D ifferential
E quations
& A pplications
Volume 3, Number 4 (2011), 503–525
WAVE EQUATION WITH p(x,t)–LAPLACIAN AND
DAMPING TERM: EXISTENCE AND BLOW–UP
S TANISLAV A NTONTSEV
Dedicated to Professor Jesús Ildefonso Dı́az
on the occasion of his 60th birthday
(Communicated by C. O. Alves)
Abstract. In this work, we consider the Dirichlet problem for equation
utt = div a(x,t)|∇u| p(x,t)−2 ∇u + α Δut + b(x,t)|u|σ (x,t)−2 u + f (x,t).
Under suitable conditions on the functions a , b , f , p , σ the local, global and blow up solutions
have been discussed.
1. Introduction
Let Ω ⊂ Rn be a bounded domain with Lipschitz-continuous boundary Γ and
QT = Ω × (0, T ]. We consider the following initial boundary value problem
utt = Lu + f (x,t), (x,t) ∈ QT = Ω × (0, T ),
Lu = div a(x,t) |∇u| p(x,t)−2 ∇u + α ∇ut + b(x,t) |u|σ (x,t)−2 u,
(1)
u(x, 0) = u0 (x), ut (x, 0) = u1 (x), x ∈ Ω,
(2)
u|ΓT = 0, ΓT = ∂ Ω × (0, T ).
(3)
Here α > 0 is a constant. The coefficients a(x,t), b(x,t), the exponents p(x,t), σ (x,t)
and the source term f (x,t) are given measurable functions of their arguments. We
assume that
u0 ∈ L2 (Ω) ∩W 1,p(·,0), u1 ∈ L2 (Ω), f ∈ L2 (QT ).
(4)
We discuss existence and blow up of solutions to problem (1)-(3), concentrating
our attention on difficulties caused by variable exponents p(x,t), σ (x,t). It should be
Mathematics subject classification (2010): 35B40, 35L70, 35L45.
Keywords and phrases: nonlinear wave equations, energy estimates, global existence, blow up, nonstandard growth conditions.
This work was partially supported by the Research Project PTDC/MAT/110613/2009, FCT, Portugal and by the
Research Project MTM2011-26119, MICINN, Spain.
c
, Zagreb
Paper DEA-03-32
503
504
S TANISLAV A NTONTSEV
mentioned that questions of existence, uniqueness and regularity of weak solutions for
parabolic and elliptic equations in the forms
ut = ai (x,t, u)|uxi | pi (x,t)−2 uxi + bi (x,t, u) + d(x,t, u),
xi
ai (x, u)|uxi | pi (x)−2 uxi + bi (x, u) + d(x, u) = 0,
xi
(5)
(6)
have been studied by many authors under various conditions on the data and by different methods- (see, e.g., [4, 8, 9, 10, 11, 12, 13, 32, 37], and the further references
therein). These equations are usually referred to as parabolic and elliptic equations with
nonstandard growth conditions. Also the localization (vanishing) and blow up properties of energy weak solutions for elliptic and parabolic equations of the type (5) and (6)
have been investigated sufficiently completely ( see, e.g., [4, 6, 7, 12]).
Such elliptic, parabolic and hyperbolic equations occur in the mathematical modelling of various physical phenomena, e.g., the flows of electro-rheological fluids or
fluids with temperature-dependent viscosity, nonlinear viscoelasticity, processes of filtration through a porous media and the image processing (see, e.g., [5, 6, 8, 35, 36] )
and the further references therein).
For hyperbolic equations in the form (1) with constant exponents p, σ local and
global existence and blow up have been investigated in many papers - see, e.g.,[18, 19,
20, 21, 23, 24, 25, 26, 28, 29, 30, 31, 33, 37, 40, 41, 42, 43, 44, 45, 46, 47] and the
further references therein.
The hyperbolic equations with nonstandard growth conditions, to the best of our
knowledge, were considered only in papers [14, 15, 22, 34].
In [22], the existence result was proved for the problem (1)-(3) with a ≡ 1, p ≡
p(x), b ≡ 0 . In the paper [34], the existence and blow of solutions were studied for
the problem (1)-(3) with a ≡ 1, p ≡ 2, b ≡ 0and the source term either is a power,
f (u) = b(x)u p(x) , or is nonlocal f (u) = b(x) Ω uq(y) (y,t)dy. Existence and blow-up
results for complete equation (1) were announced in [1, 2, 3].
The present paper is organized as follows. In Section 2 we introduce the function
spaces of Orlicz-Sobolev type and a brief description of their main properties. Section
3 is devoted to proof the existence of local and global energy weak solutions to problem
(1)-(3). The weak solution is obtained as the limit of the sequence of Galerkin’s approximations. First we derive estimates for an energy functional. As in last section we
will consider the blow-up for energy weak solutions with nonpositive energy functional.
Next under suitable conditions we obtain estimates for Galerkin’s approximations
in any finite time or only for small time. Further, we pass to the limit, using a standard
monotonicity argument. Finally we prove existence theorems for small and any finite
time. Section 4 is devoted to the investigation of the blow up of energy weak solutions.
We consider separately two cases. First, we take p(x,t) = p(x), σ (x,t) = σ (x), α > 0 .
Second, we take p = p(x,t), σ = σ (x,t) and α > 0 . Also we consider the case α =
0, p = p(x), σ = σ (x) and establish conditional results, assuming that the problem (1)(3) has at least one local energy solution for suitable initial data.
505
WAVE EQUATION WITH p(x,t) -L APLACIAN AND DAMPING TERM
2. The function spaces
1,p(·)
2.1. Spaces L p(·) (Ω) and W0
(Ω)
The definitions of the function spaces used throughout the paper and a brief description of their properties follow [17].
Let Ω ⊂ Rn be a bounded domain, ∂ Ω be Lipschitz-continuous, and let p(x) ∈
−
[p , p+ ] ⊂ (1, ∞) be log-continuous in Ω: ∀ x, y ∈ Ω such that |x − y| < 12 ,
|p(x) − p(y)| ω (|x − y|),
where
limτ →0+ ω (τ ) ln
(7)
1
= C < ∞.
τ
By L p(·) (Ω) we denote the space of measurable functions f (x) on Ω such that
A p(·) ( f ) =
| f (x)| p(x) dx < ∞.
Ω
The space L p(·) (Ω) equipped with the norm
f p(·),Ω ≡ f L p(·) (Ω) = inf λ > 0 : A p(·) ( f /λ ) 1
1, p(·)
becomes a Banach space. The Banach space W0
(Ω) with p(x) ∈ [p− , p+ ] ⊂ (1, ∞)
is defined by
⎧
⎨ W 1, p(·) (Ω) = u ∈ W 1,1 (Ω) : (|u|, |∇ u|) ∈ L p(·) (Ω) ,
0
0
(8)
⎩ u 1,p(·) = ∇u p(·),Ω + u p(·),Ω.
