R E S E A R C H
Open Access
The stability of the anisotropic parabolic
equation with the variable exponent
Huashui Zhan
**Correspondence:
[email protected] School of Applied Mathematics, Xiamen University of Technology, Xiamen, 361024, P.R. China
Abstract
Consider the equation
ut= div
(
−−−−−−−−−−→
a(x)|∇u|p(x)–2∇u
)
=N
i=1
(
ai(x)|uxi|pi(x)–2u
xi
)
xi,withai(x),pi(x)∈C1(
),pi(x) > 1,ai(x)≥0. Basing on the weighted variable exponent
Sobolev space, a new kind of weak solutions of the equation is introduced. Whether the usual Dirichlet homogeneous boundary value condition can be imposed
depends on whetherai(x) is degenerate on the boundary or not. If some of{ai(x)}are
degenerate on the boundary, a partial boundary value condition is imposed. If every
ai(x) is degenerate on the boundary, by the new definition of a weak solution, the
stability of weak solutions can be proved without any boundary value condition.
MSC: 35K55; 35K92; 35K85; 35R35
Keywords: anisotropic parabolic equation; boundary value condition; the stability
1 Introduction
In recent years, the parabolic equation with variable nonlinearities
ut=div
|∇u|p(x)–∇u, (x,t)∈Q
T=×(,T), (.)
coming from the so-called electrorheological fluids theory (see [, ]), has been researched widely [–]. Here,⊂RNis a bounded domain with smooth boundary∂,p(x) > is a measurable function. Recently, we have studied the equation
ut=div
a(x)|∇u|p(x)–∇u, (x,t)∈QT, (.)
with
u(x, ) =u(x), x∈, (.)
u(x,t) = , (x,t)∈∂×(,T), (.)
in our previous works [, ]. Herea(x)∈C(), and whenx∈,a(x) > . The main dedi-cations of [, ] are that, ifa(x)|x∈∂= , then the stability of weak solutions can be proved
without the boundary value condition, provided that the diffusion coefficienta(x) satisfies some other restrictions.
In this paper, we consider an anisotropic parabolic equation of the type
ut=div
−−−−−−−−−−−→ a(x)|∇u|p(x)–∇u=
N
i=
ai(x)|uxi|
pi(x)–u
xi
xi, (x,t)∈QT, (.)
we denote that
p=min
x∈
p(x),p(x), . . . ,pN–(x),pN(x)
, p> ,
p=max
x∈
p(x),p(x), . . . ,pN–(x),pN(x)
.
We assume thatai(x)∈C(), and whenx∈,ai(x) > . If for everyi= , , . . . ,N,
ai(x) = , x∈∂, (.)
then we can expect that similar conclusions as those in [, ] are true. While only some of
ai(x) satisfy (.), the situations may be different. To see that, let us give a simple example to show the difference. LetN= ,p(x) =p(x)≡p, the domainbe a square,
=(x,x) : <x< , <x< .
Suppose thata(x)≡ and
a(x) = , x∈∂, (.)
consider the equation
ut= ∂ ∂x
a(x)|ux| p–u
x
+ ∂
∂x
|ux| p–u
x
. (.)
To obtain the stability of weak solutions of (.), initial value condition (.) is indispens-able. However, instead of the usual Dirichlet boundary value condition (.), we may con-jecture that only a partial boundary value condition
u(x,t) = , (x,t)∈×(,T), (.)
is required. Here
=
(x,x) : <x< ,x= ∪(x,x) : <x< ,x= . (.)
