0 < α < 4/(N − 4)+ . Then, for any K > 0, there exists a
number ε0 ≡ ε0 (K) > 0 such that if k∆x uk ≤ K and k∇x uk ≤ ε0 , we have
J(u) ≥
m0
k∇x uk22
4
and
m0
˜
J(u)
≥
k∇x uk22 .
2
(2.4)
6
Existence of global solutions
Proof:
EJDE–2002/91
We see from the Gagliardo-Nirenberg inequality that
(α+2)(1−θ)
kukα+2
α+2 ≤ Ckuk
2N
(N −2)
(α+2)θ
k∆x uk2
(α+2)(1−θ)
≤ Ck∇x uk2
(α+2)θ
k∆x uk2
(2.5)
with
θ=
N − 2
1 + 2
N − 2 1 −1
((N − 2)α − 4)+
−
+
−
=
(≤ 1). (2.6)
2N
α+2
N
2N
2
2(α + 2)
Here, we note that
(α + 2)(1 − θ) − 2 =

α




(4−N )α+4


2


if 0 < α ≤ N 4−2
(0 < α < ∞ for N = 1, 2),
if N 4−2 < α < N 4−4
( N 4−2 < α < ∞ for N = 3, 4).
(2.7)
Hence, if k∆x uk2 ≤ K, we have
m0
k2
k∇x uk22 −
kukα+2
α+2
2
α+2
m0
(α+2)(1−θ)
(α+2)θ
≥
k∇x uk22 − Ck∇x uk2
k∆x uk2
2
m0
(α+2)(1−θ)−2 ≥
− CK (α+2)θ k∇x uk2
k∇x uk22 .
2
J(u) ≥
(2.8)
Using (2.7), we define ε0 ≡ ε0 (K) by
(α+2)(1−θ)−2
CK (α+2)θ ε0
=
m0
.
4
Thus, we obtain
J(u) ≥
m0
k∇x uk22
4
(2.9)
˜
if k∇x uk2 ≤ ε0 . It is clear that (2.9) is valid for J(u).
Let us define a stable in
H01
2
∩ H as follows: For some K > 0,
n
WK ≡ (u, v) ∈ (H01 ∩ H 2 ) × H01 : k∆x uk2 < K,
q
o
k∇x vk2 < K and 4m−1
E(u,
v)
<
ε
0
0
Remark 2.8 If f (u)u ≥ 0, we do not need ε0 (K), and WK is replaced by
W̃K ≡ {(u, v) ∈ (H01 ∩ H 2 ) × H01 : k∆x uk2 < K, k∇x vk2 < K}
EJDE–2002/91
3
Mohammed Aassila & Abbes Benaissa
7
Global Existence and Asymptotic Behavior
A simple computation shows that
0
E (t) = −
Z
u0 g(u0 ) dx ≤ 0,
Ω
hence the energy is non-increasing and in particular E(t) ≤ E(0) for all t ≥ 0.
Lemma 3.1 Let u(t) be a strong solution satisfying (u(t), u0 (t)) ∈ WK on [0, T [
for some K > 0. Then we have
1 2
E(t) ≤ cE(0) G−1
t
on [0, T [,
where c is a positive constant independent of the initial energy E(0) and G(x) =
xg(x). Furthermore, if x 7→ g(x)/x is non-decreasing on [0, η] for some η > 0,
then
1 2
E(t) ≤ cE(0) g −1
on [0, T [,
t
where c is a positive constant independent of the initial energy E(0).
Proof of lemma 3.1 For the rest of this article, we denote by c various
positive constants which may be different at different occurences. We multiply
the first equation of (1.1) by Eφ0 u, where φ is a function satisfying all the
hypotheses of lemma 2.6, we obtain
Z
0=
T
0
Eφ
S
Z
u(u00 − Φ(k∇x uk22 )∆u + g(u0 ) + f (u)) dx dt
Ω
Z
h
iT Z
0
0
= Eφ
uu dx −
S
Ω
+
Z
+
Z
T
Eφ0
Z
Eφ0
Z
S
T
S
Ω
T
0 0
00
(E φ + Eφ )
Z
S
0
uu dx dt − 2
Z
Ω
T
0
Eφ
Z
S
u02 dx dt
Ω
u02 + Φ(k∇x uk22 )|∇u|2 + f (u)u dx dt
ug(u0 ) dx dt .