(Ω)
W0
1, p(·)
An equivalent norm of W0
is given by
u
1,p(·)
W0
(Ω)
= ∇u p(·),Ω.
1, p(·)
• If condition (7) is fulfilled, then C0∞ (Ω) is dense in W0
1, p(·)
W0
(Ω)
C0∞ (Ω)
(Ω). The space
can be defined then as the closure of
with respect to the norm (8) –
see [17, 38, 48].
• The space W 1,p(·) (Ω) is separable and reflexive provided that p(x) ∈ C0 (Ω).
• Let
1 < q(x) sup q(x) < inf p∗ (x),
Ω
with
⎧
⎨ p(x)n
p∗ (x) = n − p(x)
⎩
∞
Ω
if p(x) < n,
if p(x) > n.
506
S TANISLAV A NTONTSEV
Then
uLq(·) (Ω) C ∇uL p(·) (Ω)
1,p(·)
and the embedding W0
(Ω) → Lq(·) (Ω) is continuous and compact.
• It follows directly from the definition that
p−
p+
p−
p+
min f p(·) , f p(·) A p(·) ( f ) max f p(·) , f p(·) .
• Hölder’s inequality. For all f ∈ L p(·) (Ω), g ∈ L p (·) (Ω) with
p(x) ∈ (1, ∞),
p
=
p
,
p−1
the following inequality holds:
1
1
| f g| dx +
f p(·) g p
(·) 2 f p(·) g p
(·) .
p− (p)
−
Ω
2.2. Spaces L p(·, ·) (QT ) and W(QT )
Let us assume that the exponent p(x,t) ∈ [p− , p+ ] ⊂ (1, ∞) is continuous in QT
with logarithmic module of continuity:
|p(x,t) − p(y, τ )| ω (|x − y| + |t − τ |),
where
lim ω (s) ln
s→+0
(9)
1
C < ∞.
s
Under condition (9) the space C∞ (0, T,C0∞ (Ω)) is dense in L p(·) (QT ) and the last one
can be defined then as the closure of C∞ (0, T,C0∞ (Ω)) (see, [17, 38, 48]).
We introduce the Banach space
1,p
Vt (Ω) = u : u ∈ L2 (Ω) ∩W0 − (Ω) ∩W01,2 (Ω), |∇u| p(·,t) ∈ L1 (Ω) ,
uVt (Ω) = u2,Ω + ∇u p(·,t),Ω,
and denote by Vt
(Ω) it’s dual. For every t ∈ [0, T ] the inclusion
−
Vt (Ω) ⊂ X = W01,p (Ω) ∩ L2 (Ω) ∩W01,2 (Ω)
holds, which is why Vt (Ω) is reflexive and separable as a closed subspace of X.
By W(QT ) we denote the Banach space
W(QT ) = u : [0, T ] → Vt (Ω)| u, ut , |∇u| p(·)/2 ∈ L2 (QT ), u = 0 on ΓT ,
uW(QT ) = ∇u p(·),QT + u2,QT + ut 2,QT + ∇ut 2,QT .
507
WAVE EQUATION WITH p(x,t) -L APLACIAN AND DAMPING TERM
W
(QT ) is the dual of W(QT ). Set
+
V+ (Ω) = u| u ∈ L2 (Ω) ∩W01,1 (Ω), |∇u| ∈ L p (Ω) .
By analogy with [11] we prove the following propositions.
(a) Let us prove first that every function u ∈ C∞ (0, T ;C0∞ (Ω)) can be approximated
2
in the norm of W(QT ) by the functions ∑m
k=1 dk (t)ψk (x) with dk (t) ∈ C [0, T ]. We
denote the set of such functions by Pm .
Let us take for {ψk } the orthonormal basis of the Hilbert space H0s (Ω) with s 1
so big that H0s (Ω) is dense in V+ (Ω). According to ([27], Ch.6), we can take a special
basis ψk (x) such that
(v, ψk )H0s (Ω) = λk (v, ψk ), ∀v ∈ H0s (Ω), λk > 0.
Let us represent
∞
u = ∑ ui (t)ψi (x),
i=1
ui (t) = (u(x,t), ψi (x))H0s (Ω) .
(10)
For every t ∈ [0, T ],
ut 2H s (Ω) (t) + ∇ut 2H s (Ω) (t) + u2H s(Ω) (t)
0
0
0
∞
= ∑ ((u
i (t))2 (1 + λi ) + u2i (t)) < ∞.
(11)
i=1
Let us consider the sequence {u(m) } , u(m) = ∑m
i=1 ui (t)ψi (x). For every t ∈ [0, T ],
(m)
ut − ut 2H s (Ω) (t) + ∇ut − ∇(m) ut 2H s (Ω) (t) + u − u(m)2H s (Ω) (t)
0
0
∞
∑
=
i=m+1
0
((u
i (t))2 (1 + λi) + u2i (t))
→0
as m → ∞, because of (11). Since for every t ∈ [0, T ] the sequence
(m)
φm (t) ≡ u − u(m)2H s (Ω) (t) + ut − ut 2H s (Ω) (t) + ∇ut − ∇(m) ut 2H s (Ω) (t)
0
0
0
is monotone decreasing, nonnegative, and tends to zero as m → ∞, by the Beppo Levi
theorem
T
0
φm (τ ) d τ → 0 as m → ∞.
Thus,
(m)
u − u(m)W(Q) C |u − u(m)| + |ut − ut |
(m) + |∇(ut − ut )| 2L2 (0,T ;H s (Ω)) → 0
0
508
S TANISLAV A NTONTSEV
as m → ∞.
(b) Let now u ∈ W(Q) and {uδ } be the sequence of mollifiers such that uδ ∈
C∞ (0, T ;C0∞ (Ω)). Given ε , we take δ such that u − uδ W(Q) < ε and approximate
uδ using (a):
(m) (m)
u − uδ W(Q) + uδ − uδ W(Q) < 2ε ,
u − uδ W(Q)
with
(m)
uδ
m
= ∑ di (t)ψi (x),
i=1
di (t) = (uδ , ψi )H0s (Ω) ∈ C∞ [0, T ].
P ROPOSITION 2.1. For every u ∈ W(QT ) there is a sequence {dk (t)} , dk (t) ∈
C2 [0, T ], such that
m
u − ∑ dk (t)ψk (x)W(QT ) → 0 as m → ∞ .
k=1
We introduce also a subset of functions u ∈ W (QT ) such that
W∞ = u : u ∈ W (QT ), u, ut , |∇u| p/2 , |u|σ /2 ∈ L∞ (0, T ; L2 (Ω)) .