By the way, we would like to suggest some of important works related to the equations
N
i=
ai(x)|uxi|
pi(x)–u
xi
with
u= , x∈∂, (.)
wheref(λ,u) appears in different forms in different papers. Di Nardo and Feo [] gave the condition of the existence and uniqueness for problems (.)-(.); Mihailescu et al. [] studied the usual eigenvalue problems. Later, Radulescu [] studied the eigenvalue prob-lems with sign-changing potential; Di Castro [, ] studied the regularity; El Hamidi and Vétois [] studied the sharp Sobolev asymptotics; Vétois [] studied the blow-up phe-nomena, Konaté and Ouaro [] studied nonlinear anisotropic problems with bounded Radon diffuse measure and variable exponent and proved the existence and uniqueness of entropy solution. Also one can refer to some other related papers [–]. In a word, the elliptic anisotropic equations with the variable exponent have been given great at-tention recently, a lot of interesting results have been obtained. More or less beyond my imagination, there are few references related to the anisotropic parabolic equations with the variable exponent. We only have found one paper by Antontsev and Shmarev [], in which the existence of weak solutions was studied whena(x) = in equation (.).
The most important characteristic of equation (.) lies in that the diffusions coefficients
aiare different from one to another. After giving an existence result, the main aim of this paper is to give a complete classification of boundary value conditions and to study the corresponding stability of weak solutions.
2 The definition of weak solutions
Set
C+() =
h∈C() :min
x∈
h(x) > .
For anyh∈C+(), we define
h+=sup
x∈h(x), h –=inf
x∈h(x).
Letabe a measurable positive and a.e. finite function inRNsatisfying
(C) a∈L
loc()anda –p(x)– ∈
L loc();
(C) a–s(x)∈L()withs(x)∈( N
p(x),∞)∩[
p(x)–,∞).
For any p∈C+(), the definitions of the weighted variable exponent Lebesgue spaces
Lp(x)(a,) and the weighted variable exponent Sobolev spacesW,p(x)(a,), and the
fol-lowing lemmas, can be found in [].
Lemma . Denote
ρ(u) =
a(x)|u|p(x)dx for all u∈Lp(x)(a,).
Then
() ρ(u) > (= ; < )if and only ifuLp(x)(a,)> (= ; < ),respectively;
() IfuLp(x)(a,)> ,thenup –
Lp(x)(a,)≤ρ(u)≤ u
() IfuLp(x)(a,)< ,thenup +
Lp(x)(a,)≤ρ(u)≤ u p– Lp(x)(a,).
Lemma . If <p≤p(x)≤p<∞,then
Lp(x)(a,)∗=Lp(x)a(x)
–p(x),,
p(x)+
p(x)= ,
u(x)v(x)dx≤ku
Lp(x)([a(x)]–p(x),)vLp(x)(a,),
where[Lp(x)(a,)]∗is the conjugate space,p(x) = p(px()–x) .
Lemma . Let p,s∈C+(),and let(C)and(C)be satisfied.Then there exists the fol-lowing compact embedding:
W,p(x)(a,)→→Lr(x)()
provided that r∈C+()and≤r(x) <p∗s(x)for all x∈.Here,
ps(x) =
p(x)s(x) +s(x)
and
p∗s(x) =
⎧ ⎨ ⎩
p(x)s(x)N
(s(x)+)N–p(x)s(x) if ps(x) <N,
+∞ if ps(x)≥N.
The basic definition is as follows.
Definition . A functionu(x,t) is said to be a weak solution of equation (.) with initial value condition (.) if
u∈L∞(QT), ∂u
∂t ∈L (Q
T), uxi∈L
∞,T;Lpi(x)(a
i,)
, (.)
and for any functionϕ∈L(,T;Lp()),ϕ|
x∈∂= ,ϕxi∈L(,T;Lpi(x)(ai,)) such that
QT
∂u
∂tϕ+
N
i=
ai(x)|uxi|
pi(x)–u
xiϕxi
dx dt= . (.)
Initial value condition (.) is satisfied in the sense of
lim
t→ u(x,t) –u(x)dx= . (.)
In this paper, we first study the existence of a weak solution.
Theorem . If pi(x) > ,ai(x)satisfies conditions(C), (C),
u∈L∞(), uxi∈L
pi(x)(a
i,), (.)
Theorem . If for every≤i≤N,a–
pi(x)–
i (x)dx<∞,then initial-boundary value
problem(.)-(.)-(.)has a solution.Here,boundary value condition(.)is satisfied in the sense of trace.
