Electronic Journal of Differential Equations, Vol. 2008(2008), No. 109, pp. 1–22.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
ENERGY DECAY FOR WAVE EQUATIONS OF φ-LAPLACIAN
TYPE WITH WEAKLY NONLINEAR DISSIPATION
ABBES BENAISSA, AISSA GUESMIA
Abstract. In this paper, first we prove the existence of global solutions in
Sobolev spaces for the initial boundary value problem of the wave equation of
φ-Laplacian with a general dissipation of the form
(|u0 |l−2 u0 )0 − ∆φ u + σ(t)g(u0 ) = 0 in Ω × R+ ,
`
´
Pn
2
where ∆φ =
i=1 ∂xi φ(|∂xi | )∂xi . Then we prove general stability estimates using multiplier method and general weighted integral inequalities
proved by the second author in [18]. Without imposing any growth condition
at the origin on g and φ, we show that the energy of the system is bounded
above by a quantity, depending on φ, σ and g, which tends to zero (as time
approaches infinity). These estimates allows us to consider large class of functions g and φ with general growth at the origin. We give some examples to
illustrate how to derive from our general estimates the polynomial, exponential
or logarithmic decay. The results of this paper improve and generalize many
existing results in the literature, and generate some interesting open problems.
1. Introduction
In this paper we investigate the existence of global solutions and their decay
properties for the initial boundary value problem of the wave equation with weak
dissipation
(|u0 |l−2 u0 )0 − ∆φ u + σ(t)g(u0 ) = 0 in Ω × R+
u = 0 on Γ × R+
(1.1)
u(x, 0) = u0 (x),
u0 (x, 0) = u1 (x)
on Ω
∗
n
where Ω is a bounded domain in R , n ∈ N , with aPsmooth boundary ∂Ω
= Γ,
n
2
l ≥ 2, φ, σ and g are given functions, and ∆φ =
∂
φ(|∂
|
)∂
. The
x
x
x
i
i
i
i=1
functions (u0 , u1 ) are the given initial data.
Concrete examples of (1.1) include the dissipative wave equation
u00 − ∆x u + g(u0 ) = 0 in Ω × R+
u = 0 on Γ × R+
u(x, 0) = u0 (x),
0
u (x, 0) = u1 (x)
on Ω
2000 Mathematics Subject Classification. 35B40, 35L70.
Key words and phrases. Wave equation; global existence; general dissipative term;
rate of decay; multiplier method; integral inequalities.
c
2008
Texas State University - San Marcos.
Submitted September 19, 2007. Published August 11, 2008.
1
(1.2)
2
A. BENAISSA, A. GUESMIA
EJDE-2008/109
where l = 2, φ ≡ 1 and σ ≡const. The degenerate Laplace operator
u00 − div(|∇u|p−2 ∇u) + g(u0 ) = 0
u = 0 on Γ × R+
u(x, 0) = u0 (x),
where l = 2, φ = s
p−2
2
0
in Ω × R+
u (x, 0) = u1 (x)
(1.3)
on Ω
with p ≥ 2 and σ ≡const. And the quasilinear wave equation
∇u
u00 − div p
+ g(u0 ) = 0
2
1 + |∇u|
u = 0 on Γ × R+
in Ω × R+
(1.4)
u(x, 0) = u0 (x), u0 (x, 0) = u1 (x) on Ω
√
when l = 2, φ = 1/ 1 + s and σ ≡const. Problem (1.4), with −∆u0 instead of
g(u0 ), describe the motion of fixed membrane with strong viscosity. This problem
with n = 1 was proposed by Greenberg [16] and Greenberg-MacCamy-Mizel [17] as
a model of quasilinear wave equation which admits a global solution for large data.
Quite recently, Kobayashi-Pecher-Shibata [25] have treated such nonlinearity and
proved the global existence of smooth solutions. Subsequently, Nakao [31] derived
a decay estimate of the solutions under the assumption that the mean curvature of
∂Ω is non positive.
Our purpose in this paper is: firstly to give an existence and uniqueness theorem
for global solutions in Orlitz-Sobolev spaces to the problem (1.1).
Secondly (for the stabilization problem), the aim of this paper is to obtain an
explicit and general decay rate, depending on σ, g and φ, for the energy of solutions
of (1.1) without any growth assumption on g and φ at the origin, and on σ at infinity.
More precisely, we intend to obtain a general relation between the decay rate for
the energy (when t goes to infinity), the functions σ, φ and g. The proof is based
on some general weighted integral inequalities proved by the second author in [18]
and some properties of convex functions, in particular, the dual function of convex
function to use the general Young’s inequality and Jensen’s inequality (instead of
Hölder inequality widely used in the classical case of linear or polynomial growth
of g at the origin) in objective to prove our general decay estimate under a general
growth of g at the origin. These arguments of convexity were used for the first time
(in our knowledge) by Liu and Zuazua [28], and then by Eller, Lagnese and Nicaise
[15] and Alabau-Boussouira [5].
In particular, we can consider the cases where g and φ degenerate near the origin
polynomially, between polynomially and exponentially, exponentially or faster than
exponentially. This kind of growth was considered by Liu and Zuazua [28] and
Alabau-Boussouira [5] for the wave equation, and Eller, Lagnese and Nicaise [15]
for Maxwell system. So we complement the results obtained in [8] and [9].
In this paper, the functions considered are all real valued. We omit the space
and time variables x and t of u(t, x), ut (t, x) and simply denote u(t, x), ut (t, x) by
u, u0 , respectively, when no confusion arises. Let p be a number with 2 ≤ p ≤ +∞.
We denote by k . kp the Lp norm over Ω. In particular, L2 norm is denoted k . k2 .
( . ) denotes the usual L2 inner product. We use familiar function spaces W01,2 .
The paper is organized as follow: in section 2, we give some hypotheses and we
announce the main results of this paper. In section 3 and section 4, we prove all the
EJDE-2008/109
ENERGY DECAY FOR SOLUTIONS
3
announced results. In section 5, we give some applications. Finally, we conclude
and give some comments and open questions in section 6.
2. Preliminaries and main results
We use the following hypotheses:
(H1) σ : R+ →]0, +∞[ is a non increasing function of class C 1 (R+ ) satisfying
Z +∞
σ(τ ) dτ = +∞.
(2.1)
0
(H2) φ : R+ → R+ is of class C 1 (]0, +∞[) ∩ C([0, +∞[) satisfying: φ(s) > 0 on
]0, +∞[ and φ is non decreasing.
(H3) g : R → R is a non decreasing function of class C(R) such that there exist
1 , c1 , c2 > 0, l − 1 ≤ r, (n − 2)r ≤ n + 2 and a convex and increasing
function G : R+ → R+ of class C 1 (R+ ) ∩ C 2 (]0, +∞[) satisfying G(0) = 0,
and G0 (0) = 0 or G is linear on [0, 1 ] such that
c1 |s|l−1 ≤ |g(s)| ≤ c2 |s|r
if |s| ≥ 1 ,
(2.2)
|s|l + g 2 (s) ≤ G−1 (sg(s)) if |s| ≤ 1 .
(2.3)
R +∞
Rs
Remark 2.1. 1. We have 0 φ(τ )dτ = +∞, and s 7→ 0 φ(τ )dτ is a bijection
from R+ to R+ .
Rs
2. The function φ̃(s) = 21 0 φ(τ ) dτ is a convex function. Indeed, let x1 6= 0 and
x1 6= 0 such that x1 < x2 , as φ is of class C 1 ([x1 , x2 ]) and a non decreasing function,
then φ̃ is a convex function. Now if x1 = 0, we have for all 0 ≤ λ ≤ 1
Z
Z x2
1
1 λx2
φ(s) ds = λ
φ(λz) dz
φ̃(λx2 ) =
2 0
2 0
where we have make the change of variable s = λz. As φ is a non decreasing
function and λx2 ≤ x2 for all λ ∈ [0, 1], then
φ̃(λx2 ) ≤ λφ̃(x2 ).
3. If g satisfies
H(|s|) ≤ |g(s)| ≤ H −1 (|s|)
if |s| ≤ 1
p
for a function H : R+ → R+ satisfying H (0) = 0 or H being linear on [0, δ1 ]
√
√
where δ = 2 max{1, l−2
sH( s) is convex and
1 } such that the function s 7→
increasing function from R+ to R+ of class C 1 (R+ )∩C 2 (]0, +∞[, then the condition
(2.3) is satisfied for
r
r
s
s
G(s) =
H(
).
δ
δ
In the other hand, g satisfies (H3) for any 01 ∈]0, 1 ] (with some c01 , c02 > 0 instead
of c1 , c2 , respectively).
Rs
Now we define (as before) φ̃(s) = 12 0 φ(τ )dτ and the energy associated to the
solution of the system (1.1) by the following formula:
Z
Z X
n
l−1
0 l
E(t) =
|u | dx +
φ̃(|∂xi u|2 )dx.
(2.4)
l
Ω
Ω i=1
0
4
A. BENAISSA, A. GUESMIA
EJDE-2008/109
By a simple computation, we have
E 0 (t) = −σ(t)
Z
u0 g(u0 )dx,
(2.5)
Ω
