#### Existence of nonnegative nontrivial periodic solutions to a doubly degenerate parabolic equation with variable exponent

Existence of nonnegative nontrivial periodic solutions to a doubly degenerate parabolic equation with variable exponent
Zhongqing Li 0
Wenjie Gao 0
0 College of Mathematics, Jilin University , Changchun, 130012 , PR China
The authors investigate a degenerate parabolic equation with delay and nonlocal term, which describes slow diffusive processes in physics or biology. The existence of a nonnegative nontrivial periodic solution is obtained through the use of the Leray-Schauder degree method. MSC: Primary 35D05; secondary 35K55
degenerate parabolic equation; periodic solution; variable exponent; topological degree; De Giorgi iteration
1 Introduction
(x, t) ∈ T ,
x ∈ .
employed for some technological applications, such as medical rehabilitation equipment
and shock wave absorber.
When p(x) is a constant and m > , p > , the model describes the slow diffusion process
in physics, which has been extensively investigated; see [–]. For example, in [], the
authors studied the following doubly degenerate parabolic equation with logistic periodic
sources:
∂u – div ∇um p–∇um = uα a – buβ .
∂t
∇ σ um + u + η
2 Preliminaries and the regularized problems of (1.1)
u(x) p(x) dx < ∞ ,
equipped with the following Luxemburg norm:
|u|Lp(x)( ) := inf λ > :
dx ≤ .
≤ |u|Lp(x)( )
≤ max
uv dx ≤
i.e. the embedding Lp(x)( ) → Lp(x)( ) is continuous.
If q ∈ C( ) and ≤ q(x) < p∗(x), for any x ∈
Lq(x)( ) is continuous and compact. Here
, then the embedding W,p(x)( ) →
p∗(x) :=
⎩ +∞,
p(x) ≥ N .
We next define the weak solutions to problem (.).
=
– auϕ + uϕ
and ϕ(·, t)|∂ = for t ∈ [, T ].
As in [], we introduce the following regularized problem:
⎧⎪ ∂∂ut – div{(|∇(σ um + u)| + η) p(x)– ∇(σ um + u)}
⎪⎪⎪⎨⎪ = [a – K (ξ , t)u(ξ , t – τ ) dξ ]u,
⎪⎪ u(·, t)|∂ = ,
⎪⎪⎪⎩ u(·, ) = u(·, T ),
=
∇ σ umη + u η + η
and ϕ(·, t)|∂ = for t ∈ [, T ].
(σ , f ) → u η = G η(σ , f ),
a.e. t ∈ (, T ),
3 A priori estimates to the regularized problem
First of all, the following modified De Giorgi iteration lemma will be useful (we give a proof
in the Appendix).
Lemma . (Iteration lemma) Suppose ϕ(t) is a nonnegative and nonincreasing function
on [K, +∞), it satisfies
ϕ(h) ≤
ϕ(k + d) = ,
≤ Ku,
mq
Proof Step . Multiplying (.) by u η , with any q > . Integrating over
and noticing that
function such that
∂u
∂t
u(·, t)|∂ = .
mq + dt
≤ K
–
Since |∇umη|p(x) ≥ |∇umη|p – , we deal with the second term on the left-hand side of (.)
as follows.
≥ q
≥ q
= q
mq + dt
≤ K
dx – q
p– + q –
dx – q
dx + q
p– + q –
dx + q
Combining (.) and (.), we have
and Young’s inequality with to deduce
dx + q
≤ K
≤ C
≤
dx + C( )C mp––m–
m(p–+q–)
dx + C( )C p– .
mq+(x, t) dx ≤ C,
u η
p– + q –
≤ K
dx dt + q
Choosing and appropriately, we have from (.) and (.)