Then we will study the stability of weak solutions according to the classification of boundary value conditions.
The first case is that we can impose the usual Dirichlet boundary value condition.
Theorem . If u and v are two solutions of equation(.)with the usual homogeneous value condition
u(x,t) =v(x,t) = , (x,t)∈∂×(,T), (.)
and with the initial values u(x),v(x),respectively,then
u(x,t) –v(x,t)dx≤
u(x) –v(x)dx. (.)
The second case is that we only can impose a partial boundary value condition. This case corresponds to that some of{ai(x)}are degenerate on the boundary, while others are not degenerate on the boundary. For simplicity, in this case, we assume that the domain is just ann-dimensional cube
=(x,x, . . . ,xN) : <x< , <x< , . . . , <xN<
. (.)
Assume thatk+l=N,I={i,i, . . . ,ik} ⊂ {, , . . . ,N},J={j,j, . . . ,jl} ⊂ {, , . . . ,N},I∩J=
∅, and
ai(x) > ,ai(x) > , . . . ,aik(x) > , x∈, (.)
aj(x) > ,aj(x) > , . . . ,ajl(x) > , x∈. (.)
Instead of the usual Dirichlet boundary value condition (.), in this case, only a partial boundary value condition is imposed.
u(x,t) = , (x,t)∈×(,T), (.)
where
= k
s=
{x∈∂:xis= or }.
Similar to the proof of Theorem ., if
a–
ps(x)–
s (x)dx<∞, s=j,j, . . . ,jl, (.)
Theorem . Besides(C)-(C), (.), (.)and(.),we assume thatsatisfies(.),
ajr(x) =ajr(xjr), r= , , . . . ,l, (.)
ajr(xjr) = , xjr= , or, (.)
and for large enough n,
n–
p+ jr
n
+
–n
ajr(x)
pjr(x) dxjr
p+
jr ≤c. (.)
If u and v are two solutions of equation(.)with the same partial boundary value condition
(.),and with the initial values u(x),v(x),respectively,then
u(x,t) –v(x,t)dx≤
u(x) –v(x)dx.
Certainly,we believe that,instead of conditions(.)-(.),only if
ajr(x) = , xjr= , or,
the same conclusion of Theorem.is also true.But we cannot obtain this result for the time being.
The last case is that all ai(x)are degenerate on the boundary.Then we can prove the
stability of weak solutions without any boundary value condition.
Theorem . If aisatisfies(C)-(C),and for large enough n,
n–
p+i \n
ai(x)pi(x)dx
p+ i ≤
c, (.)
let u,v be two solutions of equation(.)with the initial values u,v,respectively.Then
u(x,t) –v(x,t)dx≤
u(x, ) –v(x, )dx, (.)
wheren={x∈:
N
i=ai(x) >n}.
3 The proof of Theorems 2.5-2.7
Leta(x) satisfy (C), (C). Consider the regularized equation
ut=div
−−−−−−−−−−−→
a(x)|∇u|p(x)–∇u+εu, (x,t)∈QT, (.)
with the initial boundary conditions
u(x, ) =u(x), x∈, (.)
Proof of Theorem. Similar to [], we can easily prove that there is a solutionuεof initial-boundary value problem (.)-(.), and there is a constantconly dependent onuL∞()
but independent ofεsuch that
uεL∞(QT)≤c, uεtL(Q
T)≤c. (.)
Multiplying (.) byuεand integrating it overQT, we have
uεdx+
n
i=
QT
ai(x)|uεxi|
pi(x)dx dt+ε
QT
|∇uε|dx dt
=
u(x)dx, (.)
then
ε
QT
|∇uε|dx≤c (.)
and
N
i=
QT
ai(x)|uεxi|
pi(x)dx dt≤c. (.)