so E is non negative and non increasing function. We first state two lemmas which
will be needed later.
Lemma 2.1 (Sobolev-Poincaré’s inequality). Let p > 1 and q > 1 with (n − p)q ≤
np, then there is a constant c∗ = c∗ (Ω, p, q) such that
kukq ≤ c∗ k∇ukp
for
u ∈ W01,p (Ω).
The case p = q = 2 gives the known Poincaré’s inequality.
Lemma 2.2 (Guesmia [18]). Let E : R+ → R+ differentiable function, λ ∈ R+
and Ψ : R+ → R+ convex and increasing function such that Ψ(0) = 0. Assume that
Z +∞
Ψ(E(t)) dt ≤ E(s), ∀s ≥ 0
s
E 0 (t) ≤ λE(t),
∀t ≥ 0.
Then E satisfies the estimate
E(t) ≤ eτ0 λT0 d−1 eλ(t−h(t)) Ψ ψ −1 h(t) + ψ(E(0))
,
∀t ≥ 0
where
1
1
ds, ∀t > 0,
ψ(t) =
Ψ(s)
t
(
Ψ(t)
if λ = 0,
d(t) = R t Ψ(s)
∀t ≥ 0,
ds
if
λ > 0,
s
0
(
K −1 (D(t)), if t > T0 ,
h(t) =
0
if t ∈ [0, T0 ]
−1
ψ
t + ψ(E(0))
eλt , ∀t ≥ 0,
K(t) = D(t) +
Ψ ψ −1 t + ψ(E(0))
Z t
D(t) =
eλs ds, ∀t ≥ 0,
0
(
E(0) 0, if t > T0 ,
−1
, τ0 =
T0 = D
Ψ(E(0))
1, if t ∈ [0, T0 ].
Z
Remark 2.2. If λ = 0 (that is E is non increasing), then we have
E(t) ≤ ψ −1 h(t) + ψ(E(0)) , ∀t ≥ 0
R1 1
E(0)
where ψ(t) = t Ψ(s)
ds for t > 0, h(t) = 0 for 0 ≤ t ≤ Ψ(E(0))
and
ψ −1 t + ψ(E(0))
−1
, t > 0.
h (t) = t +
Ψ ψ −1 t + ψ(E(0))
(2.6)
This particular result generalizes the one obtained by Martinez [29] in the particular
case Ψ(t) = dtp+1 with p ≥ 0 and d > 0, and improves the one obtained by Eller,
Lagnese and Nicaise [15].
EJDE-2008/109
ENERGY DECAY FOR SOLUTIONS
5
Proof of Lemma 2.2. Because E 0 (t) ≤ λE(t) implies E(t) ≤ eλ(t−t0 ) E(t0 ) for all
t ≥ t0 ≥ 0, then, if E(t0 ) = 0 for some t0 ≥ 0, then E(t) = 0 for all t ≥ t0 , and
then there is nothing to prove in this case. So we assume that E(t) > 0 for all t ≥ 0
without lose of generality. Let
Z +∞
L(s) =
Ψ(E(t)) dt, ∀s ≥ 0.
s
We have L(s) ≤ E(s), for all s ≥ 0. The function L is positive, decreasing and of
class C 1 (R+ ) satisfying
−L0 (s) = Ψ(E(s)) ≥ Ψ(L(s)),
The function ψ is decreasing, then
0
−L0 (s)
≥ 1,
ψ(L(s)) =
Ψ(L(s))
∀s ≥ 0.
∀s ≥ 0.
Integration on [0, t], we obtain
ψ(L(t)) ≥ t + ψ(E(0)),
∀t ≥ 0.
(2.7)
Since Ψ is convex and Ψ(0) = 0, we have
Ψ(s) ≤ Ψ(1)s, ∀s ∈ [0, 1]
and
Ψ(s) ≥ Ψ(1)s, ∀s ≥ 1,
then limt→0 ψ(t) = +∞ and [ψ(E(0)), +∞[⊂ Image (ψ). Then (2.7) implies that
L(t) ≤ ψ −1 t + ψ(E(0)) , ∀t ≥ 0.
(2.8)
Now, for s ≥ 0, let
−λt
Z
fs (t) = e
t
eλτ dτ,
∀t ≥ s.
s
The function fs is increasing on [s, +∞[ and strictly positive on ]s, +∞[ such that
fs (s) = 0
and fs0 (t) + λfs (t) = 1,
∀t ≥ s ≥ 0,
and the function d is well defined, positive and increasing such that
d(t) ≤ Ψ(t)
and λtd0 (t) = λΨ(t),
∀t ≥ 0,
then
∂τ fs (τ )d(E(τ )) = fs0 (τ )d(E(τ )) + fs (τ )E 0 (τ )d0 (E(τ ))
≤ 1 − λfs (τ ) Ψ(E(τ )) + λfs (τ )Ψ(E(τ ))
= Ψ(E(τ )),
∀τ ≥ s ≥ 0.
Integrating on [s, t], we obtain
Z t
L(s) ≥
Ψ(E(τ )) dτ ≥ fs (t)d(E(t)),
∀t ≥ s ≥ 0.
(2.9)
s
Since limt→+∞ d(s) = +∞, d(0) = 0 and d is increasing, then (2.8) and (2.9) imply
E(t) ≤ d−1
inf
s∈[0,t[
ψ −1 s + ψ(E(0)) fs (t)
,
∀t > 0.
(2.10)
6
A. BENAISSA, A. GUESMIA
EJDE-2008/109
Now, let t > T0 and
ψ −1 s + ψ(E(0))
J(s) =
fs (t)
,
∀s ∈ [0, t[.
The function J is differentiable and we have
i
h
.
J 0 (s) = fs−2 (t) e−λ(t−s) ψ −1 s + ψ(E(0)) − fs (t)Ψ ψ −1 s + ψ(E(0))
Then
J 0 (s) = 0 ⇔ K(s) = D(t)
and J 0 (s) < 0 ⇔ K(s) < D(T ).
E(0)
Since K(0) = Ψ(E(0))
, D(0) = 0 and K and D are increasing (because ψ −1 is
s
decreasing and s 7→ Ψ(s)
, s > 0, is non increasing thanks to the fact that Ψ is
convex). Then, for t > T0 ,
inf J(s) = J K −1 (D(t)) = J(h(t)).
s∈[0,t[
0
Since h satisfies J (h(t)) = 0, we conclude from (2.10) our desired estimate for
t > T0 .
For t ∈ [0, T0 ], we have just to note that E 0 (t) ≤ λE(t) and the fact that d ≤ Ψ
imply
E(t) ≤ eλt E(0) ≤ eλT0 E(0) ≤ eλT0 Ψ−1 eλt Ψ(E(0)) ≤ eλT0 d−1 eλt Ψ(E(0)) .
Remark 2.3. Under the hypotheses of Lemma 2.2, we have limt→+∞ E(t) = 0.
Indeed, we have just to choose s = 12 t in (2.10) instead of h(t) and note that
d−1 (0) = 0, limt→+∞ ψ −1 (t) = 0 and limt→+∞ f 12 t (t) > 0.
Before stating the global existence theorem, we will give some notions of the theory of Orlitz spaces (see [2] and [27]) which is suitable for a large class of quasilinear
equations.
Definition 2.1. A function Φ : R+ → R+ is called an N-function if it is continuous,
convex, strictly increasing and such that
lim
s→0
Φ(s)
Φ(s)
= 0 and lim
= +∞.
s→0 s
s
The N-function complementary to Φ is defined by Φ̃(s) = maxσ≥0 (sσ − Φ(σ)).
The Simonenko indices p(Φ) and q(Φ) are defined by
tΦ0 (t)
,
t>0 Φ(t)
p(Φ) = inf
q(Φ) = sup
t>0
tΦ0 (t)
.
Φ(t)
Clearly, 1 ≤ p(Φ) ≤ q(Φ) ≤ ∞, and if q(Φ) < ∞, then
Φ(t)
Φ(t)
≤ Φ0 (t) ≤ q(Φ)
for all t > 0.
t
t
Integrating these inequalities, one sees that (2.11) is equivalent to
p(Φ)
Φ(t)
Φ(t)
is increasing, and q(Φ) is decreasing for all t > 0.
tp(Φ)
t
(2.11)
(2.12)
EJDE-2008/109
ENERGY DECAY FOR SOLUTIONS
7
We say that two N-function Φ and Φ1 are equivalent if there exists two constants
C1 and C2 such that
C1 Φ1 (t) ≤ Φ(t) ≤ C2 Φ1 (t)
for all t ≥ 0.
We denote by i(Φ) and I(φ) the reciprocal Boyd indices of Φ. Sometimes, i(Φ)
is called the lower index and I(Φ) the upper index of Φ. We have the following
characterisations:
i(Φ) = sup p(Φ1 )
and I(Φ) = inf q(Φ1 ).
Φ1 ∼Φ
Φ1 ∼Φ
Φ
Φ
Let Φ be an N-function satisfying I(Φ) < ∞. The Orlitz space
R L = L (Ω) is the
space of all measurable functions f defined on Ω such that Ω Φ(|f |) dx < +∞. It
is endowed with the norm
Z
|f | dx ≤ 1
kf kΦ = inf λ > 0;
Φ
λ
Ω
For every f ∈ LΦ and every g ∈ LΦ̃ the following Hölder type inequality holds:
Z
|f g| dx ≤ 2kf kΦ kgkΦ̃ .
Ω
1,Φ
1,Φ
Let us denote by W
= W (Ω) the space of all functions in LΦ such that the
distributional partial derivatives belong to LΦ , and by W01,Φ (Ω) the closure of the
test functions in this space. Such spaces are well known in the literature as OrlitzSobolev spaces (see [2]). We have Poincaré’s inequality for Orlitz-Sobolev spaces
kukΦ ≤ Ck∇x ukΦ ,
u ∈ W01,Φ (Ω),
W01,Φ (Ω).
(2.13)
W0−1,Φ̃
W0−1,Φ̃ (Ω)
so that k∇x ukΦ defines an equivalent norm in
By
=
1,Φ
we denote the dual space of W0 (Ω).
The classical Sobolev embedding theorem has been extended into Orlitz setting.
n
In the following we only need that if Φ is an N-function such that for n0 = n−1
(n > 1)
Z +∞
Φ̃(s)
ds = +∞,
(2.14)
n0 +1
s
1
then it is possible to define an optimal N-function Φ∗ such that the embedding
W01,Φ (Ω) ,→ LΦ
∗
(2.15)
∗
holds; optimality means that LΦ is the smallest Orlitz space for which (2.15) holds.
If the integral in (2.14) is finite or n = 1, then
W01,Φ (Ω) ,→ L∞ (Ω).
We assume that Φ is N-function satisfying
2n
, +∞ ∩]1, +∞[
i(Φ) ∈
n+2
and I(Φ) < +∞.
(2.16)
(2.17)
and where Φ∗ is the N-function from (2.15), and identifying L2 with its dual, we
have
W01,Φ ,→ L2 ,→ W −1,Φ̃ .
8
A. BENAISSA, A. GUESMIA
EJDE-2008/109
These dense inclusions also hold by the Sobolev embedding (2.16) in the case that
the integral in (2.14) is finite. Set
Z 2
1 s
Φ(s) =
φ(t) dt.
2 0
Theorem 2.1. Assume that (u0 , u1 ) ∈ W01,Φ (Ω) × L2 (Ω). Then problem (1.1)
admits a unique strong solution on Ω × [0, ∞[ in the class
C([0, ∞[, W01,Φ (Ω)) ∩ C 1 ([0, ∞[, L2 (Ω))
Proof. The theory of maximal monotone operators associated with subdifferentials
(see [21], [22], [10] and [4]) imply that, for every (u0 , u1 ) ∈ W01,φ (Ω) × L2 (Ω), the
problem (1.1) admits a unique global strong solution.
Our main result on stabilization is the following.
Theorem 2.2. Assume that (H1)-(H3) hold. Let σ̃(t) =
exist ω, 0 > 0 such that the energy E satisfies
E(t) ≤ ϕ1 ψ −1 h(σ̃(t)) + ψ(ϕ1 −1 (E(0))) ,
Rt
σ(τ )dτ . Then there
0
∀t ≥ 0
(2.18)
where
Z
1
E(0)
for t > 0; h(t) = 0 for 0 ≤ t ≤
;
ωϕ(E(0))
t
ψ −1 t + ψ(E(0))
, for t > 0;
h−1 (t) = t +
ωϕ ψ −1 t + ψ(E(0))