–
for any q > , where C depends on q, p , m, and .
dx dt ≤
dx dt ≤ C.
considering (.), we obtain
Similarly to (.), we obtain
dx dt ≤ C,
mq+(x, t) dx dt ≤ C,
u η
mq+(x, t) dx ≤ C.
u η
From (.) and (.), we conclude
mq+(x, t) dx ≤ C + C(t – t),
u η
mq+(x, τ ) dx =
u η
mq+(x, τ + T ) dx ≤ C + CT .
u η
We finally arrive at
for any q > , where C depends on q, p–, m, T and .
Step . Let
Ak(t) = x ∈ ; u η(x, t) > k ,
Ak(t) ,
∇ umη + u η + η
≤ K
∇ umη + u η + η
m(u η – k)+m–∇(u η – k)+ p(x) dx
∇(u η – k)+m p(x) dx
∇(u η – k)m p– dx – Ak( ) .
≤ K
∇ umη + u η + η
p(x)–
∇ umη + u η ∇(u η – k)+m dx dt
∇ umη + u η + η
p(x)–
∇ umη + u η ∇(u η – k)+m dx dt
Substituting (.) into (.), we have
dx ≤ K
≤ S
p–(N+p–)
if p– < N ,
where S is the Sobolev embedding constant, and
r =
≤ K
≤ C
Nr–N–p–
≤ C Ak( ) r(N+p–)
Let Jk( ) =
Nr–N–p–
Utilizing Young’s inequality with , we obtain from (.)
Nr–N–p–
Nr–N–p–
≤ C Jk + C( )μk(p––)(N+p–) + Cμkp– .
Nr–N–p–
Upon choosing appropriately, one obtains
Jk( ) ≤ C μ (p––)(N+p–) + μ p–
k k
Nr–N–p–
For any h > k > , it is easy to see
Jk( ) ≥ Ah( ) (h – k)rm.
The relationships (.) and (.) above imply that
deg u – T η , u+ , BR, = ,
where u+ = max{u, }.
Proposition . Assume that a ∈ L∞(QT ), K ∈ L∞(QT ). If u η solves u = G η(σ , ρf (u+) +
– σ ), for some σ ∈ [, ] and ρ ∈ [, ], then u η ≥ for any (x, t) ∈ QT . Moreover, if u η = ,
then u η > in QT .
⎧⎨ – u = μu, x ∈ ,
⎩ u = , x ∈ ∂ ,
γp+ = max γp(x) = p–p–– ,
M = K L(QT ) + p+–
γp– = min γp(x) = p+p–+ ,
Cp+,p(x) max |∇e|| | p+
× max
γp±–
a L(QT ) + | |T γp± T γp±
= a –
∇ σ umη + u η + η
∇ σ umη + u η + η
= –
:= (R).
(R) =
QT
∇ σ umη + u η + η
∇ σ umη + u η + η
∇ σ umη + u η + η
(R) ≤
≤
=
× max
Since r < r ≤ ( m ) m– and < < , we have σ mumη– +
≤ mumη– + < + < . Hence
Noting that γ ± < and using Hölder’s inequality, we have
p
≤ T
Integrating (.) over [, T ] and noting (.), we get
× max
QT
QT
∇ σ umη + u η p(x)
u η Lm∞+(QT ) + u η L∞(QT ) a L(QT )
≤ rm+ + r
a L(QT ) + rm + r | |T
≤ r a L(QT ) + | |T .
Substituting this inequality into (.), we have
× max r a L(QT ) + | |T
× max
p+– γ–
Substituting (.) into (.) and noticing that p ≤ p+–, we get
(R) ≤ p+–
× max
a L(QT ) + | |T γp± T γp±
× max
|∇φ| dx dt + p+–
Step . We claim
≤ p+–
× max
≤ p+–
× max
× max
γp±–
a L(QT ) + | |T γp± T γp±
e dx dt
γp±–
a L(QT ) + | |T γp± T γp±
Cp+,p(x) max |∇e|| | p+
γp±–
a L(QT ) + | |T γp± T γp±
Now the definition of r and (.) yield
r ≤
≤ r,
4 Existence of nontrivial nonnegative solution to (1.1)
Theorem . Assume K (x, t) ≥ for a.e. (x, t) ∈ QT and T QT ae dx dt > μ. Then
problem (.) has a nontrivial nonnegative periodic solution.