Hence, by (.), (.), (.), using Lemma ., there exists a function u and ann
-dimensional vector−→ζ = (ζ, . . . ,ζn) satisfying
u∈L∞(QT), ∂u
∂t ∈L (Q
T), ζi∈L
,T;L pi
(x)
pi(x)–a –pi(x) i ,
,
anduε→u a.e.∈QT,
uεu, weakly star inL∞(QT),
uε→u, inL,T;Lrloc(),
∂uε
∂t
∂u
∂t, inL (Q
T),
ε∇uε, inL(QT),
ai(x)|uεxi|
pi(x)–uε
xi ζi, inL
,T;Lpipi(x(x)–) a –pi(x) i ,
.
Herer< Np N–p.
Now, similar to the general evolutionaryp-Laplacian equation, we are able to prove that (the details are omitted here)
lim
t→ u(x,t) –u(x)dx= ,
and
N
i=
QT
ai(x)|uxi|
pi(x)–u
xiϕxidx dt=
N
i=
QT
for any functionϕ∈L(,T;Lp()),ϕ|
x∈∂= ,ϕxi∈L(,T;Lpi(x)(ai,)).
Then we have
QT
∂u
∂tϕ+
N
i=
ai(x)|uxi|
pi(x)–u
xiϕxi
dx dt= . (.)
Thususatisfies equation (.) with initial value (.) in the sense of Definition ..
Remark . In general, we only can obtain (.). It seems impossible to prove that, for every giveni,
QT
ai(x)|uxi|
pi(x)–u
xiϕxidx dt=
QT
ζi(x)ϕxidx dt.
Lemma . If for any given i∈ {, , . . . ,N},a–
pi(x)–
i (x)dx<∞,then
|uxi|dx≤c. (.)
Proof
QT
|uxi|dx dt
=
{(x,t)∈QT;a
pi(x)–
i |uxi|≤}
|uxi|dx dt+
{(x,t)∈QT;a
pi(x)–
i |uxi|>}
|uxi|dx dt
≤
QT a–
pi(x)–dx dt+
Q
ai(x)|uxi|
pi(x)dx dt
≤c.
By this lemma, if for every ≤i≤N,a–
pi(x)–
i (x)dx<∞, then
|∇u|dx≤c. (.)
Thus, we can define the trace ofuon the boundary∂. Accordingly, Theorem . is ob-viously true.
Now, we will prove Theorem .. For any given positive integern, letgn(s) be an odd function, and
gn(s) =
⎧ ⎨ ⎩
, s>n,
nse–ns
, ≤s≤n.
Clearly,
lim
n→gn(s) =sgn(s), s∈(–∞, +∞). (.)
Proof of Theorem. Letuandvbe two weak solutions of equation (.) with the same boundary value condition (.) and with the initial valuesu(x, ),v(x, ), respectively. By a process of limit, since (.)u=von the boundary, we can chooseϕ=gn(u–v) as the test function, then
gn(u–v) ∂(u–v)
∂t dx
+ N
i= ai(x)
|uxi|
pi(x)–u
xi–|vxi|
pi(x)–v
xi
(uxi–vxi)g
n(u–v)dx
= . (.)
Thus
lim
n→∞ gn(u–v) ∂(u–v)
∂t dx= d
dtu–vL(), (.)
ai(x)
|uxi|
pi(x)–u
xi–|vxi|
pi(x)–v
xi
(uxi–vxi)g
n(u–v)dx≥. (.)
Now, letn→ ∞in (.). Then
d
dtu–vL()≤.
It implies that
u(x,t) –v(x,t)dx≤
u(x) –v(x)dx, ∀t∈[,T).
4 The stability of the partially boundary value conditions
Proof of Theorem. Letuandvbe two weak solutions of equation (.) with the initial valuesu(x, ),v(x, ), respectively.
For ≤r≤l, let
φn,jr(xjr) = ⎧ ⎨ ⎩
ifxjr >
n, or –xjr>
n,
najr(xjr) ifxjr ≤n, or –xjr≤n.