Ω
˜
Under the assumption (u(t), u0 (t)) ∈ WK , the functionals J(u(t)) and J(u(t))
2
are both equivalent to k∇x u(t)k2 , by lemma 2.7. So we deduce that
Z
T
S
Z
h
iT Z
E 2 φ0 dt ≤ − Eφ0
uu0 dx +
S
Ω
+2
Z
T
Eφ0
S
h
Z
T
(E 0 φ0 + Eφ00 )
S
u02 dx dt −
Ω
0
≤ − Eφ
Z
Ω
S
Z
T
Eφ0
Z
ug(u0 ) dx dt
Ω
T
S
uu0 dx dt
Ω
S
iT Z
0
uu dx +
Z
0 0
00
(E φ + Eφ )
Z
Ω
uu0 dx dt
8
Existence of global solutions
+2
Z
T
Z
0
Eφ
S
+ε
Z
02
u dx dt + c(ε)
T
|u0 |≤1
S
0
Eφ
Z
u2 dx dt −
Z
g(u0 )2 dx dt
|u0 |≤1
S
Z
Eφ0
T
Z
Ω
EJDE–2002/91
T
Eφ0
Z
ug(u0 ) dx dt
|u0 |>1
S
for all ε > 0. Choosing ε small enough, we deduce that
Z T
E 2 φ0 dt
S
Z
h
iT Z
≤ − Eφ0
uu0 dx +
S
Ω
T
Z
≤ cE(S) −
0
Eφ
(E 0 φ0 + Eφ00 )
Z
S
Z
Also, we have
Z T
Z
Eφ0
uu0 dx dt + c
Z
Ω
0
ug(u ) dx dt + c
|u0 |>1
S
Z
T
Eφ0
Z
S
T
0
Eφ
Z
S
u02 dx dt
Ω
02
u dx dt.
Ω
ug(u0 ) dx dt
|u0 |>1
S
≤
T
T
Z
Eφ0
S
≤c
Z
≤c
Z
Z
Ω
T
E
3/2 0
φ
|u0 |>1
Z
0
0
u g(u ) dx
|u0 |>1
S
T
φ0 (E
S
≤c(ε0 )
1/(q+1) Z
|u|q dx
q
3
2 − q+1
) (−E 0 )
q
(q+1)
T
Z
φ0 (−E 0 E) dt + ε0
T
Z
S
|g(u0 )|
q/(q+1)
E
q
q+1
≤
(q+1)
q
Z
q/(q+1)
dx
T
q
φ0 E 3/2 (−E 0 ) (q+1)
S
q
3
φ0 E (q+1)( 2 − (q+1) ) dt
S
≤c(ε0 )E(S)2 + ε0 E(0)(q−1)/2
Z
T
φ0 E 2 dt
S
for every ε0 > 0. Choosing ε0 small enough, we obtain
Z T
Z
Z T
E 2 φ0 dt ≤ cE(S) + c
Eφ0
u02 dx dt
S
S
Ω
We want to majorize the last term of the above inequality, we have
Z T
Z
Z T
Z
Z T
Z
Eφ0
u02 dx dt =
Eφ0
u02 dx dt +
Eφ0
u02 dx dt
S
Ω
S
+
Ω1
Z
T
S
Eφ0
S
Z
Ω2
u02 dx dt,
Ω3
where, for t ≥ 1,
Ω1 := {x ∈ Ω : |u0 | ≤ h(t)},
Ω2 := {x ∈ Ω : h(t) < |u0 | ≤ h(1)},
EJDE–2002/91
Mohammed Aassila & Abbes Benaissa
9
Ω3 := {x ∈ Ω : |u0 | > h(1)},
and h(t) := g −1 (φ0 (t)), which is a positive non-increasing function and satisfies
h(t) → 0 as t → +∞. Because
T
Z
Eφ0
Z
S
u02 dx dt ≤c
T
Z
Ω1
Z
E(t)φ0 (t)
S
Ω1
≤cE(S)
h(t)2 ds dt
T
Z
φ0 (t)(g −1 (φ0 (t)))2 dt ≤ cE(S),
S
we have the following: Since g is non-decreasing, for x ∈ Ω2 we have φ0 (t) =
g(h(t)) ≤ |g(u0 )|, and hence
Z
T
Eφ0
Z
S
Z
u02 dx dt ≤
Ω2
T
E
S
≤h(1)
Z
Z
|g(u0 )|u02 dx dt
Ω2
T
E
Z
S
u0 g(u0 ) dx dt ≤
Ω2
h(1)
E(S)2 ;
2
and since g(x) ≥ cx for x ≥ h(1), we have
Z
T
Eφ0
S
Z
u02 dx dt ≤c
Z
≤c
Z
Ω3
T
Eφ0
Z
S
T
u0 g(u0 ) dx dt
Ω
E(−E 0 ) dx dt ≤ cE(S)2 .