φ̃(s)
if r = 1 and G is linear on [0, 1 ],


1
(φ̃(s))1+ r
ϕ(s) =
s1/r
if r 6= 1 and G is linear on [0, 1 ],


 20 s2 G0 20 s2
if G0 (0) = 0;
φ̃−1 (s)
φ̃−1 (s)
(
φ̃(s) if G is linear on [0, 1 ],
ϕ1 (s) =
s
if G0 (0) = 0.
ψ(t) =
1
dτ
ωϕ(τ )
Remark 2.4. 1. Under the hypotheses of Theorem 2.2 and thanks to Remark 2.3,
we have strong stability of (1.1); that is,
lim E(t) = 0.
(2.19)
t→+∞
2. Thanks to (H2) and (H3), the function ϕ (defined in Theorem 2.2) is of class
C 1 (R+ ) and satisfies the same hypotheses as the function Ψ in Lemma 2.2. Then
we can apply Lemma 2.2 for Ψ = ωϕ.
3. We obtain same results for the problem
u00 − ∆φ u − σ(t) div(ψ(|∇x u0 |2 )∇x u0 ) = 0
such that
c1 ≤ ψ(s2 ) ≤ c2
2
2
|s| + ψ (s) ≤ G
−1
if |s| ≥ 1 ,
(sg(s))
if |s| ≤ 1 .
(2.20)
(2.21)
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ENERGY DECAY FOR SOLUTIONS
9
Using (2.18), we give several significant examples of growth at the origine of g
and φ, and the corresponding decay estimates. Some of these examples were given
(in less general form) by Liu and Zuazua [28] and Alabau-Boussouira [5] for the
wave equation, and Eller, Lagnese and Nicaise [15] for Maxwell system.
Polynomial
growth for φ and polynomial growth for g. If
or logarithmic
q
p+1
φ(t) = ctm ln(t + 1) (degeneracy of finite order) and G(t) = c0 t 2 for c, c0 > 0
1
l(p+1)−2
(that is c01 |s| 2
≤ |g(s)| ≤ c02 |s| p on [−1 , 1 ] for some c01 , c02 > 0), m ≥ 0,
q ≥ −m and p ≥ 1 (note that c3 sm+q+1 ≤ φ̃(s) ≤ c03 sm+q+1 for c3 , c03 > 0 when s
is near 0), then there exists α > 0 such that for all t ≥ 0