3. Local and global existence
3.1. Definition. Main results
D EFINITION 3.1. A function u : ΩT → R is called a energy weak solution to (1)(3) if:
QT
u ∈ W (QT ) ∩W∞ (QT );
(12)
u(·,t) → u0 in W01,2 (Ω) ∩W 1,p(·,0) (Ω), ut (·,t) → u1 in L2 (Ω);
(13)
−ut ϕt + a |∇u| p(·)−2 ∇u + α ∇ut · ∇ϕ − b |u|σ (·)−2 uϕ dxdt
=
Ω
u1 ϕ (·, 0)dx +
QT
f ϕ dx,
(14)
for all ϕ ∈ C∞ (0, T ;C0∞ (Ω)) , ϕ (x, T ) = 0, x ∈ Ω.
Let us assume that the coefficients of the problem (1)-(3), in addition to (9), satisfy
0 < a− a(x,t) a+ < ∞, |at | Ca , 0 < α ,
(15)
WAVE EQUATION WITH p(x,t) -L APLACIAN AND DAMPING TERM
509
1 < p− p(x,t) p+ < ∞, |pt | = −pt C p ,
(16)
1 < σ− σ (x,t) σ+ < ∞, 0 σt Cσ ,
(17)
and that one of the following conditions be true: either
or
0 < b− −b(x,t) b+ < ∞, 0 bt ,
(18)
0 < b− b(x,t) b+ < ∞, 0 bt and σ+ 2 or σ+ < p− .
(19)
Also it is assumed that
u0 ∈ L2 (Ω) ∩W01,2 (Ω) ∩W 1,p(·,0)(Ω), u1 ∈ L2 (Ω), f ∈ L2 (QT ).
(20)
In this section we prove global and local in time existence theorems.
T HEOREM 3.1. (Global existence in time) Under conditions (9), (15)-(20), the
problem (1)-(3) has at least one energy weak solution in the sense of Definition 3.1
which is global in time (for any t ∈ [0, T ], T < ∞).
Let us assume that
0 < a− a(x,t) a+ < ∞, |at | Ca ,
(21)
0 < b− b(x,t) = b(x,t) b+ < ∞, |bt | Cb ,
(22)
pt 0, |pt | C p , 0 σt Cσ ,
(23)
and
2 < σ− σ+ <
n+2
2n
p− ,
< p− .
n
n+2
(24)
T HEOREM 3.2. (Local existence in a small time) Under conditions (9), (21)-(24),
the problem (1)-(3) has at least one weak solution in the sense of Definition 3.1 for a
small time t ∈ [0, T0 ), ( T0 > 0 is small).
3.2. Step 1. Galerkin’s approximations
The Galerkin’s approximations of solutions to problem (1)-(3) are sought in the
form
m
∑
u(m) ≡
k=1
uk (t)ψk (x), uk (t) = (u(x,t), ψk (x))H0s (Ω) .
(25)
We assume also
(m)
u1
(m)
→ u1 strongly in L2 (Ω), u0
→ u0 strongly in W01,2 (Ω).
(26)
The coefficients uk (t) are defined from the relations
Ω
(m)
(utt − Lu(m) − f )ψk = 0, k = 1, ..., m.
(27)
510
S TANISLAV A NTONTSEV
Last equalities and the initial conditions lead us to the Cauchy problem for the system
of m ordinary differential equations of the second order for the coefficients uk (t)
uk (0) =
u
k = Fk (t, u1 (t), ..., um (t)),
(28)
u0 ψk , u
k (0) =
(29)
Ω
Ω
u1 ψk , k = 1, ..., m,
where
Fk =
Ω
−
p(·,t)−2 (m)
(m)
∇ψk dx
∇u + α ∇ut
a∇u(m) (·,t)
σ (·,t)−2 (m)
bu(m) +
u ψk + f ψk dx.
Ω
By Peano’s Theorem, for every finite m the problem (28), (29) has a solution
uk (t), k = 1, ..., m on an interval (0, Tm ) for each m. The estimates below allow one to
take Tm = T for all m.
3.3. Step 2. A priori estimates
Here we derive estimates for an energy functional and for approximated solutions
which do not depend on m.
3.3.1. Energy relation
Multiplying each of equations (28) by c
k (t) and summing over k = 1, ..., m, we
arrive at the relation
p
(m) 2
ut
a∇u(m) d
b|u(m) |σ
(m) 2
+
−
dx + α ∇ut dx
dt Ω
2
p
σ
Ω
(m)
p
(m)
p
a|∇u |
|∇u |
∇u(m) pt dx
1
−
p
ln
+
=
at
p
p2
Ω
bt |u(m) |σ b|u(m) |σ (m)
(m)
1 − σ ln|u | σt dx + f ut dx. (30)
+
−
σ
σ2
Ω
Ω
Omitting the index m for simplicity and introducing the energy functional
E(t) =
Ω
|ut (·,t)|2
|∇u| p(·,t)
|u|σ (·,t)
+ a(·,t)
− b(·,t)
dx,
2
p(·,t)
σ (·,t)
(31)
we can rewrite (30) in the form
E (t) + α
where
Ω
|∇ut (·,t)|2 dx = Λ,
Λ(t) = Λ1 + Λ2 + Λ3,
(32)
(33)
WAVE EQUATION WITH p(x,t) -L APLACIAN AND DAMPING TERM
|∇u | p a|∇u | p
+
Λ1 =
(1 − pln|∇u |) |pt | dx,
at
p
p2
Ω
bt |u |σ b|u |σ
Λ2 = −
+
(1
−
σ
ln|u
|)
σ
t dx,
σ
σ2
Ω
511
Λ3 =
Ω
f ut dx .
(34)
(35)
(36)
3.3.2. Estimates of the energy functional
In this Section we analyze different conservation laws for energy functional and
solution estimates which do not depend on m .
In particular, we derive some estimates for energy functional which will be important to prove the blow-up.
Let us assume that
1 < p− p(x,t) p+ < ∞, 1 < σ− σ (x,t) σ+ < ∞,
(37)
0 < a− a(x,t) a+ < ∞, |at | Ca ,
(38)
0 < b− b(x,t) = b(x,t) < b+ < ∞, |bt | Cb ,
(39)
at 0, 0 bt , pt 0, 0 σt , |pt | C p , |σt | Cσ .
(40)
L EMMA 3.1. Let (37)-(39) be fulfilled and in addition
Then
E(t) + α
t
0
Ω
pt = σt = f = 0.
(41)
|∇ut (x, s)|2 dxds E(0), ∀t 0.
(42)
The inequality (42) transforms to the equality if at = bt = 0 .
Proof. To prove this Lemma it is enough to apply the formulas (32)-(35).
2
L EMMA 3.2. Let (37)-(40) be fulfilled and f = 0. Then
t
E(t) +
0
Ω
α |∇ut (x, s)|2 dxds E(0) + Ct,
with the constant C = e (a+C p + b+Cσ )|Ω|.