By (.), (.), (.), lr=φn,jr(xjr)gn(u–v)(x) = when x ∈∂, we can choose l
r=φn,jr(xjr)gn(u–v) as the test function, then
l
r=
φn,jrgn(u–v)
∂(u–v) ∂t dx
+ N
i= ai(x)
|uxi|
pi(x)–u
xi–|vxi|
pi(x)–v
xi
·(uxi–vxi)g
n(u–v) l
r=
φn,jrdx
+ N
i= ai(x)
|uxi|
pi(x)–u
xi–|vxi|
pi(x)–v
xi
gn(u–v)
l
r=
φn,jr
xi
Thus lim n→∞ l r=
φn,jrgn(u–v) ∂(u–v)
∂t dx= d
dtu–vL() (.)
and
ai(x)
|uxi|
pi(x)–u
xi–|vxi|
pi(x)–v
xi
·(uxi–vxi)g
n(u–v) l
r=
φn,jrdx≥. (.)
Moreover, sinceφn,jr(xjr) is only dependent on the variablexjr, we have
N
i= ai(x)
|uxi|
pi(x)–u
xi–|vxi|
pi(x)–v
xi
gn(u–v)
l
r=
φn,jr xi dx = k s= ais(x)
|uxis|pis(x)–uxis–|vxis|pis(x)–vxis
gn(u–v)
l
r=
φn,jr xis dx + l r= ajr(x)
|uxjr|pjr(x)–uxjr–|vxjr|pjr(x)–vxjr
gn(u–v)
l
r=
φn,jr xjr dx = l r= ajr(x)
|uxjr|pjr(x)–uxjr–|vxjr|pjr(x)–vxjr
gn(u–v)
l
r=
φn,jr
xjr
dx. (.)
Clearly,
l
r=
φn,jr xjr = l
r=,r=r φn,jrφ
n,jr(xjr),
and|φn,jr(xjr)|=najr(xjr) whenxjr<n or –xjr<n, in other places, it is identical to zero.
Accordingly, for anyjr∈ {j,j, . . . ,jl}, if we notice that
= (, )×(, )× · · · ×(, ),
by conditions (.)-(.), we have
ajr(x)
|uxjr
| pjr
(x)–ux
jr–|vxjr| pjr
(x)–vx
jr
gn(u–v)
l
r=
φn,jr
xjr
dxjr
= n + n
ajr(xjr)
|uxjr
| pjr(x)–
uxjr –|vxjr|
pjr(x)– vxjr
gn(u–v)
×
l
r=,r=r φn,jrφ
n,jr(xjr)dxjr
≤cn n + n
≤cn|uxjr |
pjr
(x)–+|∇vx
jr| pjr
(x)– Lqjr(x)([ajr
(xjr)] –pjr
(x),(,n)∪(–n,))
×ajr
(xjr)Lpjr(x)(ajr (xjr),(,
n)∪(–n,))
. (.)
By Lemma ., we have
najr
(xjr)Lpjr(x)(ajr (xjr),(,
n)∪(–n,))
≤n n + n
ajr(xjr)ajr(xjr) pjr
(x)dx
p+ jr
≤cn –p+
jr n + n
ajr
(xjr) pjr
(x)dx
p+
jr ≤c. (.)
Then, by (.)-(.), we have
ajr(x)
|uxjr
| pjr
(x)–ux
jr–|vxjr| pjr
(x)–vx
jr
×(u–v)gn(u–v)
l
r=
φn,jr
xjr
dxjr
≤c n + n
ajr(xjr)|uxjr| pjr
(x)dxj
r q+ jr +c n + n
ajr(xjr)|vxjr|pjr(x)dxjr
q+
jr, (.)
which goes to asn→ ∞. Now, letn→ ∞in (.). Then
d
dtu–vL()≤.
It implies that
u(x,t) –v(x,t)dx≤
|u–v|dx, ∀t∈[,T).
5 The stability without the boundary value condition
Proof of Theorem. Letuandvbe two weak solutions of equation (.) with the initial valuesu(x, ) andv(x, ), respectively, but without any boundary value condition.