−ωσ̃(t)

if (m + q, p) = (0, 1),

αe

r(m+q+1)
− (r+1)(m+q)

if m + q > 0, p = 1, r > 1,
E(t) ≤ α σ̃(t) + 1

2(m+q+1)

− 2p(m+q)+p−1


α σ̃(t) + 1
otherwise.
Moreover, we can obtain more precise rate of decay, in the case φ(s) = sm with
m ≥ 0 and
c1 |s|p ≤ |g(s)| ≤ c2 |s|θ if |s| ≤ 1
where p1 ≤ θ ≤ p. We have the following estimates: If l ≥ p + 1, then for all t ≥ 0,
(
αe−ωσ̃(t)
if 2m + 1 ≤ θ,
E(t) ≤
2θ(m+1)
− 2m+1−θ
α(σ̃(t) + 1)
if 2m + 1 > θ.
If l < p + 1, then for all t ≥ 0,
(
2θ(m+1)
α(σ̃(t) + 1)− 2m+1−θ
E(t) ≤
l
α(σ̃(t) + 1)− p+1−l
if l ≥
if l <
2θ(m+1)(p+1)
(θ+1)(2m+1) ,
2θ(m+1)(p+1)
(θ+1)(2m+1) .
Polynomial
growth for φ and exponential growth for g. If
or logarithmic
q
m
φ(t) = ct ln(t + 1) (degeneracy of finite order) and H(|s|) ≤ |g(s)| ≤ H −1 (|s|)
1 −s−γ
, m ≥ 0, q ≥ −m and c, γ > 0 (note that
se
γ
γ
−
m+q+1
0 m+q+1
c3 s
≤ φ̃(s) ≤ c3 s
and G(s) = e−2 2 s 2 for c3 , c03 > 0 when s is near 0,
γ
0 −
and ψ(s) ≤ c01 ec2 s 2 on ]0, 1] for c01 , c02 > 0), then there exist α, β > 0 such that
on [−1 , 1 ] where H(s) =
−2(m+q+1)
γ(2(m+q)+1)
E(t) ≤ β ln αh(σ̃(t)) + 2
,
∀t ≥ 0.
Polynomial or logarithmic
growth
q for φ and faster than exponential
m
growth for g. If φ(t) = ct ln(t + 1) (degeneracy of finite order) and H(|s|) ≤
|g(s)| ≤ H −1 (|s|) on [−1 , 1 ] where H(s) = 1s Hn (s), m ≥ 0, q ≥ −m, c, γ > 0 and
H1 (s) = e−s
−γ
−H
and Hn (s) = e
1
n−1 (s)
, n = 2, 3, · · · ,
then (as in the example 2) there exist α, β, δ > 0 such that
−2(m+q+1)
γ(2(m+q)+1)
E(t) ≤ β H̄n (h(σ̃(t)))
, ∀t ≥ 0.
where
H̄1 (t) = ln(αt + δ)
and H̄n (t) = ln(H̄n−1 (t)), n = 2, 3, · · · .
10
A. BENAISSA, A. GUESMIA
EJDE-2008/109
Polynomial or logarithmic growth for
φ andqbetween polynomial and
m
exponential growth for g. If φ(t) = ct ln(t + 1)
(degeneracy of finite order)
and H(|s|) ≤ |g(s)| ≤ H −1 (|s|) on [−1 , 1 ] where H(s) =
m ≥ 0, q ≥ −m, c > 0 and
1 −(Hn (s))γ
,
se
γ > 1,
H1 (s) = − ln s and Hn (s) = ln(Hn−1 (s)), n = 2, 3, · · ·
(then G(s) = e−(
−1
2
ln
s γ
2)
when s is near 0), then there exist α, β, δ > 0 such that
2(m+q+1)
E(t) ≤ βe− 2(m+q)+1 H̄n (h(σ̃(t))) ,
∀t ≥ 0.
where
1
H̄1 (t) = ln(αt + δ) γ
and H̄n (t) = eH̄n−1 (t) , n = 2, 3, · · · .
Exponential growth for φ (degeneracy of infinite order) and linear growth
−γ
−γ
−γ
for g. If φ(t) = e−t , γ > 0, (note that c01 tγ+1 e−t ≤ φ̃(t) ≤ c02 tγ+1 e−t for
c01 , c02 > 0 when s is near 0) then there exist α, β > 0 such that for all t ≥ 0

− γ+1
−r
γ

β(h(σ̃(t))) r+1
ln(αh(σ̃(t)) + 2)
, if r > 1,
E(t) ≤
γ+1
−
γ

β(h(σ̃(t)))−1 ln(αh(σ̃(t)) + 2)
, if r = 1.
Faster than exponential growth for φ (degeneracy of infinite order) and
linear growth for g. If φ(t) = e−e
−γ
t
c02 tγ+1 e−e
−t−γ
c01 ,
t−γ
, γ > 0, (note that c01 sγ+1 e−e
t−γ
−γ
e−t
≤
c02
φ̃(t) ≤
e
for
> 0 when t is near 0) then there exist α, β > 0
such that for all t ≥ 0,

−1 − γ+1
−r
γ

r
β(h(σ̃(t))) r+1
, if r > 1,
ln(αh(σ̃(t)) + 2)
ln r+1
ln(αh(σ̃(t)) + 3)
E(t) ≤
γ+1
−1
−
γ

β(h(σ̃(t)))−1 ln(αh(σ̃(t)) + 2)
,
if r = 1.
ln ln(αh(σ̃(t)) + 3)
Faster than polynomials, less than exponential growth for φ (degeneracy
γ
of infinite order) and linear growth for g. If φ(t) = e−(− ln t) , γ ≥ 1, (note
0 −(− ln t)γ
1−γ
0 −(− ln t)γ
1−γ
0
that c1 te
(− ln t)
≤ φ̃(t) ≤ c2 te
(− ln t)
for c1 , c02 > 0 when t
is near 0) then there exist α, β > 0 such for all t ≥ 0, that

− γ−1
1 −r
r
γ
γ

β(h(σ̃(t))) r+1
e−( r+1 ln(αh(σ̃(t))+2)) ln(αh(σ̃(t)) + 2)
, if r > 1,
E(t) ≤
γ−1
1 −
γ

γ

β(h(σ̃(t)))−1 e−(ln(αh(σ̃(t))+2)) ln(αh(σ̃(t)) + 2)
,
if r = 1.
Slow than polynomials for φ (slow degeneracy) and linear growth for
g. If φ(t) = | ln t|−γ near of 0 where γ > 0, (note that c01 s(− ln s)−γ ≤ φ̃(s) ≤
c02 s(− ln s)−γ for c01 , c02 > 0 when s is near 0) then there exists α > 0 such that for
all t ≥ 0,