Proof. We evaluate Λ1 , Λ2 in the following way:
|∇u | p a|∇u | p +
1 − p ln|∇u | |pt | dx
2
p
p
Ω
a|∇u | p 1 − p ln|∇u | |pt |dx
2
p
Ω∩(pln|∇u|1)
Λ1 =
at
(43)
512
S TANISLAV A NTONTSEV
ea+C p |Ω| = C1 ,
(44)
and
bt |u |σ
b|u |σ 1
−
σ
ln
|u
|
σt dx
σ
σ2
Ω
b|u |σ 1 − σ ln |u | σt dx
2
Ω∩(σ ln|u|1) σ
Λ2 = −
+
eb+Cσ |Ω| = C2 .
(45)
Integrating the energy relation (32) with respect to t, we obtain that
E(t) +
t
0
Ω
α |∇ut |2 dxds E(0) + tC,C = C1 + C2 .
(46)
2
3.3.3. A priori estimates of solutions
L EMMA 3.3. (Global estimates of solutions (b(x,t) 0 )). Let the conditions (9),
(15)-(20) be fulfilled. Then for any finite T < ∞,
Ψ(T ) = sup 0tT
Ω
|ut |2 + |∇u| p(·) + |u|σ (·) dx
+α
T
0
Ω
|∇ut |2 dxds C
(47)
with a constant C which depends on:
f 22,QT , u1 L2 (Ω) , u0 Lσ (·,0) (Ω) , u0 W 1,2 (Ω) , |Ω|, T
and does not depend on m.
Proof. Recall that in this case all terms of the energy functional E are nonnegative.
Then we obtain
|∇u | p a|∇u | p
Λ1 =
(1
−
p
ln|∇u
|)|p
|
dx
at
+
t
p
p2
Ω
a|∇u | p Ca
1 − p ln|∇u | |pt |dx
E(t) +
2
a−
p
Ω∩(p ln|∇u|1)
Ca
a+
E(t) + 2 e |Ω| C(E(t) + 1),
a−
p−
and
Λ2 = −
bt |u |σ
Ω
σ
+
b|u |σ 1
−
σ
ln
|u
|
σt dx
σ2
WAVE EQUATION WITH p(x,t) -L APLACIAN AND DAMPING TERM
b|u |σ 1 − σ ln |u | σt dx
2
Ω∩(σ ln|u|1) σ
Cb
E(t) +
σ−
Cb
b+
E(t) + 2 e |Ω| C(E(t) + 1),
σ−
σ−
and
513
|Λ3 | E(t) + f 22,Ω .
Finally,
Λ C(E(t) + f 22,Ω + 1).
Hence we arrive at the inequality
E (t) + α
Ω
|∇ut (·,t)|2 dx C(E(t) + f 22,Ω + 1).
Applying Gronwall’s lemma, we conclude the proof of the Lemma.
2
L EMMA 3.4. (Global estimates of solutions ( 0 b(x,t), σ+ 2 or 2 < σ+ <
p− .)) The estimate (47) remains valid if the condition b(x,t) 0 is replaced by the
following
0 b(x,t), σ+ 2 or 2 < σ+ < p− .
Proof. In this case, we rewrite the energy relation (32) in the form
|u|σ
2
E (t) + α |∇ut (·,t)| dx =
b
dx + Λ,
σ
Ω
Ω
where
=
E(t)
Ω
|ut |2
|∇u| p(·,t)
+ a(·,t)
dx.
2
p(·,t)
Integrating last one with respect to t, we obtain
t
t
|u|σ
t
+α
|∇ut (·, s)|2 dxds =
b
E(t)
dx + Λds + E(0),
0
σ
Ω
0 Ω
0
with
(48)
|u|σ t σ
σ
Ω b σ dx 0 C Ω |u| dx + 1 , |Λ| CE(t) + C Ω |u| dx.
Now it is enough to evaluate the term
inequalities
σ
Ω |u|
dx. If σ+ 2 we use the chain of
|u|σ C
b(·,t)
dx
|u|σ dx
σ
Ω
Ω
C 1 + |u|2 dx
Ω
514
S TANISLAV A NTONTSEV
t
2
2
2C 1 + t
|ut | dxds + |u0 | dx
Ω
0 Ω
t
C 1 + t E(s)ds
.
0
If 2 < σ+ < p− we use the embedding inequality
σ+
Ω
|u |
dx C
C
σ+
p−
Ω
σ+
p
Ω
p−
|∇u | dx
|∇u | dx
p−
+C
+ C(ε ), ε ∈ (0, 1), σ+ < p− < np− /(n − p−).
ε E(t)
For suitable ε > 0 we come to the inequality
+α
E(t)
t
Ω
0
|∇ut (·, s)|2 dxds C
t
0
E(s)ds
+1
and Gronwall’s lemma and (48) lead us to the desired estimate.
2n
L EMMA 3.5. (Local estimates 2 < σ− σ+ < n+2
n p− , n+2 < p− ) Let the conditions (21)-(24) be fulfilled. Then there exists a small T0 > 0 such that
Ψ(t) = sup 0st
Ω
|ut (x, s)|2 + |∇u| p(·) + |u|σ (·) dx
+α
t
Ω
0
|∇ut |2 dxds C, 0 t < T0
(49)
with a constant C which depends on:
T0 , f 22,QT , u1 L2 (Ω) , u0 Lσ (·,0) (Ω) , u0 W 1,p(·,0) (Ω)
but does’nt depend on m.
Proof. We will use the energy relation (48) and inequalities
t t
Λ C
+ |u|σ dx ds,
E(s)
0
Ω
|u|σ (·,t) dx σ+
Ω
|u|
dx C
Ω
Ω
0
|u|σ+ dx + |Ω|,
Ω
|∇u| p− dx σ+ θ p−
2
Ω
|∇u| p dx + |Ω|,
dx
Ω
(51)
σ+ (1−θ )
2
|u| dx
Ω
Ω
γ
ε |∇u| p(·,t) dx + Cε
|u|2 dx + C, ε ∈ (0, 1),
Ω
|∇u|
p−
(50)
(52)
515
WAVE EQUATION WITH p(x,t) -L APLACIAN AND DAMPING TERM
where
σ+
(σ+ − 2)n
n+2
< 1, if σ+ <
p− ,
θ=
p−
np− − 2(n − p−)
n
σ+ p−
> 1.
