Existence of weak solutions for viscoelastic hyperbolic equations with variable exponents
Boundary Value Problems
Existence of weak solutions for viscoelastic hyperbolic equations with variable exponents
Yunzhu Gao 0 1 3
Wenjie Gao 0 2
0 Statistics, Beihua University , Jilin, P.R
1 Department of Mathematics
2 Institute of Mathematics, Jilin University , Changchun , P.R. China
3 Department of Mathematics and Statistics, Beihua University , Jilin , P.R. China
The authors of this paper study a nonlinear viscoelastic equation with variable exponents. By using the FaedoGalerkin method and embedding theory, the existence of weak solutions is given to the initial and boundary value problem under suitable assumptions.
existence; weak solutions; viscoelastic; variable exponents

1 Introduction
Let
⊂ RN (N ≥ ) be a bounded Lipschitz domain and < T < ∞. Consider the
following nonlinear viscoelastic hyperbolic problem:
⎧ utt –
⎪⎪⎨
⎪
⎪⎩ u(x, ) = u(x),
ut(x, ) = u(x),
u –
t
utt + g(t – τ ) u(τ ) dτ + utm(x)–ut = up(x)–u, (x, t) ∈ QT ,
(.)
where QT =
× (, T ], ST denotes the lateral boundary of the cylinder QT .
It will be assumed throughout the paper that the exponents m(x), p(x) are continuous
in
with logarithmic module of continuity:
< m– = inf m(x) ≤ m(x) ≤ m+ = sup m(x) < ∞,
x∈ x∈
< p– = inf p(x) ≤ p(x) ≤ p+ = sup p(x) < ∞,
x∈ x∈
∀z, ξ ∈ , z – ξ  < ,
m(z) – m(ξ ) + p(z) – p(ξ ) ≤ ω z – ξ  ,
where
τ→+
lim sup ω(τ ) ln
τ
= C < +∞.
And we also assume that
(H) g : R+ → R+ is C function and satisfies
g() > , –
g(s) ds = l > ;
© 2013 Gao and Gao; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
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(H) there exists η > such that
g (t) < –ηg(t), t ≥ .
In the case when m, p are constants, there have been many results about the existence
and blowup properties of the solutions, we refer the readers to the bibliography given in
[–].
In recent years, much attention has been paid to the study of mathematical models of
electrorheological fluids. These models include hyperbolic, parabolic or elliptic equations
which are nonlinear with respect to gradient of the thought solution and with variable
exponents of nonlinearity; see [–] and references therein. Besides, another important
application is the image processing where the anisotropy and nonlinearity of the diffusion
operator and convection terms are used to underline the borders of the distorted image
and to eliminate the noise [, ].
To the best of our knowledge, there are only a few works about viscoelastic hyperbolic
equations with variable exponents of nonlinearity. In [] the authors studied the finite
time blowup of solutions for viscoelastic hyperbolic equations, and in [] the authors
discussed only the viscoelastic hyperbolic problem with constant exponents. Motivated
by the works of [, ], we shall study the existence and energy decay of the solutions to
Problem (.) and state some properties to the solutions.
The outline of this paper is the following. In Section , we introduce the function spaces
of OrliczSobolev type, give the definition of the weak solution to the problem and prove
the existence of weak solutions for Problem (.) with Galerkin’s method.
2 Existence of weak solutions
In this section, the existence of weak solutions is studied. Firstly, we introduce some
Banach spaces
Lp(x)( ) = u(x) : u is measurable in , Ap(·)(u) =
up(x) dx < ∞ ,
u p(·) = inf λ > , Ap(·)(u/λ) ≤ .
Lemma . [] For u ∈ Lp(x)( ), the following relations hold:
() u p(·) < (= ; > )+⇔ Ap(·)(u) < (= ;–> ); + –
() u p(·) < ⇒ u pp(·) ≤ Ap(·)(u) ≤ u pp(·); u p(·) > ⇒ u pp(·) ≥ Ap(·)(u) ≥ u pp(·);
() u p(·) → ⇔ Ap(·)(u) → ; u p(·) → ∞ ⇔ Ap(·)(u) → ∞.
Lemma . [, ] For u ∈ W,p(·)( ), if p satisfies condition (.), the p(·)Poincaré
inequality
u p(x) ≤ C ∇u p(x)
holds, where the positive constant C depends on p and .
Remark . Note that the following inequality
up(x) dx ≤ C
∇up(x) dx
does not in general hold.
Lemma . [] Let be an open domain (that may be unbounded) in RN with cone
property. If p(x) : → R is a Lipschitz continuous function satisfying < p– ≤ p+ < Nk and
r(x) : → R is measurable and satisfies
p(x) ≤ r(x) ≤ p∗(x) = NN–pk(px()x)
a.e. x ∈ ,
then there is a continuous embedding W k,p(x)( ) → Lr(x)( ).
The main theorem in this section is the following.
Theorem . Let u, u ∈ H( ), the exponents m(x), p(x) satisfy conditions (.)(.).
Then Problem (.) has at least one weak solution u : × (, ∞) → R in the class
u ∈ L∞ , ∞; H( ) ,
u ∈ L∞ , ∞; H( ) ,
u ∈ L , ∞; H( ) .
And one of the following conditions holds:
(A) < p– < p+ < max{N , NN–pp–– }, < m– < m+ < p–;
(A) max{, NN+ } < p– < p+ < , < m– < m+ < pp––– < .
Proof Let {wj}j∞= be an orthogonal basis of H( ) with wj
– wj = λjwj, x ∈ ,
wj = , x ∈ ∂ .