Letn={x∈:
N
i=ai(x) >n}, and
φn(x) =
⎧ ⎨ ⎩
ifx∈n,
nNi=ai(x) ifx∈\n.
We can chooseχ[τ,s]φngn(u–v) as the test function, whereχ[τ,s]is the characteristic
func-tion of [τ,s)⊆[,T), then
s
τ
φngn(u–v) ∂(u–v)
∂t dx dt
+ N i= s τ ai(x)
|uxi|
pi(x)–u
xi–|vxi|
pi(x)–v
xi
(uxi–vxi)g
n(u–v)φn(x)dx dt
+ N i= s τ ai(x)
|uxi|
pi(x)–u
xi–|vxi|
pi(x)–v
xi
(uxi–vxi)gn(u–v)φnxidx dt
= . (.)
As usual, we have
ai(x)
|uxi|
pi(x)–u
xi–|vxi|
pi(x)–v
xi
(uxi–vxi)g
n(u–v)φn(x)dx≥. (.)
At the same time, sinceut∈L(QT), using the Lebesgue dominated theorem, we have
lim
n→∞
s
τ
φn(x)gn(u–v) ∂(u–v)
∂t dx dt
=
|u–v|(x,s)dx–
|u–v|(x,τ)dx. (.)
Thus, we only need to deal with the last term on the left-hand side of (.). Obviously, φnxi=n(
N
j=aj(x))xiwhenx∈\n, in other places, it is identical to zero. By condition
(.), we have
ai(x)
|uxi|
pi–u
xi–|vxi|
pi–v
xi
φηxign(u–v)dx
=
\n ai(x)
|uxi|
pi(x)–u
xi–|vxi|
pi(x)–v
xi
φnxign(u–v)dx
≤n \n
ai(x)
|uxi|
pi(x)–+|v
xi|
pi(x)– N j= aj(x)
xi
gn(u–v)
dx
≤cn \n
ai(x)
|uxi|
pi(x)+|v
xi|
pi(x)dx
q+ i
\n ai(x)
N j= aj(x)
xi
pi(x) dx
p+i
≤c
\n
ai(x)|uxi|
pi(x)dx
q+ i
+
\n
ai(x)|vxi|
pi(x)dx q+ i ×
η \nai(x)
N j= aj(x)
xi
pi(x) dx
p+i
≤c \n
ai(x)|uxi|
pi(x)dx
q+i
+c \n
ai(x)|vxi|
pi(x)dx
q+i
. (.)
Then
lim η→
τs ai(x)
|uxi|
pi(x)–u
xi–|vxi|
pi(x)–v
xi
φnxign(u–v)dx dt
≤clim
η→ \nai(x)|uxi|
pi(x)dx
q+ i
+
\n
ai(x)|vxi|
pi(x)dx
q+ i
= . (.)
Now, letη→ in (.). Then
u(x,s) –v(x,s)dx≤
u(x,τ) –v(x,τ)dx, (.)
by the arbitrariness ofτ, we have
u(x,s) –v(x,s)dx≤
u(x) –v(x)dx.
6 Conclusion
Different from the usual evolutionaryp(x)-Laplacian equation, the anisotropic parabolic equation is considered in this paper. Due to the anisotropic character, it admits that there are different diffusion coefficients corresponding to different directions. If in some direc-tions the diffusion coefficients are degenerate on the boundary, while in other direcdirec-tions they are not degenerate, how to give a suitable partial boundary value condition to match the equation is a very interesting problem. The paper first researches this problem. By introducing a new kind of weak solutions, which are based on the weighted variable ex-ponent Sobolev spaces, the existence and stability of the weak solutions are proved in this paper.
Acknowledgements
The paper is supported by NSF of Fujian Province (Grant No. 2015J1092), supported by SF of Xiamen University of Technology (Grant No. XYK201448), China.
Competing interests
The author declares that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 28 July 2017 Accepted: 6 September 2017
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