γ=
2(p− − θ σ+ )
On the other hand we have that
t t
E(s)ds
+1 .
|u|2 dx C
|ut |2 dxds + |u0 |2 dx C
Ω
0
Ω
Ω
0
Combining (48), (50)-(52) with a suitable ε , we come to the inequality
t
γ
E(s)ds + 1 , γ > 1,
E(t) C
0
which gives the estimate
− 1
+ 1 E(0)
+ 1 1 − tC(E(0)
+ 1)γ −1(γ − 1) γ −1 < ∞,
E(t)
−1
+ 1)γ −1(γ − 1)
t < T0 = C(E(0)
.
2
The Lemma is proved.
R EMARK 3.1. The estimates of the Lemmas 3.3 and 3.4 imply that
2n
Λ(T ) = sup
|u|2 + |u| n−2 +|u|q dx C,
(53)
t∈[0,T ] Ω
where q np− /(n − p−) if p− < n and q < ∞ if p− n.
Also we have that
Ω
|∇u(x,t)|2 dx 2
T 0
Ω
Ω
|∇u0 (x)|2 dx + t
|ut |q dx
2q
dt C
QT
T
0
Ω
|∇ut |2 dxds ,
|∇ut |2 dxdt,
2n
, n > 2, 1 q < ∞, n = 2.
n−2
Finally we arrive at the estimate
1q
Λ(T ) + sup
0sT Ω
|ut (x, s)|2 + |∇u|2 + |∇u| p + |u|σ dx
+
T
0
2
Ω
|∇ut | dxds +
T 0
Ω
q
|ut | dx
2
q
dt K
(54)
with a constant K independent on m. Last estimate is valid for any finite interval
[0, T ] (under conditions of Lemmas 3.3 and 3.4) or only for a small interval of the time
[0, T0 ) (under conditions of Lemma 3.5).
516
S TANISLAV A NTONTSEV
(m)
3.3.4. Compactness of ut
L EMMA 3.6. Assume that the estimate (54) is valid. Then
(m)
C||ϕ ||W (Q ) , T < T0
u
ϕ
dxdt
tt
T
(55)
QT
with a constant C independent of m.
Proof. Let dk (t) ∈ C2 (0, T ) be arbitrary functions. Multiplying (27) by dk (t),
integrating over [0, T ] and summing with respect to k, we arrive at the identity
QT
(m)
(m)
utt ϕ + a|∇u(m) | p(x,t)−2 ∇u(m) + α ∇ut
∇ϕ dxdt
−
QT
(m) σ −2 (m)
u + f ϕ dxdt = 0
b|u |
(56)
which is valid for any function
ϕ=
N
∑ dk (t)ψk (x), N m.
(57)
k=1
We introduce
p(·)−2 (m)
−
→
(m)
G m = a∇u(m) .
∇u + α ∇ut
Then (56) takes the form
(m)
QT
utt ϕ dxdt =
QT
→
−
− G m ∇ϕ + b|u(m) |σ −2 u(m) + f ϕ dxdt = J.
Using results of Section 2 it’s easy to verify that
p(·)−1
|J| a+ |∇u(m) p(·)
,Q
p(·)−1 T
∇ϕ p(·),QT
(m) + α ∇ut 2,Q ∇ϕ 2,Q + b+|u(m) |σ −1 T
T
σ (·)
,Q
σ (·)−1 T
||ϕ ||σ (·),QT
+ || f ||2,QT ||ϕ ||2,QT
C(K)||ϕ ||W (QT ) .
The Lemma is proved.
R EMARK 3.2. Applying the known inequality
w(x + h) − w(x) 2
dx C||∇w||2L2 (Ω) , ∀w ∈ W01 (Ω)
h
Ω
2
517
WAVE EQUATION WITH p(x,t) -L APLACIAN AND DAMPING TERM
(m)
to the function w = ut
, we derive from (54),
T (m)
2
(m)
ut (x + h,t) − ut (x,t) dxdt C(K)|h|2 .
Ω
0
(58)
According to results of [39], Lemma 3.6 and estimate (58) we can conclude the compactness of the sequence umt in the following sense
(m)
ut
→ ut strongly in L2 (QT ) ∩ L2 (Ω).
(59)
(m)
In certain papers (see, for example [43, 44]) to prove the compactness ut
used the following
, they
L EMMA 3.7. ([16]) Let Ω be any bounded domain in Rn , {wk }∞
k=1 be an orthogonal basis in L2 (Ω). Then for every ε > 0 , there exists a positive number N1ε such
that
Nε
1
2
+ ε u1,q
u2,Ω = u ∑ (u, wk )2Ω
k=1
1,q
for all u ∈ W0 (Ω), (2 q < ∞).
By this Lemma and estimates (54), for every ε > 0 , there exist positive constants
N1ε and N2ε independent of m such that, as m → ∞,
u
(m)
(t) − u(t) N1ε
2
∑ u(m) − u, wk Ω
1/2
k=1
+ ε u(m)(t) − u(t)1,2 C(T )ε , t ∈ [0, T ], (60)
and
T
0
(m)
ut
− ut 2 2
N2ε ∑
k=1 0
T
(m)
2
ut − ut , wk Ω
+ 2ε 2
T
(m)
− ut 21,2 Cε 2 .
(61)
u(m) → u strongly in L∞ (0, T ; L2 (Ω) and a.e. in QT ,
(62)
0
||ut
Since ε is arbitrary, we obtain
(m)
ut
2
→ ut strongly in L (QT ) and a.e. in QT .
(63)
518
S TANISLAV A NTONTSEV
3.4. Step 3. Passage to the limit as m → ∞
A weak solution of problem (1)-(3) will be obtained as the limit of the sequence of
Galerkin’s approximations u(m) as m → ∞. Let us establish the passage to the limit as
m → ∞ on the intervals of the time when the estimate (54)
isvalid. Last estimateallows
us conclude that there exist u and a subsequence of u(m) , still denoted by u(m) ,
such that (see also, e.g., [39, 42, 43]).
u(m) → u strongly in L∞ (0, T ; L2 (Ω)), a.e. on QT
(m)
→ ut strongly in L2 (QT ), a.e. on QT
(m)
ut weakly in L∞ (0, T ; L2 (Ω)), a.e. t ∈ [0, T ]
ut
ut
∇u(m) ∇u weakly in L p(·) (QT )∩L∞ (0, T ; L2 (Ω)),
(m)
∇ut (m)
utt utt
(64)
∇ut weakly in L2 (QT ),
weakly in W (QT ),
a|∇u(m) | p−2∇u(m) η weakly in L p (·) (QT ).