deg u – G η , f u+ , BR\Br, =
∇ umη + u η + η
Multiplying (.) by umη + u η, integrating over QT and noting the T -periodicity of u η
and the boundedness of u η, we have
≤ C
∇ umη + u η + η
∇ umη + u η dx dt
umη+ + uη dx dt ≤ M,
[,T]∩{t:|∇umη|Lp(x)( )>}
[,T]∩{t:|∇umη|Lp(x)( )≤}
[,T]∩{t:|∇umη|Lp(x)( )>}
∇umη p(x) dx dt + T
∇umη p(x) dx dt + T ≤ M + T .
∇ umη + u η + η
∇ umη + u η dx dt
∇ umη + u η p(x) dx dt.
From (.) and (.), we have
∇ umη + u η p(x) dx dt ≤ C.
≤ C,p(x) max
∇ umη + u η p(x) dx
So umη ∈ Lp– (, T ; W,p(x)( )) and umη is uniformly bounded in the space Lp– (, T ;
W,p(x)( )). Thus, up to subsequence if necessary, we may assume that umη um ∈
Lp– (, T ; W,p(x)( )). In what follows, our main goal is to prove that u is a weak solution of
problem (.).
Step . The following relation is obvious:
∇ umη + u η p(x) dx dt p± T – p± .
QT
QT QT
∇ umη + u η L( ) dt
≤ C,p(x) max
∇ umη + u η dx dt ≤ C.
In the following, we prove
First, denote
∇ umη + u η + η
p(x)– p(x)
∇ umη + u η p(x)– dx dt ≤ C.
K–p = min Kp(x),
K+p = max Kp(x),
K–p = min Kp(x),
K+p = max Kp(x),
K–p = min Kp(x),
K+p = max Kp(x).
A straightforward computation shows that
≤ p+– K+p
≤ p+– K+p K+p
≤ Mp
QT
By the p(x)-Hölder’s inequality, we have
p(x)– dx dt
p(x)– dx dt
m m
∇ u η + u η + ∇ u η + u η
p(x)– dx dt
p(x)– dx dt
dx dt +
p(x)– dx dt .
× max
p(x)– dx dt
× max
∂t
H ∈ (L p(x)– (QT ))N such that
–
in Lp (, T ; Lp(x)( )). Thus, we have
Step . Using a method analogous to [], we get
– ,p(x)
of and η. Since umη is uniformly bounded in Lp (, T ; W
=
∂t
+ H∇ϕ – auϕ + uϕ
∇um p(x)–∇um∇ϕ dx dt =
H∇ϕ dx dt.
– (∇v), ∇ umη + u η – ∇v
which gives
≤
∇ umη + u η + η
p(x)–
∇v ∇ umη + u η – v dx dt
p(x)–
∇ umη + u η + η
∇ umη + u η dx dt
∇ umη + u η + η
Multiplying the equation
= a –
∇ umη + u η + η
∇ umη + u η + η
∇ umη + u η dx dt
∇ umη + u η ∇v dx dt
∇v∇
H∇um dx dt =
H∇v dx dt +
|∇v|p(x)–∇v∇ um – v dx dt
QT QT
Thus, (.) and (.) imply
≤
≤
≤
≥
QT
Combining (.) with (.), we obtain
∇ϕ dx dt.
H – ∇u
m p(x)–
∇u
m ∇ϕ dx dt.
H – ∇u
m p(x)–
∇u
m ∇ϕ dx dt.
Proof of Lemma . Define the following sequence:
ks = k + d –
, s = , , , . . . ,
s = , , , . . . .