1
γ
−
γ(1+ 1 )+1

r
γ(1+ 1 )+1 −(h(σ̃(t)))
r
α(h(σ̃(t)))
e
, if r > 1,
E(t) ≤
1
γ

− γ+1
−(h(σ̃(t))) γ+1
α(h(σ̃(t)))
e
,
if r = 1.
EJDE-2008/109
ENERGY DECAY FOR SOLUTIONS
11
3. Proof of Theorem 2.2
For the rest of this article, we denote by c various positive constants which may
be different at different occurrences.
If E(t0 ) = 0 for some t0 ≥ 0, then E(t) = 0 for all t ≥ t0 , and then we have
nothing to prove in this case. So we assume that E(t) > 0 for all t ≥ 0 without loss
of generality.
We multiply the first equation of (1.1) by σ(t) ϕ̃(E)
E u where ϕ̃ : R+ → R+ is
convex, increasing and of class C 1 (]0, +∞[) such that ϕ̃(0) = 0, and we integrate
by parts, we have, for all 0 ≤ S ≤ T ,
T
Z ϕ̃(E)
u (|u0 |l−2 u0 )0 − ∆φ u + σ(t)g(u0 ) dx dt
E
S
Z Ω
iT
h
ϕ̃(E)
uu0 |u0 |l−2 dx
= σ(t)
E
S
Ω
Z TZ
ϕ̃(E) 0
ϕ̃(E) 0 ϕ̃(E)
−
u + σ(t)
u + σ(t)(
) u dx dt
u0 |u0 |l−2 σ 0 (t)
E
E
E
S
Ω
Z T
Z T
Z X
Z
n
ϕ̃(E)
ϕ̃(E)
φ(|∂xi u|2 )|∂xi u|2 dx dt +
σ 2 (t)
+
σ(t)
ug(u0 ) dx dt.
E
E
S
S
Ω i=1
Ω
Z
σ(t)
0=
Using Lemma 2.1 for p = 2 and q = l and the definition of E, we have (note also
that φ̃ is convex and defines a bijection from R+ to R+ )
Z
Z
1/l Z
l−1
l
|u|l dx
|u0 |l dx
uu0 |u0 |l−2 dx ≤
Ω
Ω
Ω
1/2 l−1
Z
E l
|∇u|2 dx
≤c
Ω
≤ cE
l−1
l
n
X
−1
φ̃
n
Z X
Ω i=1
i=1
≤ cE
l−1
l
(3.1)
1/2
φ̃(|∂xi u| )dx
2
q
φ̃−1 (E).
In the other hand, we have sφ(s) ≥ 2φ̃(s), l ≥ 2, φ̃−1 is non decreasing and ϕ̃ is
l−1
convex, increasing and of class C 1 (]0, +∞[) such that ϕ̃(0) = 0 (then s 7→ s l ,
s 7→ φ̃−1 (s) and s 7→ ϕ̃(s)
s are non decreasing). Then we deduce that
Z
T
σ(t)ϕ̃(E(t))dt
S
Z T
Z
q
ϕ̃(E(S))
ϕ̃(E)
−1
≤ cE (S) φ̃ (E(S))
+c
σ(t)
(|u0 |l + |ug(u0 )|) dx dt.
E(S)
E
S
Ω
(3.2)
To estimate the last integral above, we distinguish three cases:
Case 1: r = 1 and G is linear on [0, 1 ]: We choose ϕ̃(s) = s. For all t ≥ 0, we
denote
l−1
l
0
Ω+
t = {x ∈ Ω : uu ≥ 0},
0
Ω−
t = {x ∈ Ω : uu ≤ 0}.
12
A. BENAISSA, A. GUESMIA
EJDE-2008/109
We have C1 |s| ≤ |g(s)| ≤ C2 |s| for all s ∈ R (because 2 ≤ l ≤ r + 1 = 2), and then
(using (2.5))
Z T
Z
Z T
Z
ϕ̃(E)
σ(t)
|u0 |l dx dt ≤ c
σ(t)
u0 g(u0 ) dx dt ≤ cE(S)
E
S
Ω
S
Ω
and (note that σ 0 ≤ 0)
Z
Z T
ϕ̃(E)
σ(t)
|ug(u0 )| dx dt
E
S
Ω
Z T
Z
≤c
σ(t)
|uu0 | dx dt
S
Ω
Z
h
≤ c σ(t)
Z
2
Z
iT
u dx + c
u dx − σ(t)
Ω−
t
Ω+
t
≤ cφ̃−1 (E(S)) + cφ̃−1 (E(S))
T
2
S
Z
Z
σ (t) −
0
Z
2
Ω−
t
Ω+
t
S
u2 dx dt
u dx +
T
(−σ 0 (t))dt ≤ cφ̃−1 (E(S)).
S
Then
Z
T
σ(t)E(t)dt ≤ c 1 +
S
E(S)
+
φ̃−1 (E(S))
s
E(S) −1
φ̃ (E(S)).
φ̃−1 (E(S))
Using the fact that φ̃ is convex, increasing and φ̃(0) = 0 (then s 7→
decreasing) we obtain from (3.2) that
Z +∞
σ(t)E(t)dt ≤ cφ̃−1 (E(S)).
s
φ̃−1 (s)
is non
S
−1
Let Ẽ = φ̃
◦ E ◦ σ̃
−1
(note that σ̃ is a bijection from R+ to R+ ). Then, for ω > 0,
Z +∞
1
φ̃(Ẽ(t))dt ≤ Ẽ(S).
ω
S
Using Lemma 2.2 for Ẽ in the particular case Ψ(s) = ω φ̃(s) and λ = 0, we deduce
from (2.6) that
Ẽ(t) ≤ ψ −1 h(t) + ψ(φ̃−1 (E(0))) , ∀t ≥ 0.
Then, using the definition of Ẽ, we obtain (2.18) in the case where r = 1 and G is
linear on [0, 1 ].
1
Case 2: r > 1 and G is linear on [0, 1 ]. We choose ϕ̃(s) =
t ≥ 0, we denote
Ω1t = {x ∈ Ω : |u0 | ≥ 1 },
s1+ r
.
(φ̃−1 (s))1/r
For all
Ω2t = {x ∈ Ω : |u0 | ≤ 1 }.
Using Young’s and Lemma 2.1 (for q = r + 1 and p = 2) and condition (2.2) we
have, for all > 0 (using also the fact that s 7→ ϕ̃(s)
s is non decreasing),
Z T
Z
ϕ̃(E)
σ(t)
(|u0 |l + |ug(u0 )|) dx dt
1
E
S
Ωt
Z T
Z
1 Z
r
r+1
r+1
r+1
ϕ̃(E) ≤
σ(t)
|u|r+1 dx
|g(u0 )| r dx
dt
E
S
Ω1t
Ω1t
EJDE-2008/109
ENERGY DECAY FOR SOLUTIONS
T
Z
Z
σ(t)
+c
T
≤
σ(t)
S
Z
T
≤
σ(t)
S
Z
u0 g(u0 ) dx dt
Ω1t
S
Z
13
ϕ̃r+1 (E)
E r+1
Z
|∇u|2 dx dt + c
T
Z
Ω1t
Z
σ(t)
ϕ̃r+1 (E)φ̃−1 (E)
dt + c
E r+1
Z
T
1
(|g(u0 )|1+ r + u0 g(u0 )) dx dt
Ω1t
S
Z
u0 g(u0 ) dx dt
σ(t)
Ω1t
S
T
≤
σ(t)ϕ̃(E)dt + cE(S).
S
Choosing small enough, we obtain from (3.2) that
Z T
q
l−1
ϕ̃(E(S)) σ(t)ϕ̃(E(t))dt ≤ c E(S) + E l (S) φ̃−1 (E(S))
E(S)
S
Z T
Z
ϕ̃(E)
+c
(|u0 |l + |ug(u0 )|) dx dt.
σ(t)
2
E
Ωt
S
On the other hand, we have C1 |s|l−1 ≤ |g(s)| ≤ C2 |s| for all s ∈ [−1 , 1 ] and then
(note that s 7→ ϕ̃(s)
s is non decreasing and follow the proof in the case 1)
Z T
Z
Z
Z T
ϕ̃(E)
(|u0 |l + |ug(u0 )|) dx dt ≤ c
σ(t) (u0 g(u0 ) + |uu0 |) dx dt
c
σ(t)
E
Ω2t
S
Ω
S
≤ c E(S) + φ̃−1 (E(S)) .
Then from (3.2) we deduce that
Z T
σ(t)ϕ̃(E(t))dt
S
≤c 1+
l−2
E(S)
ϕ̃(E(S)))
+ E 2l (S)
E(S))
φ̃−1 (E(S))