Integrating by parts in the first term in (56), we can rewrite last one in the form
Qt
(m)
(m)
− ut ϕt + a||∇u(m) || p(x,t)−2 ∇u(m) + α ∇ut
∇ϕ dxd τ
(m) t
b||u(m) ||σ −2 u(m) + f ϕ dxd τ − ut , ϕ Ω = 0, t ∈ [0, T ]. (65)
−
0
Qt
Taking into account (64) and passing to the limit in (65) as m → ∞, we obtain for
any ϕ ∈ Pm ,
Qt
− ut ϕt + (η + α ∇ut )∇ϕ dxd τ
−
Qt
t
b||u||σ −2u + f ϕ dxd τ − (ut , ϕ )Ω = 0, a.e. t ∈ [0, T ]. (66)
0
Now we prove that
t
0
(η , ∇ϕ )Ω d τ =
t
0
a |∇u| p(·)−2 ∇u, ∇v
Ω
dτ .
Substituting ϕ in (65) by u(m) and by u in (66) and integrating by parts in the terms
with α we obtain
Qt
(m) 2
− ut
+ a||∇u(m)|| p(x,t)−2 ∇u(m) ∇u(m) dxd τ
t + α ∇u(m) , ∇u(m) Ω −
b||u(m) ||σ −2 u(m) + f u(m) dxd τ
0
Qt
WAVE EQUATION WITH p(x,t) -L APLACIAN AND DAMPING TERM
t
(m)
− ut , u(m) Ω = 0,
0
519
(67)
and
Qt
t
− ut2 + η ∇u dxd τ + α ∇u, ∇u)Ω −
0
Qt
t
b||u||σ −2 u + f udxd τ − ut , u Ω = 0, a.e. t ∈ [0, T ]. (68)
0
According to the monotonicity of the operator A(u) = a |∇u| p(·)−2 ∇u , we have that
0
t
0
A(u(m) ) − A(v), ∇u(m) − ∇v
Ω
dτ .
(69)
Taking into account (64), (67) and (68), (69), we derive that
0
t
0
(η − A(v), ∇u − ∇v)Ω d τ , ∀v ∈ W (QT ).
(70)
Choosing v = u − λ w, where λ > 0 is a real number and w ∈ W (QT ), and substituting
it into (70), we have
0
t
0
(η − A(u − λ w), ∇w)Ω d τ .
Letting λ → 0 in last inequality and using (70), we come at the inequality
0
t
0
(η − A(u), ∇w)Ω d τ .
Taking into account the density of the functions w, we conclude the proof of the Theorems 3.1 and 3.2.
2
4. Blow up of solutions
First we consider equation (1) with α > 0 , assuming that conditions of Lemma
3.1 are fulfilled. Assume that
E(0) 0, 0 < (u0 , u1 )L2 (Ω) , 2 p− p+ < σ− .
(71)
T HEOREM 4.1. Let u be an energy weak solution to problem (1)-(3). Let the
conditions of the Lemma 3.1 be fulfilled and (71) hold. Then there exists a finite time
tmax < ∞ such that
Φ(t) = u(t)22,Ω + α
t
0
Ω
|∇u|2 dxds → ∞ if t → tmax .
(72)
520
S TANISLAV A NTONTSEV
Proof. It’s easy to verify that
Φ
= 2(u, ut )Ω + α
Φ
= 2||ut ||22,Ω + 2
Ω
||∇u||2 dx,
Ω
− a||∇u|| p + b||u||σ dx.
Using the inequality (42) of the Lemma 3.1, we calculate
Φ
−a |∇u| p + b |u|σ dx
Ω
t
+ 2λ E(t) + α
|∇ut |2 dxds
2 ut (t)22,Ω + 2
0
Ω
2
= (2 + λ ) ut (t)
λ
λ
||u||σ dx
− 1 a||∇u|| p + b 1 −
+2
p
σ
Ω
t
+ 2λ α
0
Ω
||∇ut ||2 dxds > 0
(73)
for some λ > 2 and p+ < λ < σ− .
It follows that
Φ
(t) > 0, if Φ
(0) 2(u0 , u1 )L2 (Ω) > 0 .
Thus we can conclude that 0 < Φ(t) 0 < Φ
(t), 0 < Φ
(t) and Φ(t) → ∞ as t → tmax .
Assume in contrast that tmax = ∞. Using properties the Orlicz-Sobolev spaces (see
Section 2), we derive the following inequalities (for any fixed t )
||u(·,t)||2,Ω C||u(·,t)||Lσ (·,t) (Ω)
1 1 σ−
σ+
σ (·,t)
σ (·,t)
C max
||u(·,t)||
dx
,
||u(·,t)||
dx
, (74)
Ω
Ω
||∇u(·,t)||2,Ω
C||∇u(·,t)||L p(·) (Ω)
p1 p1 −
+
C max
||∇u(·,t)|| p(·,t) dx
,
||∇u(·,t)|| p(·,t) dx
.
Ω
Ω
Notice that, according to (73),
Ω
|ut |2 dx CΦ
,
Ω
|∇u(·,t)| p(·,t) dx CΦ
,
σ (·,t) dx
Ω
|u(·,t)|
Then we can rewrite (74), (75) in the forms
1 1 ||u(·,t)||2,Ω C max Φ
σ− , Φ
σ+ ,
CΦ
.
(75)
WAVE EQUATION WITH p(x,t) -L APLACIAN AND DAMPING TERM
||∇u(·,t)||2,Ω C max
521
1 1 Φ
p− , Φ
p+ .
Hence we arrive at the inequality
0 Φ
= 2
Ω
u ut dx +
α
2
Ω
|∇u|2 dx
α
|∇u|2 dx
2 u(·,t)2,Ω ut (·,t)2,Ω + +
2 Ω
1 1 1 1 2 2 +
+
C max Φ
σ− 2 , Φ
σ+ 2 + max Φ
p− , Φ
p+ .
Taking into account that 0 < Φ
(t), we assume, without loss of generality, that Φ
1
and come at the ordinary differential inequality
μ
(76)
C Φ
Φ
,
where
1
1
1 2 < 1 if σ− > 2, p− > 2.
= max
+ ,
μ
σ− 2 p−
The last inequality leads us to estimate
1
t(μ − 1) μ −1 − μ −1
Φ (0)
→∞,
Φ
(t) Φ
(0) 1 −
C
as
t → tmax =
(77)
C −μ +1
Φ (0)
< ∞.
μ −1
2
The theorem is proved.
R EMARK 4.1. Let us notice that constants μ and C (and respectively tmax ) in
(77) depend only on |Ω|, n, a± , b± , p± , σ± .
Now we assume that the exponents p, σ weakly dependent on t , that is, the constants C p ,Cσ are small. The proof the blow up is the same as in the previous Theorem
if we guarantee that
E(t) + α
t
0
Ω
|∇ut |2 dxds 0, 0 t tmax
(78)
with tmax already defined in Theorem 4.1. According to the Lemma 3.2 (see inequality
(43)), we have that
E(t) + α
t
0
Ω
|∇ut |2 dxds E(0) + tmaxe(a+C p + b+Cσ )|Ω|.