ϕ(ks+) ≤
By induction, one has
ϕ(ks) ≤
, s = , , , . . . ,
ϕ(ks+) ≤
where r > is to be chosen. In fact, if (.) is right, then
duced the result.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and approved the final version.
Acknowledgements
The authors would like to thank the anonymous referees for their valuable comments on and suggestions regarding the
original manuscript. This work was supported by National Science Foundation of China (11271154), by Key Lab of
Symbolic Computation and Knowledge Engineering of Ministry of Education and by the 985 program of Jilin University.
1. Ru˚žicˇka , M: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics, vol. 1748 . Springer, Berlin ( 2000 )
2. DiBenedetto , E: Degenerate Parabolic Equations . Universitext. Springer, New York ( 1993 )
3. Fragnelli , G, Nistri , P, Papini, D: Non-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms . Discrete Contin. Dyn. Syst . 31 , 35 - 64 ( 2011 )
4. Ladyženskaja , OA, Solonnikov, VA, Ural'ceva, NN: Linear and Quasilinear Equations of Parabolic Type . Translations of Mathematical Monographs, vol. 23. Am. Math. Soc., Providence ( 1968 ) Translated from the Russian by S . Smith
5. Sun , J, Y in, J, Wang, Y : Asymptotic bounds of solutions for a periodic doubly degenerate parabolic equation . Nonlinear Anal . 74 , 2415 - 2424 ( 2011 )
6. Tsutsumi , M: On solutions of some doubly nonlinear degenerate parabolic equations with absorption . J. Math. Anal. Appl . 132 , 187 - 212 ( 1988 )
7. Wang , J, Gao, W: Existence of nontrivial nonnegative periodic solutions for a class of doubly degenerate parabolic equation with nonlocal terms . J. Math. Anal. Appl . 331 , 481 - 498 ( 2007 )
8. Allegretto , W, Nistri , P: Existence and optimal control for periodic parabolic equations with nonlocal terms . IMA J. Math. Control Inf . 16 , 43 - 58 ( 1999 )
9. Huang , R, Wang, Y , Ke , Y : Existence of non-trivial nonnegative periodic solutions for a class of degenerate parabolic equations with nonlocal terms . Discrete Contin. Dyn. Syst ., Ser . B 5, 1005 - 1014 ( 2005 )
10. Nakao , M: Periodic solutions of some nonlinear degenerate parabolic equations . J. Math. Anal. Appl . 104 , 554 - 567 ( 1984 )
11. Pang , PYH, Wang , Y , Yin , J: Periodic solutions for a class of reaction-diffusion equations with p-Laplacian . Nonlinear Anal., Real World Appl . 11 , 323 - 331 ( 2010 )
12. Diening , L, Harjulehto , P, Hästö , P, Ru˚žicˇka , M: Lebesgue and Sobolev Spaces with Variable Exponents . Lecture Notes in Mathematics, vol. 2017 . Springer, Heidelberg ( 2011 )
13. Fragnelli , G: Positive periodic solutions for a system of anisotropic parabolic equations . J. Math. Anal. Appl . 367 , 204 - 228 ( 2010 )
14. Guo , B, Gao , W: Study of weak solutions for parabolic equations with nonstandard growth conditions . J. Math. Anal. Appl . 374 , 374 - 384 ( 2011 )
15. Wu , Z, Yin , J, Wang, C: Elliptic & Parabolic Equations . World Scientific , Hackensack ( 2006 )
16. Simon , J: Compact sets in the space Lp(0, T; B). Ann. Mat. Pura Appl. (4) 146 , 65 - 96 ( 1987 )
17. Porzio , MM, Vespri , V: Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations . J. Differ. Equ . 103 , 146 - 178 ( 1993 )
Cite this article as: Li and Gao: Existence of nonnegative nontrivial periodic solutions to a doubly degenerate parabolic equation with variable exponent . Boundary Value Problems 2014 , 2014 : 77