S
Then we deduce that
Z
T
E 2 φ0 dt ≤ cE(S),
S
and thanks to Lemma 2.6, we obtain
E(t) ≤
c E(0)
,
φ(t)
∀t ≥ 1.
Let s0 be such that g(1/s0 ) ≤ 1, since g is non-decreasing we have
ψ(s) ≤ 1 + (s − 1)
1
1
1
≤s
=
g(1/s)
g(1/s)
G(1/s)
hence s ≤ φ 1/G(1/s) and
1
1
≤
φ(t)
s
Thus
with t :=
1
.
G(1/s)
1
≤ G−1 (1/t).
φ(t)
∀s ≥ s0 ,
10
Existence of global solutions
EJDE–2002/91
Now define H(x) := g(x)/x, H is non-decreasing, H(0) = 0, then we use the
function h(t) := H −1 (φ0 (t)). On Ω2 it holds that
φ0 (t)(u0 )2 ≤ |H(u0 )|(u0 )2 = u0 g(u0 ).
The same calculations as above with
φ−1 (t) = 1 +
Z
1
t
1
ds
H(1/s)
2
yield E(t) ≤ c E(0) g −1 (1/t) .
Lemma 3.2 Let u(t) be a strong solution satisfying (u(t), u0 (t)) ∈ WK on [0, T [
for some K > 0. Assume that
Z
0
+∞
min{γ+1,α(1−θ0 )}
g −1 (1/t)
dt < +∞.
Then we have
k∇u0 (t)k22 + k∆u(t)k22 ≤ Q21 (I0 , I1 , K),
with limI0 →0 Q21 (I0 , I1 , K) = I12 and where we set
I02 = E(0) =
1
ku1 k22 + J(u0 ),
2
I12 = k∇u1 k22 + Φ(k∇x u0 k22 )k∆u0 k22
Proof Multiplying the first equation of (1.1) by −∆u0 (t) and integrating over
Ω, we obtain
i 1 dh
k∇u0 (t)k22 + Φ(k∇x uk22 )k∆u(t)k22 + ∇g(u0 (t)), ∇u0 (t)
2 dt
Z
=−
f 0 (u)∇u.∇u0 (t) dx + Φ0 (k∇x uk22 )(∇u0 (t), ∇u(t))k∆x uk22 .
Ω
We set
E1 (t) ≡ k∇x u0 k22 + Φ(k∇x uk22 )k∆x uk22
Using the assumptions on Φ, g et f , we have
Z
d
0
2
E1 (t) ≤Ck∇x ukγ+1
k∇
u
k
k∆
uk
+
2k
|u|α |∇x u||∇x u0 | dx
x
2
x
2
2
2
dt
Ω
n
Z
1/2 Z
1/2 o
(γ+1)/2 3
2α
2
≤C E(t)
K +
|u| |∇x u| dx
|∇x u0 | dx
Ω
Ω
(3.1)
EJDE–2002/91
Mohammed Aassila & Abbes Benaissa
11
Here, we see from the Gagliardo-Nirenberg inequality that
Z
1/2
|u|2α |∇x u|2 dx
≤ ku(t)kα
N α k∇x u(t)k 2N
(N −2)
Ω
≤ Cku(t)k
≤
α(1−θ0 )
2N
(N −2)
0
k∆x u(t)kαθ
2 k∆x u(t)k2
(3.2)
α(1−θ0 )
0 +1
Ck∇x u(t)k2
k∆x u(t)kαθ
2
α(1−θ0 ) αθ0 +1
≤ CE(t)
K
with
N − 2
1 +
((N − 2)α − 2)+
−
=
2
α
2α
Hence, it follows from (3.1) and (3.2) that
θ0 =
(≤ 1).