s
E(S) −1
φ̃ (E(S)).
φ̃−1 (E(S))
l−2
Finally (note that s →
7 s 2l , s 7→ ϕ̃(s)
and s 7→ φ̃−1s(s) are non decreasing), we
s
obtain
Z +∞
σ(t)ϕ̃(E(t))dt ≤ cφ̃−1 (E(S)).
S
−1
Let Ẽ = φ̃
◦ E ◦ σ̃
−1
. Then we deduce from this inequality that, for ω > 0,
Z +∞ 1
ϕ̃ φ̃(Ẽ(t)) dt ≤ Ẽ(S).
ω
S
1+ 1
r
and
Using Lemma 2.2 for Ẽ in the particular case Ψ(s) = ω ϕ̃(φ̃(s)) = ω φ̃(s)
s1/r
λ = 0, we deduce from (2.6) our estimate (2.18). 2
20 s2
0
0s
Case 3: G0 (0) = 0. We choose ϕ̃(s) = φ̃2
G
. Using the fact that
−1 (s)
φ̃−1 (s)
s 7→ G0 (s), s 7→
Z
T
σ(t)
S
ϕ̃(E)
E
s2
φ̃−1 (s)
Z
Ω1t
and s 7→
ϕ̃r−1 (s)
sr−1
are non decreasing, we obtain (as in case 2)
(|u0 |l + |ug(u0 )|) dx dt ≤ Z
T
σ(t)
S
ϕ̃r+1 (E)φ̃−1 (E)
dt + cE(S)
E r+1
14
A. BENAISSA, A. GUESMIA
Z
EJDE-2008/109
T
≤
σ(t)
S
Z
ϕ̃2 (E)φ̃−1 (E)
dt + cE(S)
E2
T
= 20
σ(t)ϕ̃(E)G0 (
S
T
20 E 2
)dt + cE(S)
φ̃−1 (E)
Z
≤ 20
σ(t)ϕ̃(E)dt + cE(S)
S
Choosing small enough, we obtain from (3.2) that
Z T
q
l−1
ϕ̃(E(S)) σ(t)ϕ̃(E(t))dt ≤ c E(S) + E l (S) φ̃−1 (E(S))
E(S)
S
Z T
Z
ϕ̃(E)
σ(t)
(|u0 |l + |ug(u0 )|) dx dt.
+c
E
S
Ω2t
Let now G1 (s) = G(s2 ) (note that G1 satisfies the same hypotheses as G) and let G∗
and G∗1 denote the dual functions of the convex functions G and G1 respectively in
the sense of Young (see Arnold [6, page 64], for the definition). Because G is convex
and G is not linear near 0, then there exists 01 > 0 such that G00 > 0 on ]0, 01 ].
Since, because G0 (0) = 0 and (2) − (3) are still satisfied for 00 = min{1 , 01 } instead
of 1 , we can assume, without lose of generality, that G0 defines a bijection from
R+ to R+ . Then G∗ and G∗1 are the Legendre transform of G and G1 respectively,
which are given by (see Arnold [6, pp. 61-62])
G∗ (s) = s(G0 )−1 (s) − G[(G0 )−1 (s)],
G∗1 (s) = s(G01 )−1 (s) − G1 [(G01 )−1 (s)].
Thanks to our choice
ϕ̃(s) =
s 20 s2 0 20 s2 s
0
G
=q
G01 q
,
φ̃−1 (s)
φ̃−1 (s)
φ̃−1 (s)
φ̃−1 (s)
we have
(G0 )−1 ( ϕ̃(s)
s )
,
s
s
q
ϕ̃(s)
s 0 s
0
G∗1
φ̃−1 (s) ≤ q
= 0 ϕ̃(s).
G01 q
s
φ̃−1 (s)
φ̃−1 (s)
G∗
ϕ̃(s) ≤ ϕ̃(s)
Then, by Poincaré’s inequality, Young’s inequality (see Arnold [6, p. 64]) and
Jensen’s inequality (see Rudin [35]), we deduce (|Ω| is the measure of Ω in Rn )
Z T
Z
ϕ̃(E)
σ(t)
(|u0 |l + |ug(u0 )|) dx dt
E
S
Ω2t
Z
Z T
ϕ̃(E) G−1 (u0 g(u0 ))dx
≤
σ(t)
E
Ω2t
S
Z
1/2 Z
1/2 +
|∇u|2 dx
G−1 (u0 g(u0 ))dx
dt
Ω2t
Ω
s
1 Z
ϕ̃(E)
≤
σ(t)
φ̃−1 (E) |Ω|G−1
u0 g(u0 )dx
E
|Ω| Ω
S
1 Z
+ |Ω|G−1
u0 g(u0 )dx dt
|Ω| Ω
Z
T
q
EJDE-2008/109
Z
T
≤c
S
T
Z
≤c
S
ENERGY DECAY FOR SOLUTIONS
15
Z
Z T
ϕ̃(E) q
ϕ̃(E) σ(t) G∗1
σ(t)
u0 g(u0 ) dx dt
φ̃−1 (E) + G∗ (
) dt + c
E
E
S
Ω
(G0 )−1 ( ϕ(E)
)
E
σ(t) 0 +
ϕ̃(E)dt + cE(S).
E
Using the fact that s 7→ (G0 )−1 (s) and s 7→
s
φ̃−1 (s)
are non decreasing, we deduce
φ̃−1 (E(0))
2E(0) ,
that, for 0 < 0 ≤
Z
Z T
Z T
ϕ̃(E)
0 l
0
(|u | + |ug(u )|) dx dt ≤ c0
σ(t)ϕ̃(E)dt + cE(S).
σ(t)
E
S
S
Ω2t
Then, choosing 0 small enough, we deduce from (3.2) that
s
Z T
l−2
φ̃−1 (E(S)) ϕ̃(E(S)) σ(t)ϕ̃(E(t))dt ≤ c 1 + E 2l (S)
E(S)
E(S)
E(S)
S
q
2 2 q
l−2
0 s
φ̃−1 (s) ϕ̃(s)
s
0
2l
and s 7→
are
Finally (note that s 7→ s
=
2
G
0
−1
s
s
φ̃ (s)
φ̃−1 (s)
non decreasing), we obtain
Z
+∞
σ(t)ϕ̃(E(t))dt ≤ cE(S).
S
Let Ẽ = E ◦ σ̃ −1 . Then we deduce from this inequality that, for ω > 0,
Z +∞
1
ϕ̃(Ẽ(t))dt ≤ Ẽ(S).
ω
S
Using Lemma 2.2 for Ẽ in the particular case Ψ(s) = ω ϕ̃(s) and λ = 0, we deduce
from (2.6) our estimate (2.18). This is completes the proof.
4. An application to wave equations of φ-Laplacian with source term
In this section we shall propose some applications of Theorem 2.2.
Example 1. Let us consider the Cauchy problem for the wave equation, in Ω×R+ ,
(|u0 |l−2 u0 )0 − e−λ(x)
n
X
∂xi eλ(x) φ(|∂xi u|2 )∂xi u + σ(t)g(u0 ) + f (u) = 0,
i=1
(4.1)
u = 0 on Γ × R+
u(x, 0) = u0 (x),
u0 (x, 0) = u1 (x)
on Ω.
We define the energy associated to the solution as
Z
Z
Z
n
X
l−1
φ̃(|∂xi u|2 )dx +
eλ(x) F (u) dx
eλ(x) |u0 |l dx +
eλ(x)
E(t) =
l
Ω
Ω
Ω
i=1
Z
l−1
=
eλ(x) |u0 |l dx + J(u)
l
Ω
Ru
where F (u) = 0 f (s) ds. For the function f ∈ C(R) we assume that there exists
an N-function ψ satisfying i(ψ), I(ψ) ∈]1, +∞[ and
|f (t)| ≤ ψ 0 (|t|)
for every t ∈ R.
(4.2)
16
A. BENAISSA, A. GUESMIA
EJDE-2008/109
If condition (2.14) is satisfied (so that Φ∗ exists), then we assume in addition that
ψ(t) ≤ Φ∗ (Ct)
for all large t > 0.
(4.3)