Assuming that
δ = max (C p ,Cσ ) |E(0)| (tmax e(a+ + b+ )|Ω|)−1 , E(0) < 0,
we arrive at (78). Then we come to
(79)
522
S TANISLAV A NTONTSEV
T HEOREM 4.2. Let u be an energy weak solution to problem (1)-(3). Let the
conditions of Lemma 3.2 and (79) (with tmax defined in Theorem 4.1) be fulfilled. Let
the conditions
E(0) < 0, 0 < (u0 , u1 )L2 (Ω) , 2 p− p+ < λ < σ−
hold. Then the solution u blows up (in the sense that Φ(t) becomes unbounded) on the
finite interval (0,tmax ).
Now we consider equation (1) with α = 0, assuming that the problem (1)-(3) has
at least one local energy solution. Here we follow the paper [19], where the authors
proved the blow up for an abstract hyperbolic equation in a Banach space which includes, as an example, the equation of the type (1) with the a = b = 1 , p = const.,
σ = const.
We assume that
E(0) 0, 0 < (u0 , u1 )L2 (Ω) , σ− > max{2, p+}.
(80)
Repeating the arguments of the paper [19], we prove the following theorems.
T HEOREM 4.3. Let u be an energy weak solution to problem (1)-(3) with α = 0 .
Let conditions of the Lemma 3.1 be satisfied and assume that (80) holds. Then u blows
up (in the sense that u(t)22,Ω becomes unbounded) on the finite interval (0,tmax ) with
tmax = 2 u0 22,Ω /(λ − 2)(u0 , u1 )Ω .
Proof. Let us introduce the function
G(t) = u(t)2 = u(t)22,Ω .
It is very easy to verify that
G
(t) = 2 (u, ut )Ω , G
(t) = 2 ut (t)2 + 2
Ω
−a |∇u| p + b |u|σ dx.
(81)
Taking into account (80), we evaluate G
(t) in the following way
−a |∇u| p + b |u|σ dx + 2λ E(t)
Ω
λ
λ
2
− 1 a |∇u| p + b 1 −
= (2 + λ ) ut (t) + 2
|u|σ dx
p
σ
Ω
G
(t) 2 ut (t)2 + 2
(2 + λ ) ut (t)2
(82)
for some λ , p+ λ σ− . Then we literally repeat the arguments of the paper [19].
From the first identity (81) we conclude that
G
2 (t) 4 ut 2 u2 = 4G ut 2 ⇒ ut 2 G
2
.
4G
(83)
WAVE EQUATION WITH p(x,t) -L APLACIAN AND DAMPING TERM
523
Combining (82) and (83), we come to the following ordinary differential inequality
G
(t) (2 + λ )
G
G
2
(2 + λ )
⇔ 2 0.
4G
G
4G
(84)
Then (80), (84) imply that G
(t) > 0 for all t > 0. Integrating (84) twice as above,
we come to the inequality
u0 22,Ω
λ − 2 (u0 , u1 )Ω
1−t
2
u0 22
−
4
λ −2
u(t)22,Ω .
2
Now we assume that the exponents p, σ weakly dependent on t , that is, the constants C p ,Cσ are small.
Repeating the above mentioned arguments (see (78), (79)), we prove
T HEOREM 4.4. Let u be an energy weak solution to problem (1)-(3) with α = 0 .
Let conditions of the Lemma 3.2 be fulfilled and (79) (with tmax defined in previous
Theorem) and (80) hold. Then the solution u blows up on the finite interval (0,tmax ).
REFERENCES
[1] S. A NTONTSEV, Wave equation with p(x,t) - Laplacian and damping term: Existence and blow-up,
in Proceedings of the International Conference ”Modern Problems of Applied Mathematics and Mechanics: Theory, Experiment and Applications” devoted to the 90th anniversary of Prof. N.N Yanenko,
Novosibirsk, Russia, May 30-June 4, 2011.
[2] S. A NTONTSEV, Wave equation with p(x,t) -Laplacian and damping term: Existence and blow-up, in
Abstracts, Nonlinear Models in Partial Differential Equations, An international congress on occasion
of J.I. Diaz’s 60th birthday, Toledo, Spain, June 14-17, 2011, p. 8.
[3] S. A NTONTSEV, Wave equation with p(x,t) -Laplacian and damping term: Blow-up of solutions, C.
R. Mecanique, (2011, in press).
[4] S. A NTONTSEV, M. C HIPOT, AND Y. X IE, Uniqueness results for equations of the p(x) -Laplacian
type, Adv. Math. Sci. Appl., 17 (2007), 287–304.
[5] S. N. A NTONTSEV, J. I. D ÍAZ , AND S. S HMAREV, Energy Methods for Free Boundary Problems:
Applications to Non-linear PDEs and Fluid Mechanics, Bikhäuser, Boston, 2002. Progress in Nonlinear Differential Equations and Their Applications, Vol. 48.
[6] S. N. A NTONTSEV AND J. F. RODRIGUES , On stationary thermo-rheological viscous flows, Ann.
Univ. Ferrara, Sez.,VII. Sci. Mat., 52 (2006), 19–36.
[7] S. N. A NTONTSEV AND S. I. S HMAREV, Elliptic equations and systems with nonstandard growth
conditions: existence, uniqueness and localization properties of solutions, Journal Nonlinear Analysis,
65 (2006), 722–755.
[8] S. N. A NTONTSEV AND S. I. S HMAREV, Elliptic equations with anisotropic nonlinearity and nonstandard growth conditions, Elsevier, 2006. Handbook of Differential Equations. Stationary Partial
Differential Equations, Elsevier, Vol. 3, Chapter 1, 1-100.
[9] S. N. A NTONTSEV AND S. I. S HMAREV, Parabolic equations with anisotropic nonstandard growth
conditions, in Internat. Ser. Numer. Math. 154, Birkhäuser, Verlag Basel/Switzerland, 2006, 33–44.
[10] S. N. A NTONTSEV AND S. I. S HMAREV, Extinction of solutions of parabolic equations with variable
anisotropic nonlinearities, Proceedings of the Steklov Institute of Mathematics, Moscow, Russia, 268
(2008), 2289–2301.
524
S TANISLAV A NTONTSEV
[11] S. N. A NTONTSEV AND S. I. S HMAREV, Anisotropic parabolic equations with variable nonlinearity,
Publ. Sec. Mat. Univ. Autònoma Barcelona, (2009), 355–399.
[12] S. N. A NTONTSEV AND S. I. S HMAREV, Localization of solutions of anisotropic parabolic equations,
Nonlinear Anal., 71 (2009), e725–e737.
[13] S. N. A NTONTSEV AND S. I. S HMAREV, Blow-up of solutions to parabolic equations with nonstandard growth conditions, Journal of Computational and Applied Mathematics, 234 (2010), 2633–
2645.