n
o
α(1−θ0 )
(γ+1)
d
E1 (t) ≤ C E(t) 2 K 3 + E(t) 2 K αθ0 +2 .
dt
(3.3)
we conclude that
k∆x u(t)k22 + k∇x u0 (t)k22
Z ∞
Z ∞
n
o
1
2
3
(γ+1)/2
αθ0 +2
≤
I1 + CK
E(t)
dt + CK
E(t)α(1−θ0 )/2 dt
min{1, m0 }
0
0
Example
Let g(x) be the inverse function of
M (0) = 0
and M (x) =
xσ
(log(− log x))β
for 0 < x < x0 ,
(β, σ > 0).
The function g exists and satisfies the hypothesis (H2), when 0 < σ < 1 (see
Appendix). So
1
g −1 (1/t) = σ
t (log(log t))β
the conditions in the Lemma 3.2 give
Z ∞
1
dt < ∞,
σ(γ+1) (log(log t))β(γ+1)
t
t
Z ∞0
1
dt < ∞,
σα(1−θ0 ) (log(log t))βα(1−θ0 )
t
t0
(3.4)
(3.5)
which are similar to Bertrand integrals. So, when γ = 0, the first integral (3.4)
is not finite, we obtain the following cases: if σ(γ + 1) > 1, the integral is finite,
if σ(γ + 1) = 1, and β(γ + 1) > 1, also the integral is finite. The second integral
(3.5), is fine under the following conditions:
σ −1 < α ≤
2
(N − 2)+
for N = 1, 2, 3
12
Existence of global solutions
or
2(1 − σ)
σ
α>
or
α = σ −1
or
α=
and β −1 < α ≤
2(1 − σ)
σ
EJDE–2002/91
for N = 3
2
(N − 2)+
and α >
for N = 1, 2, 3
2(1 − β)
β
for N = 3.
Hence, we must restrict ourselves to 1 ≤ N ≤ 3.
Remark 3.3 When Φ ≡ 1, g(x) = |x|p−1 x, p ≥ 1, and f (y) = −|y|q−1 y with
q ≥ 1, we obtain
E(t) ≤ cE(0)e−ωt
E(t) ≤
∀t ≥ 0, c > 0, ω > 0,
cE(0)
(1 + t)2/(p−1)
∀t ≥ 0, c > 0
if p = 1
if p > 1.
Also
Q21 (I0 , I1 , K) = I12 + cK 2 I0q−1 ,
(q−1)(1−θ)
Q22 (I0 , I1 , K) = I12 + cK (q−1)θ+2 I0
.
When g(x) = |x|p−1 x, p ≥ 1, f (y) ≡ 0, and p < γ + 2, we obtain the same above
results (see [1]).
Theorem 3.4 Under the hypotheses of lemma 3.1 and 3.2 there exists an open
set S1 ⊂ (H 2 (Ω) ∩ H01 (Ω)) × H01 (Ω), which includes (0, 0) such that if (u0 , u1 ) ∈
S1 , the problem (1.1) has a unique global solution u satisfying
u ∈ L∞ ([0, ∞[; H 2 (Ω) ∩ H01 (Ω)) ∩ W 1,∞ ([0, ∞[; H01 (Ω)) ∩ W 2,∞ ([0, ∞[; L2 (Ω)),
furthermore we have the decay estimate
2
E(t) ≤ c E(0) g −1 (1/t)
∀t > 0.
(3.6)
Proof of theorem 3.4
Let K > 0. Put
SK ≡ {(u0 , u1 ) ∈ WK | Q1 (I0 , I1 , K) < K},
S1 ≡
[
SK .
K>0
Note that if E0 , E1 are sufficiently small, then SK is not empty.