Vk = span{wi, . . . , wk} is the subspace generated by the first k vectors of the basis {wj}j∞=. By
normalization, we have wj = . Let us define the operator
t
Lu,
=
utt + ∇u∇
–
g(t – τ )∇u∇
dτ + utm(x)–ut
– αup(x)–u
dx,
∈ Vk.
For any given integer k, we consider the approximate solution
k
i=
uk =
cik(t)wi,
which satisfies
⎧⎨ Luk, wi = , i = , , . . . k,
⎩ uk() = uk,
ukt() = uk,
here uk = ik=(u, wi)wi, uk = ik=(u, wi)wi and uk → u, uk → u in H( ).
Here we denote by (·, ·) the inner product in L( ).
(.)
Problem (.) generates the system of k ordinary differential equations
⎧⎪ (cik(t)) = –λicik(t) + λi t g(t – τ )cik(τ ) dτ
⎪⎪⎨⎪⎪
⎪⎪⎪⎪⎪⎩ cik() = (u, wi),
+ ( ik=(cik(t)) , wi)m(x)–( ik=(cik(t)) , wi)
– α( ik= cik(t), wi)p(x)–( ik= cik(t), wi),
(cik()) = (u, wi), i = , , . . . , k.
By the standard theory of the ODE system, we infer that problem (.) admits a unique
solution cik(t) in [, tk], where tk > . Then we can obtain an approximate solution uk(t) for
(.), in Vk , over [, tk). And the solution can be extended to [, T ], for any given T > , by
the estimate below. Multiplying (.) (cik(t)) and summing with respect to i, we conclude
that
ddt uk + ∇uk – t
– α ddt p(x) ukp(x) dx = .
By simple calculation, we have
t
–
∇uk)(t) – g
∇uk(τ )∇uk(t) dx dτ +
uk m(x) dx
d
dt
t
∇uk (t) –
g(s) ds ∇uk
p(x) ukp(x) dx
(.)
(.)
(.)
(.)
t
t
(ϕ
∇ψ )(t)
ϕ(t – τ ) ∇ψ (t) – ∇ψ (τ ) dτ .
Combining (.)(.) and (H)(H), we get
ddt uk + ∇uk
+ –
= g
∇uk (t) – g(t) ∇uk –
∇uk)(t) – α
uk m(x) dx.
uk + ∇uk + –
t
+ (g
∇uk)(t) – α p(x) ukp(x) ≤ C,
Integrating (.) over (, t), and using assumptions (.)(.), we have
where C is a positive constant depending only on u H , u H .
Hence, by Lemma ., we also have
uk +
∇uk +
t
–
Next, multiplying (.) by (cik(t)) and then summing with respect to i, we get that the
following holds:
(.)
Note that
–
t
From Lemma ., we have
uk ≤ C
∇uk ,
d
dt
t
uk dx + ∇uk +
m(x)
uk
m(x)
= –
∇uk∇uk dx +
α ukp(x)–ukuk ≤ αε uk + αε ukp(x)–uk
≤ αε uk +
ukp(x)–uk
dx.
t
t
where C, C∗ are embedding constants. From (.)(.), we obtain that
uk dx + ( – ε – αεC) ∇uk + ddt
m(x) uk m(x)
≤ ε ∇uk +
( – l)g()
ε
∇uk(τ ) dτ
+ max C∗ (p––) ∇uk p–– , C∗ (p+–) ∇uk p+– .
Integrating (.) over (, t) and using (.), Lemma ., we get
uk dτ + ( – ε – αεC)
∇uk dτ +
≤ ε C + ( – l)g()T + C,
where C is a positive constant depending only on u H .
Taking α, ε small enough in (.), we obtain the estimate
m(x) uk m(x) dx
ukp(x)–uk dx =
uk(p(x)–) dx
≤ max
uk(p––) dx,
uk(p+–) dx
Hence, by Lemma ., we have
uk dτ +
m(x) uk m(x) dx ≤ C.
uk dτ + min
m+
m+ uk mm(–x), m+ uk m(x) ≤ C,
where C is a positive constant depending only on u H , u H , l, g(), T .
From estimate (.), we get
uk is uniformly bounded in L , T ; H( ) .
By (.)(.) and (.), we infer that there exist a subsequence {ui} of {uk} and a function
u such that
ui
ui
ui
ui
u
u
u
u
weakly star in L∞ , T ; H( ) ,
weakly in Lp– , T ; W ,p(x)( ) ,
weakly star in L∞ , T ; H( ) ,
weakly in L , T ; H( ) .
Next, we will deal with the nonlinear term. From the AubinLions theorem, see Lions [,
pp.], it follows from (.) and (.) that there exists a subsequence of {ui}, still
(.)
(.)
(.)
(.)
(.)
(.)
(.)
(.)
represented by the same notation, such that
ui → u
strongly in L , T ; L( ) ,
which implies ui → u almost everywhere in
× (, T ). Hence, by (.)(.),
uip(x)–ui
up(x)–u
for arbitrary T > . In view of (.)(.) and Lemma .. in [], we obtain
uk ()
u()
weakly in H( ),
uk ()
u ()
weakly in H( ).
Hence, we get u() = u, u() = u. Then, the existence of weak solutions is established.
(.)
(.)
(.)
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
YG carried out the study of existence of weak solutions for viscoelastic hyperbolic equations with variable exponents and
drafted the manuscript. WG participated in the discussion of existence of weak solutions for viscoelastic hyperbolic
equations with variable exponents.
Acknowledgements
Supported by NSFC (11271154) and by Department of Education for Jilin Province (2013439).
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