[14] G. AUTUORI , P. P UCCI , AND M. C. S ALVATORI , Asymptotic stability for anisotropic Kirchhoff systems, J. Math. Anal. Appl., 352 (2009), 149–165.
[15] G. AUTUORI , P. P UCCI , AND M. C. S ALVATORI , Global nonexistence for nonlinear Kirchhoff systems, Arch. Rational Mech. Anal., 196 (2011), 489–516.
[16] J. C LEMENTS , Existence theorems for a quasilinear evolution equation, SIAM J. Appl. Math., 26
(1974), 745–752.
[17] L. D IENING , P. H ARJULEHTO , P. H ASTO , AND M. R Ú ŽI ČKA, Lebesgue and Sobolev Spaces with
Variable Exponents, Springer, Berlin, 2011. Series: Lecture Notes in Mathematics, Vol. 2017,1st Edition.
[18] M. D REHER, The wave equation for the p -Laplacian, Hokkaido Math. J., 36 (2007), 21–52.
[19] V. A. G ALAKTIONOV AND S. I. P OHOZAEV, Blow-up and critical exponents for nonlinear hyperbolic equations, Nonlinear Analysis, 53 (2003), 453–466.
[20] H. G AO AND T. F. M A, Global solutions for a nonlinear wave equation with p− Laplacian operator,
EJQTDE., (1999), 1–13.
[21] V. G EORGIEV AND G. T ODOROVA, Existence of a solution of the wave equation with nonlinear
damping and source terms, J. Differential Equations, 109 (1994), 295–308.
[22] J. H AEHNLE AND A. P ROHL, Approximation of nonlinear wave equations with nonstandard
anisotropic growth conditions, Math. Comp., 79 (2010), 189–208.
[23] M. JAZAR AND R. K IWAN, Blow-up results for some second-order hyperbolic inequalities with a
nonlinear term with respect to the velocity, J. Math. Anal. Appl., 327 (2007), 12–22.
[24] M. JAZAR AND R. K IWAN, Blow-up of a non-local semilinear parabolic equation with Neumann
boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 215–218.
[25] T. K ATO, Blow up of solutions of some nonlinear hyperbolic equations, Manuscripta Math., 28 (1980),
235–268.
[26] H. L EVINE AND G. T ODOROVA, Blow-up of solutions of the cauchy problem for a wave equations
with nonlinear damping and source terms and positive initial energy, in Proceedings of the American
Mathematical Society, 129 (2000), 793–805.
[27] J.-L. L IONS , Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, 1969.
[28] S. A. M ESSAOUDI , Blow up in a nonlinearly damped wave equation, Math. Nachr., 231 (2001), 105–
111.
[29] S. A. M ESSAOUDI , Blow up in the Cauchy problem for a nonlinearly damped wave equation, Commun. Appl. Anal., 7 (2003), 379–386.
[30] S. A. M ESSAOUDI , On the decay of solutions for a class of quasilinear hyperbolic equations with
non -linear damping and source terms, Mathematical Methods in the Applied Sciences, 28 (2005),
1819–1828.
[31] S. A. M ESSAOUDI AND B. S AID H OUARI , Global non-existence of solutions of a class of wave
equations with non-linear damping and source terms, Math. Methods Appl. Sci., 27 (2004), 1687–
1696.
[32] C. M U , R. Z ENG , AND B. C HEN, Blow-up phenomena for a doubly degenerate equation with positive
initial energy, Nonlinear Anal., 72 (2010), 782–793.
[33] M. N AKAO AND Y. Z HIJIAN, Global attractors for some quasi-linear wave equations with a strong
dissipation, Adv. Math. Sci. Appl., 17 (2007), 89–105.
[34] J. P. P INASCO, Blow-up for parabolic and hyperbolic problems with variable exponents, Nonlinear
Anal., 71 (2009), 1094–1099.
[35] K. R AJAGOPAL AND M. R Ú ŽI ČKA, Mathematical modelling of electro-rheological fluids, Cont.
Mech. Therm., 13 (2001), 59–78.
[36] M. R Ú ŽI ČKA, Electrorheological fluids: modeling and mathematical theory, Springer, Berlin, 2000.
Lecture Notes in Mathematics, 1748.
WAVE EQUATION WITH p(x,t) -L APLACIAN AND DAMPING TERM
525
[37] A. A. S AMARSKII , V. A. G ALAKTIONOV, S. P. K URDYUMOV, AND A. P. M IKHAILOV, Blow-up
in quasilinear parabolic equations, Walter de Gruyter & Co., Berlin, 1995. Translated from the 1987
Russian original by Michael Grinfeld and revised by the authors.
[38] S. G. S AMKO, Density C0∞ (Rn ) in the generalized Sobolev spaces W m,p(x) (Rn ) , Dokl. Akad. Nauk,
369 (1999), 451–454.
[39] J. S IMON, Compact sets in the space l p (0,t;b) , Ann. Mat. Pura Appl., IV. Ser., 146 (1952), 65–96.
[40] G. T ODOROVA AND E. V ITILLARIO, Blow-up for nonlinear dissipative wave equations in Rn , J.
Math. Anal. Appl., (2005), 242–257.
[41] Z. W ILSTEIN, Global well-posedness for a nonlinear wave equation with p Laplacian damping,
Dissertation,
University of Nebraska-Lincoln,
downloaable at
:http://digitalcommons.unl.edu/mathstudent/24, (2011), 1–116.
[42] Z. YANG, Cauchy problem for quasi-linear wave equations with viscous damping, J. Math. Anal.
Appl., 320 (2006), 859–881.
[43] Z. YANG AND G. C HEN, Global existence of solutions for quasi-linear wave equations with viscous
damping, J. Math. Anal. Appl., 285 (2003), 604–618.
[44] Y. Z HIJIAN, Existence and asymptotic behaviour of solutions for a class of quasi-linear evolution
equations with non-linear damping and source terms, Mathematical Methods in the Applied Sciences,
25 (2002), 795–814.
[45] Y. Z HIJIAN, Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear
wave equations with dissipative term, J. Differential Equations, 187 (2003), 520–540.
[46] Y. Z HIJIAN, Initial boundary value problem for a class of non-linear strongly damped wave equations,
Math. Methods Appl. Sci., 26 (2003), 1047–1066.
[47] Y. Z HIJIAN, Cauchy problem for a class of nonlinear dispersive wave equations arising in elastoplastic flow, J. Math. Anal. Appl., 313 (2006), 197–217.
[48] V. V. Z HIKOV, On the density of smooth functions in Sobolev-Orlich spaces, Zap. Nauchn. Sem.
S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 310 (2004), 1–14.
(Received August 31, 2011)
S. Antontsev
CMAF
University of Lisbon
Portugal
e-mail: [email protected]
Differential Equations & Applications
www.ele-math.com
[email protected]