If (u0 , u1 ) ∈ SK for some K > 0, then an assumed strong solution u(t) exist
globally and satisfies (u(t), u0 (t)) ∈ WK for all t ≥ 0. Let {wj }∞
j=1 be the basis
EJDE–2002/91
Mohammed Aassila & Abbes Benaissa
13
of H01 consisted by the eigenfunction of −∆ with Dirichlet condition. We define
the approximation solution um (m=1, 2, . . . ) in the form
um =
m
X
gjm wj
j=1
where gjm (t) are determined by
(u00m (t), wj ) + Φ(k∇x um (t)k22 )(∇x um (t), ∇x wm )
+(g(u0m (t)), wj ) + (f (um (t)), wj ) = 0
(3.7)
for j ∈ {1, 2, . . . , m} with the initial data where um (0) and u0m (0) are determined
in such a way that
um (0) = u0m =
m
X
(u0 , wj )wj → u0 strongly in H01 ∩ H 2 as m → ∞,
j=1
u0m (0) = u1m =
m
X
(u1 , wj )wj → u1 strongly in H01 as m → ∞.
j=1
By the theory of ordinary differential equations, (3.7) has a unique solution
um (t). Suppose that (u0 , u1 ) ∈ SK for K > 0. Then, (um (0), u0m (0)) ∈ SK for
large m. It is clear that all the estimates obtained above are valid for um (t) and,
in particular, um (t) exists on [0, ∞[. Thus, we conclude that (um (t), u0m (t)) ∈
WK for all t ≥ 0 and all the estimates are valid for um (t) for all t ≥ 0.
Thus, um (t) converges along a subsequence to u(t) in the following way:
1
2
um (.) → u(.) weakly * in L∞
loc ([0, ∞); H0 ∩ H ),
1
u0m (.) → ut (.) weakly * in L∞
loc ([0, ∞); H0 ),
2
um (.) → utt (.) weakly * in L∞
loc ([0, ∞); L ),
and hence,
1
Φ(k∇x um (.)k22 )∇x um (.) → Φ(k∇x u(.)k22 )∇x u(.) weakly * in L∞
loc ([0, ∞); H0 ),
1
g(um (.)) → g(u(.)) weakly * in L∞
loc ([0, ∞); H0 ),
Therefore, the limit function u(t) is a desired solution belonging to
L∞ ([0, ∞[; H01 ∩ H 2 ) ∩ W 1,∞ ([0, ∞[; H01 ) ∩ W 2,∞ ([0, ∞[; L2 )
The uniqueness can be proved by use of the monotonicity of g, nα < 2n/(n−
4) and sup0≤t≤T (ku(t)kH 2 + ku0 (t)kH01 ) ≤ C(T ) < ∞ (see [2]).
4
The case α = 0
In this section we shall discuss the existence of a global solution to the problem
(1.1) with f (u) ≡ −u. More precisely, we impose an assumption on f (u) instead
of (H3) as follows:
14
Existence of global solutions
EJDE–2002/91
(H.3)’ f (.) satisfies f (u) = −k3 u for u ∈ R with k3 C(Ω) < m0 , k3 > 0, where
C(Ω) is a quantity such that
C(Ω) =
kuk2
u∈H01 \{0} k∇x uk2
sup
(4.1)
Remark 4.1 The condition k3 C(Ω) < m0 implies that |Ω| is small in some
sense. On the other hand, if f (u) = u, we need not take C(Ω) into consideration.
Our result reads as follows.
Theorem 4.2 Under the hypotheses of Lemma 3.1 (we replace (H.3) by (H.3)’)
and 3.2 , there exists an open unbounded set S2 in (H 2 ∩H01 )×H01 , which includes
(0, 0), such that if (u0 , u1 ) ∈ S2 , the problem (1.1) has a unique solution u in
the sense of theorem 3.4 which satisfies the decay estimate (3.6).
Proof of theorem 4.2
This proof is also given in parallel way to the proof of theorem 3.4 so se just
sketch the outline.
First, let k3 C(Ω) < m0 . Then, by (4.1,)
1
J(u) =
2
k∇x uk22
Z
Φ(s) ds −
0
k3
1
kuk22 ≥ (m0 − k3 C(Ω))k∇x uk22 .