Thus, we can verify that for all u ∈ W01,Φ with norm small enough that
Z
Z
1
Φ(|∇x u|) dx ≤ |J(u)| ≤ C
Φ(|∇x u|) dx.
C Ω
Ω
(4.4)
So, we obtain same results as in the theorem 2.1.
Proof of the example 1. We prove only the second part. The proof of the first
part is a direct application of the theorem 2.2. We make an additional assumption
on g(v):
(H3’) Suppose that there exist ci > 0; i = 1, 2, 3, 4 such that
c1 |v|p ≤ |g(v)| ≤ c2 |v|θ if |v| ≤ 1,
(4.5)
c3 |v|s ≤ |g(v)| ≤ c4 |v|r for all |v| ≥ 1,
(4.6)
n+2
.
where 1 ≤ m ≤ r, θ ≤ p, l − 1 ≤ s ≤ r ≤ n−2
Proof of the energy decay. We denote by c various positive constants which
may be different at different occurrences. We multiply the first equation of (1.1) by
E q σ̃ 0 u, where σ̃ is a function satisfying all the hypotheses of lemma 2.2, we obtain
Z T
Z
q 0
0=
E σ̃
u((|u0 |l−2 u0 )t − ∆φ u + σ(t)g(u0 )) dx dt
S
Ω
Z
= E q σ̃ 0
0
0 l−2
uu |u |
Ω
T
Z
E q σ̃ 0
−
S
Z T
+
Z
T
dx S −
|u0 |l dx dt +
S
Z
0
(qE E
S
Z T
Ω
E q σ̃ 0
T
Z
q−1 0
Z
q 00
uu0 |u0 |l−2 dx dt
σ̃ + E σ̃ )
Ω
2(γ+1)
E q σ̃ 0 k∇uk2
dx dt
S
σ(t)ug(u0 ) dx dt.
Ω
we deduce that
Z T
2(m + 1)
E q+1 σ̃ 0 dt
S
≤ − E q σ̃ 0
Z
Ω
T
uu0 |u0 |l−2 dx S +
2(l − 1)(m + 1) + l
+
l
Z
Z
T
(qE 0 E q−1 σ̃ 0 + E q σ̃ 00 )
Z
S
T
q 0
uu0 |u0 |l−2 dx dt
Ω
Z
0 l
T
|u | dx dt −
E σ̃
S
Z
Ω
q 0
Z
E σ̃
S
σ(t)ug(u0 ) dx dt.
Ω
(4.7)
Since E is non-increasing, σ̃ 0 is a bounded nonnegative function on R+ (and we
denote by µ its maximum) and using Hölder inequality, we have
Z
l−1
1
E(t)q σ̃ 0
uu0 |u0 |l−2 dx dt ≤ cµE(S)q+ l + 2(m+1)
∀t ≥ S.
Ω
and
Z
T
(qE 0 E q−1 σ̃ 0 + E q σ̃ 00 )
S
Z
uu0 |u0 |l−2 dx dt, dx dt
Ω
Z
T
≤ cµ
0
1
q− 1l + 2(m+1)
−E (t)E(t)
S
Z
T
dt + c
S
E(t)q+
l−1
1
l + 2(m+1)
(−σ̃ 00 (t)) dt
EJDE-2008/109
ENERGY DECAY FOR SOLUTIONS
≤ cµE(S)q+
l−1
1
l + 2(m+1)
17
.
Using these estimates we conclude from the above inequality that
Z T
2(m + 1)
E(t)1+q σ̃ 0 (t) dt
S
≤ cE(S)q+
l−1
1
l + 2(m+1)
T
Z
q 0
−
Z
E σ̃
S
+
2(l − 1)(m + 1) + l
l
Z
T
E q σ̃ 0
Z
S
|u0 |l dx dt
Ω
0
σ(t)ug(u ) dx dt
Ω
Z
Z
2(l − 1)(m + 1) + l T q 0
E σ̃
+
|u0 |l dx dt
l
S
Ω
Z T
Z
σ(t)ug(u0 ) dx dt −
E q σ̃ 0
σ(t)ug(u0 ) dx dt.
1
q+ l−1
l + 2(m+1)
≤ cE(S)
T
Z
E q σ̃ 0
−
Z
|u0 |≤1
S
|u0 |>1
S
Now, we estimate each terms on the right-hand side of the above inequality, to
apply Lemma 2.2. Using Hölder inequality, we obtain
Z T
Z
E q σ̃ 0
|u0 |l dx dt
S
Ω
Z T
Z l
(p+1)
1 0
1 0
0
0
q 0
u ρ(t, u ) dx dt + C
u ρ(t, u0 )
≤C
E σ̃
dx dt
E σ̃
S
Ω σ(t)
S
Ω σ(t)
Z
Z T
T
l
σ̃ 0
σ̃ 0
(−E 0 ) p+1 dt
(−E 0 ) dt + C 0 (Ω)
Eq l
≤C
Eq
σ(t)
S
S
σ p+1 (t)
Z T
l
0
p+1−l
l
σ̃ p+1
≤ CE q+1 (S) + C 0 (Ω)
(−E 0 ) p+1 dt.
E q σ̃ 0 p+1
σ(t)
S
Z
T
q 0
Z
Now, fix an arbitrarily small ε > 0 (to be chosen later), by applying Young’s
inequality, we obtain
Z T
Z
E q σ̃ 0
|u0 |l dx dt
S
Ω
≤ CE q+1 (S) + C 0 (Ω)
(p+1)
p + 1 − l (p+1−l)
ε
p+1
Z
T
p+1
E q p+1−l σ̃ 0 dt + C 0 (Ω)
S
1
l
E(S).
(p+1)
p+1ε l
(4.8)
If l ≥ p + 1, from (4.5) and (4.6) we obtain easily that
Z T
Z
E q σ̃ 0
|u0 |l dx dt ≤ CE q+1 (S).
S
Thanks to Young’s inequality,
Z T
Z
Z
E q σ̃ 0
σ(t)ug(u0 ) dx dt
|u0 |≤1
S
Z
T
E q σ̃ 0
×
T
E q σ̃ 0
Z
T
≤c
S
Z
σ(t)k∇x uk2m+2
σ(t)kuk2
Z
|u0 |≤1
S
Z
Z
|u0 |≤1
S
(4.9)
Ω
|u0 |≤1
2θ
(u0 g(u0 )) θ+1
1/2
dx
dt
|u0 |≤1
Z
1
1
E q+ 2(m+1) σ̃ 0 σ (θ+1) (t)
|u0 |<1
θ/(θ+1)
dt
σu0 g(u0 ) dx
12
|g(u0 )|2 dx dt
18
A. BENAISSA, A. GUESMIA
T
Z
1
1
EJDE-2008/109
θ
E q+ 2(m+1) σ̃ 0 σ (θ+1) (t)(−E 0 ) θ+1 dt.
≤c
S
Applying Young’s inequality, we obtain
Z
Z T
Z
E q σ̃ 0
σ(t)ug(u0 ) dx dt ≤ C(Ω)εθ+1
2
|u0 |≤1
S
T
θ+1
1
1
E q+ 2(m+1) σ̃ 0 σ (θ+1) (t)
dt
S
+ C(Ω)
1
θ+1
θ
ε2
Z
T
(−E 0 ) dt
S
(4.10)
and
Z
T
q 0
Z
σ(t)ug(u0 ) dx dt
E σ̃
|u0 |≥1
S
≤ C(Ω)
1
(r+1)
ε
(r + 1) 1
Z
T
E (q+ 2(γ+1) )(r+1) σ̃ 0 σ(t)r+1 dt +
1
S
C(Ω)r 1
E(S),
(r + 1) ε r+1
r
1
(4.11)
The case l ≥ p + 1. We consider two subcases
• θ ≥ 2m + 1. Choose q = 0 and we have
1
2(γ+1)
(θ + 1) = 1 + α, where
θ−(2m+1)
2(m+1)
α=
≥ 0.
• θ < 2m + 1. Choose q such that q +
2m−θ+1
q = 2θ(m+1)
.
1
2(m+1)
(θ + 1) = q + 1. Thus,
The case l < p + 1.
• 2m + 1 > θ If l ≥
q + 1. Thus, q =
α=
2θ(m+1)(p+1)
1
(θ+1)(2m+1) , we choose q such that q + 2(m+1) (θ + 1)
p+1
2m−θ+1
2θ(m+1) and q p+1−l = q + 1 + α with
=
l(2m + 1)(θ + 1) − 2θ(m + 1)(p + 1)
≥ 0.
2‘θ(m + 1)(p + 1 − l)
p+1
If l < 2θ(m+1)(p+1)
, we choose q such that q p+1−l
= q + 1. Thus q =
(θ+1)(2m+1)
1
and q + 2(m+1) (θ + 1) = q + 1 + α, where