2
2
(4.2)
˜
We may assume J(u)
also satisfies (4.2). If u(t) is a strong solution satisfying
k∇x u(t)k2 < K and k∇x u0 (t)k2 < K on [0, T [ for some K > 0, then as in lemma
3.1, we derive the decay estimate
2
E(t) ≤ c g −1 (1/t) .
(4.3)
Multiplying the equation by −∆x u0 , we see
1 d
k3 d
E1 (t) ≤ |Φ0 (k∇x u(t)k22 )|(∇x u(t), ∇x u0 (t))k∆x u(t)k22 +
k∇x u(t)k22
2 dt
2 dt
k3 d
≤ CK 3 E(t)(γ+1)/2 +
k∇x u(t)k22
2 dt
(4.4)
where we set
E1 (t) = Φ(k∇x u(t)k22 )k∆x u(t)k22 + k∇x u0 (t)k22 .
we integrate (4.4) to obtain
k∆x u(t)k22 + k∇x u(t)k22
Z ∞
n
o
(γ+1)
1
2
3
≤
I1 + CK
E(t) 2 dt + k3 k∇x u(t)k22 − k3 k∇x u0 k22
min{1, m0 }
0
EJDE–2002/91
Mohammed Aassila & Abbes Benaissa
≤
n
1
I 2 + CI02 + C I0γ+1 K 3
min{1, m0 } 1
≡
Q22 (I0 , I1 , K)
Z
0
∞
15
(γ+1) o
g −1 (1/t)
dt
on [0, T [.
Defining
SK ≡ {(u0 , u1 ) ∈ H01 ∩ H 2 : Q2 (I0 , I1 , K) < K},
S2 ≡
[
SK
K>0
we conclude that if (u0 , u1 ) ∈ S2 , the corresponding solution to the problem
(1.1) exists globally and satisfies the estimate
2
E(t) ≤ c g −1 (1/t)
and k∆x u(t)k22 + k∇x u0 (t)k22 < K 2 ,
for all t > 0. The proof of theorem 4.2 is complete.
5
The case Φ ≡ 1
Usually, we study global existence for Kirchhoff equation (i.e. when Φ 6≡ 1)
in the class H 2 ∩ H01 (also when f ≡ g ≡ 0). Thus the condition in Lemma
3.2 excludes some functions g which verify (H2), for example g(x) = e−1/x or
1/x
g(x) = e−e
or the example above. We consider the case Φ ≡ 1 (or a constant
function) and we prove a global H01 solution that decays. Here we do not need
the condition of Lemma 3.2 and we will take only α ≤ 4/(n − 2)+ because we
work only in H01 (Ω).
Now, we consider the initial boundary-value problem
u00 − ∆x u + g(u0 ) + f (u) = 0 in Ω × [0, +∞[,
u = 0 on Γ × [0, +∞[,
u(x, 0) = u0 (x), u0 (x, 0) = u1 (x) on Ω,
(5.1)
First, we shall construct a stable set in H01 . For this, we need define the following
functionals:
Z Z u
1
J(u) ≡ k∇x uk22 +
f (η) dη dx for u ∈ H01 ,
2
Ω 0
Z
2
˜
J(u) ≡ k∇x uk2 +
f (u)u dx for u ∈ H01 ,
Ω
1
E(u, v) ≡ kvk22 + J(u)
2
for (u, v) ∈ H01 × L2 .
Then we can define the stable set
W = {u ∈ H01 (Ω) : k∇x uk22 − k1 kukα+2
α+2 > 0} ∪ {0}
16
Existence of global solutions
EJDE–2002/91
Lemma 5.1 (i) If α < 4/[n − 2]+ , then W is an open neighborhood of 0 in
H01 (Ω).
(ii) If u ∈ W, then
k∇x uk22 ≤ d∗ J(u)
Proof.
with
d∗ =
2(α + 2)
.
α
(5.2)
(i) From the Sobolev-Poincaré inequality (see lemma 2.1) we have
α
2
k1 kukα+2
α+2 ≤ Ak1 k∇x uk2 k∇x uk2
(5.3)
where A = cα+2
. Let
∗
U (0) ≡ u ∈ H01 (Ω) : k∇x ukα
2 <
1 .
Ak1
Then, for any u ∈ U (0)\{0}, we deduce from (5.3) that
2
k1 kukα+2
α+2 < k∇x uk2 ,
that is, K(u) > 0. This implies U (0) ⊂ W.
(ii) By the definition of K(u) and J(u) we have the inequality
J(u) ≥
1
k1
α
k∇x uk22 −
kukα+2
k∇x uk22
α+2 ≥
2
α+2
2(α + 2)
Lemma 5.2 Let u(t) be a strong solution of (5.1). Suppose that
1
˜
u(t) ∈ W and J(u(t))
≥ k∇x u(t)k22
2
(5.4)
for 0 ≤ t < T . Then we have
2
E(t) ≤ cE(0) G−1 (1/t)
on [0, T [,
where c is a positive constant independent of the initial energy E(0) and G(x) =
xg(x). Furthermore, if x 7→ g(x)/x is non-decreasing on [0, η] for some η > 0,
then we have
2
E(t) ≤ cE(0) g −1 (1/t)
on [0, T [,
where c is a positive constant independent of the initial energy E(0).
Examples
p
1) If g(x) = e−1/x for 0 < x < 1, p > 0, then E(t) ≤ c/(ln t)2/p .
2) If g(x) = e−e
1/x
for 0 < x < 1, then E(t) ≤ c/(ln(ln t))2 .
EJDE–2002/91
Mohammed Aassila & Abbes Benaissa
17
˜
Proof of lemma 3.1 The functionals J(u(t)) and J(u(t))
are both equivalent
to k∇x u(t)k22 , indeed we have
Z
2
f (u)u dx ≤ k1 kukα+2
α+2 ≤ k∇x u(t)k2
Ω
So, we have
1
3
k∇x uk22 ≤ K(u(t)) ≤ k∇x uk22 .
2
2
Also, we have
|J(u(t))| ≤
1
1
α+4
k∇x u(t)k22 +
k∇x uk22 ≤
k∇x u(t)k22 .
2
α+2
2(α + 2)
Therefore,
K(u(t)) ≥
1
α+2
k∇x uk22 ≥
J(u).
2
α+4
(5.5)
Now, we can derive the decay estimate (3.6) by similar argument as lemma 3.1.
Theorem 5.3 Suppose that α ≤ 4/(n − 2) (α < ∞ if n ≤ 2), and suppose
that initial data {u0 , u1 } belongs to W, and its initial energy E(0) is sufficiently
small such that
C4 E(0)α/2 < 1,
(5.6)
α/2
where C4 = 2k1 cα+2
d∗ . Then, Problem (5.1) has a unique global solution
∗
u ∈ W satisfying
u ∈ L∞ ([0, ∞[; H01 (Ω)) ∩ W 1,∞ ([0, ∞[; L2 (Ω));
furthermore, we have the decay estimate
2
E(t) ≤ c E(0) g −1 (1/t)
∀t > 0 .
(5.7)
Proof of Theorem 3.4
Since u0 ∈ W and W is an open set, putting
T1 = sup{t ∈ [0, +∞) : u(s) ∈ W for 0 ≤ s ≤ t},
we see that T1 > 0 and u(t) ∈ W for 0 ≤ t < T1 . If T1 < Tmax < ∞, where
Tmax is the lifespan of the solution, then u(T1 ) ∈ ∂W; that is
K(u(T1 )) = 0 and u(T1 ) 6= 0.
(5.8)
We see from lemma 2.2 and lemma 5.1 that
k1 ku(t)kα+2
α+2 ≤
1
B(t)k∇x u(t)k22
2
(5.9)
18
Existence of global solutions
EJDE–2002/91
for 0 ≤ t ≤ T1 , where we set
B(t) = C4 E(0)α/2
(5.10)
α/2
with C4 = 2k1 cα+2
d∗ . Next, we put
∗
T2 ≡ sup{t ∈ [0, +∞) : B(s) < 1 for 0 ≤ s < t},
and then we see that T2 > 0 and T2 = T1 because B(t) < 1 by (5.6). Then
1
1
K(u(t)) ≥ k∇x u(t)k22 − B(t)k∇x u(t)k22 ≥ k∇x u(t)k22
2
2
(5.11)
for 0 ≤ t ≤ T1 . Moreover, (5.8) and (5.11) imply
K(u(T1 )) ≥
1
k∇x u(T1 )k22 > 0
2
which is a contradiction, and hence, it might be T1 = Tmax . Therefore, (5.7)
hold true for 0 ≤ T ≤ Tmax , and such estimate give the desired a priori estimate;
that is, the local solution u can be extended globally (i.e., Tmax = ∞). The
proof of theorem 5.3 is now complete.