2θ(m + 1)(p + 1) − l(2m + 1)(θ + 1)
> 0.
2l(m + 1)
p+1
• 2m + 1 ≤ θ, we choose q such that q p+1−l
= q + 1, thus q =
1
(q + 2(γ+1)
)(m + 1) = q + 1 + α with
p+1−l
l
α=
α=m
p+1−l
l
and
m + 1 − l m − (2γ + 1)
+
> 0.
l
2(γ + 1)
We may thus complete the proof by applying Lemma 2.2 with Ẽ = E ◦ σ̃ −1 instead
of E and Ψ(s) = sq .
EJDE-2008/109
ENERGY DECAY FOR SOLUTIONS
19
5. Comments and open questions
1. It is interesting to study the asymptotic behavior of solutions for Klein-Gordon
nonlocal equation
(|u0 |l−2 u0 )0 − φ1 (k∇uk22 , kuk22 )∆u + φ2 (k∇uk22 , kuk22 )u + σ(t)g(u0 ) = 0
u = 0 on Γ0 × R+
u0 (x, 0) = u1 (x)
u(x, 0) = u0 (x),
in Ω × R+
on Ω,
1
in particular when there exists a continuous function E(w, r, s) = l−1
l w + 2 L(r, s)
defined for w, r, s ≥ 0 such that for all solutions,
Z
Z t
0 l
2
2
σ(t)
u0 g(u0 ) dx ds = E(ku1 kll , k∇u0 k22 , ku0 k22 ).
E(ku kl , k∇uk2 , kuk2 ) +
0
Ω
For example when φ1 (r, s) = m(r) and φ2 (r, s) = n(s) where m and n are two
continuous positives functions. We can take
Z
Z
l−1
1 r
1 s
E(w, r, s) =
w+
m(τ ) dτ +
n(τ ) dτ.
l
2 0
2 0
R
So, E 0 (t) = −σ(t) Ω u0 g(u0 ) dx ds.
r
r2 s
As another example, when φ1 (r, s) = 1+s
2 and φ2 (r, s) = − (1+s2 )2 we can take
E(w, r, s) =
r2
l−1
w+
.
l
4(1 + s2 )
R
So, E 0 (t) = −σ(t) Ω u0 g(u0 ) dx.
As another example when φ1 (r, s) =
E(w, r, s) =
s
1+r
and φ2 (r, s) = arctan(r) we can take
1
l−1
w + arctan(r) s.
l
2
R
So, E 0 (t) = −σ(t) Ω u0 g(u0 ) dx.
2. An interesting problem is to study the asymptotic behavior of solutions for
Kirchhoff type systems,
(|v|l−2 v)0 = ψ1 (kv(t)k22 , kw(t)k22 )vx + φ1 (kv(t)k22 , kw(t)k22 )wx − ρ1 (t)g(v)
(|w|r−2 w)0 = φ2 (kv(t)k22 , kw(t)k22 )vx + ψ2 (kv(t)k22 , kw(t)k22 )wx − µ2 (t)h(w)
where φ1 , φ2 , ψ1 and ψ2 are real and continuous functions on R2+ , φ1 φ2 ≥ 0, ρ1
and µ2 are two positives and decreasing functions, g, h : R → R are non-decreasing
functions of class C(R).
If there is a C 1 function L(r, s) defined on R2+ , with
r ∂L
l ∂L
φ1 =
φ2 ,
l − 1 ∂r
r − 1 ∂r
∂L
≥ 0,
∂r
∂L
≥ 0.
∂s
We define the energy function E(t) = L(kv(t)kll , kw(t)krr ). So that
Z 2π
Z 2π
l
r
E 0 (t) = −
ρ1
g(v)v dx −
µ2
h(v)v dx ≤ 0.
l−1
r−1
0
0
3. Another interesting problem is to study the asymptotic behavior of solutions
for Kirchhoff equation with memory,
Z t
0 l−2 0 0
2
(|u | u ) − φ1 (k∇uk )∆u −
a(t − s)φ2 (k∇uk2 )∆uds = 0 in Ω × R+
0
20
A. BENAISSA, A. GUESMIA
u=0
u(x, 0) = u0 (x),
EJDE-2008/109
on Γ0 × R+
u0 (x, 0) = u1 (x)
on Ω.
In the non-degenerate case, the global existence in H 2 (Ω) ∩ H01 (Ω) was treated
by Abdelli and Benaissa [1] when φ2 ≡const and the function a is a polynomial.
The asymptotic behaviour of the energy play an important role to prove global
existence.
In the degenerate case, when φ1 ≥ φ2 ≥ 0, Dix and Torrejon [13] proved a global
existence of the (−∆)-analytic solution. It is an interesting question to study the
decay rate of the energy (the energy is a decreasing function). It is clear that the
energy decay rate depends on the order of degeneracy of φ1 , φ2 and the form of a.
4. Another interesting problem is to study global existence and asymptotic
behaviour for the following Kirchhoff equation with dissipation and source term
with initial data less regular than as in the classical case (i.e (u0 , u1 ) ∈ H 2 (Ω) ∩
H01 (Ω)),
(|u0 |l−2 u0 )0 − φ1 (k∇uk2 )∆u + σ(t)g(u0 ) + f (u) = 0
u = 0 on Γ0 × R+
u(x, 0) = u0 (x),
u0 (x, 0) = u1 (x)
in Ω × R+
on Ω.
This study makes possible to consider the case when g and f are not Lipschitz
functions (see Serrin, Todorova and Vitillaro [36] and Panizzi [34]). A convenient
κ−1
space is D((−∆)κ/2 ) ∩ D((−∆) 2 ) where κ ≥ 3/2, in particular when κ = 2,
κ−1
we find D((−∆)κ/2 ) ∩ D((−∆) 2 ) = H 2 (Ω) ∩ H01 (Ω). When 1 ≤ κ < 3/2, the
problem of local existence is open for the non-degenerate Kirchhoff equation without
dissipation and source term.
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Abbes Benaissa
Department of Mathematics, Djillali Liabes University, P. O. Box 89, Sidi Bel Abbes
22000, Algeria
E-mail address: benaissa abbes@yahoo.com
22
A. BENAISSA, A. GUESMIA
EJDE-2008/109
Aissa Guesmia
LMAM, ISGMP, Bat. A, UFR MIM, Université Paul Verlaine - Metz, Ile du Saulcy,
57045 Metz Cedex 01, France
E-mail address: guesmia@univ-metz.fr