Remarks: a) By a similar argument as the proof of Theorem 4.2, we can
extend Theorem 5.3 to the case α = 0.
b) It seems to be interesting to study a global decaying H 2 solution for Kirchhoff
equation with nonlinear source and boundary damping terms or with nonlinear
boundary damping and source terms, also in the case of polynomial damping
term i.e. the following problems
u00 − Φ(k∇x uk22 )∆x u + f (u) = 0 in Ω × [0, +∞[,
u = 0 on Γ0 × [0, +∞[,
∂u
= −Q(x)g(u0 ) on Γ1 × [0, +∞[,
∂ν
u(x, 0) = u0 (x), u0 (x, 0) = u1 (x) on Ω,
and
u00 − Φ(k∇x uk22 )∆x u = 0 in Ω × [0, +∞[,
u = 0 on Γ0 × [0, +∞[,
∂u
= −Q(x)g(u0 ) + f (u) on Γ1 × [0, +∞[,
∂ν
u(x, 0) = u0 (x), u0 (x, 0) = u1 (x) on Ω,
We plan to address these questions in a future investigation.
EJDE–2002/91
Mohammed Aassila & Abbes Benaissa
19
Appendix
Let g(x) be the inverse of the function M (x) defined by
M (0) = 0,
M (x) =
xσ
(log(− log x))β
for 0 < x < x0 ,
(σ, β > 0).
For x = 1/t(0 < x < x0 ) we have
g −1 (1/t) =
1
tσ (log(log t))β
(t ≥ t0 ).
Now, we prove that the function g(x) exists and verifies the hypothesis (H2).
Indeed,
h
i
xσ σ(log(− log x)) − logβ x
(M (x))0 =
, (σ, β > 0).
(log(− log x))β+1
When x is near 0 (0 < x < x0 ), it is clear that (M (x))0 ≥ 0, so M (x) is an
increasing continuous function. Thus the function g exists. We have also
x
(log(− log x))β
=
→0
M (x)
xσ−1
as x → 0 if 0 < σ < 1, so M (x) → 0 (as x → 0) not faster than x (near 0).
We deduce that g(x) → 0 as x → 0 faster than x i.e. |g(x)| ≤ c|x|. We obtain
hypothesis (H2). Now, M (x)/x is a decreasing function; indeed,
i
h
M (x) 0
xσ−2 (σ − 1)(log(− log x)) − logβ x
.
=
x
(log(− log x))β+1
0
For x = e−n , and n big, we see that (M (x)/x) ≤ 0. g is a bijective and
decreasing function, so for each x and y near 0, such that x ≤ y, we have
M (x)/x ≥ M (y)/y, also there exist unique x0 and y 0 such that M (x) = x0
and M (y) = y 0 (because M is a bijective function), also M (x) is an increasing
function, thus, we have
x ≤ y ⇐⇒ M (x) = x0 ≤ M (y) = y 0
Therefore,
x0
y0
≥
g(x0 )
g(y 0 )
0
g(x )
g(y 0 )
⇐⇒
≤
x0
y0
x0 ≤ y 0 ⇐⇒
for 0 < x < x0 .
Acknowledgments. The authors wish to thank the anonymous referee for
his/her valuable suggestions.
20
Existence of global solutions
EJDE–2002/91
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22
Existence of global solutions
Mohammed Aassila
Mathematisches Institut, Universität zu Köln
Weyertal 86-90, D-50931 Köln, Germany.
e-mail: aassila@mi.uni-koeln.de
Abbes Benaissa
Université Djillali Liabes, Faculté des Sciences,
Département de Mathématiques,
B. P. 89, Sidi Bel Abbes 22000, Algeria
e-mail: benaissa abbes@yahoo.com
EJDE–2002/91