R E S E A R C H
Open Access
Remark on the Cauchy problem for the
evolution
p-Laplacian equation
Liangwei Wang
1*, Jngxue Yin
2and Jinde Cao
3*Correspondence:
1School of Mathematics and
Statistics, Chongqing Three Gorges University, No. 666, Tian Xing Road, Wanzhou District, Chongqing, 404100, China
Full list of author information is available at the end of the article
Abstract
In this paper, we prove that the semigroupS(t) generated by the Cauchy problem of the evolutionp-Laplacian equation∂∂ut – div(|∇u|p–2∇u) = 0 (p> 2) is continuous form
a weightedL∞space to the continuous spaceC0(RN). Then we use this property to
reveal the fact that the evolutionp-Laplacian equation generates a chaotic dynamical system on some compact subsets ofC0(RN). For this purpose, we need to establish
the propagation estimates and the space-time decay estimates for the solutions first.
MSC: 35B40; 35K65
Keywords: chaos; evolutionp-Laplacian equation; Cauchy problem; propagation estimate; decay estimate
1 Introduction
In this paper, we consider the Cauchy problems of the evolutionp-Laplacian equation
∂u
∂t –div
|∇u|p–∇u= in (,∞)×RN, (.)
u(x, ) =u inRN, (.)
where p> and the nonnegative initial value u belongs to the weighted L∞ space Wσ(RN)≡ {ϕ;|x|σϕ(x)∈L∞(RN)}with the normϕ(·)Wσ(RN)=| · |σϕ(·)L∞(RN).
The evolutionp-Laplacian equation, as an important class of parabolic equations, comes from the compressible fluid flows in a homogeneous isotropic rigid porous medium. Com-paring to the classical linear heat equation, this equation, to a certain extent, reflects even more exactly physical reality [, ]. So the studies of this equation have attracted a large number of mathematicians and remarkable progress has been achieved []. Among all of the progress, the semigroup method given by Bénilan and Véron [, ] is a successful and effective method to treat the evolutionp-Laplacian equation.
Using the concepts of dynamical systems to study partial differential equations has also attracted much attention in recent decades. Such concepts, like orbit,ω-limit, attractor and chaos, were introduced to investigate the finite dimensional instances of dynamical systems of ordinary differential equations. In , it was Vázquez and Zuazua [] who first successfully used theω-limit set of the rescaled solutionsu(t·,t) to study the com-plicated asymptotic behavior of solutions for the problem (.)-(.). Subsequently, we [– ] investigated the complicated asymptotic behavior of solutions of the porous medium
equation by using theω-limit set of the rescaled solutionstμu(tβ·,t) with <μ,β<∞. Usingω-limit set to research other partial differential equations, one can refer to [–]. The theory of chaos on some partial differential equations has also been well developed since the pioneering work of Li, Wiggins, Shatah and McLaughlin; see [, ]. Li [] re-vealed the fact that around the Silnikov homoclinic orbits, the existence of chaos on Euler equations can be proved by constructing horseshoe. Cazenave, Dickstein and Weisslerit [] found that the discrete dynamical system generated by the heat equation in some sense is an example of chaos. In , Battellia and Fečkan proved the existence of homo-clinic and chaotic solutions of beam equations in []. In , Lan and Li [] found that the numerical Melnikov integral can be used as a tool for both predicting and controlling chaos on Euler equations. The chaos theory on other partial differential equations, one can see [–].
Inspired by the above papers, especially by [], we focus our attention on the semigroup and the chaos theory for the evolutionp-Laplacian equation. To overcome the difficulties caused by the degeneracy and nonlinearity of this equation, we first establish the propa-gation estimate and the decay estimate for the solutions of the problem of (.)-(.). By using the propagation estimate and the decay estimate of the solutionsu(x,t), we see that the semigroupS(t) generated by the evolutionp-Laplacian equation is continuous from the compact setBσM,+to the spaceC(RN), where
BσM,+≡ϕ∈Wσ
RN;ϕW
σ(RN)≤Mandϕ≥
with the weak-star topology ofWσ(RN). Then using the definition of chaos, the commu-tative relation between the delation operatorDσ
λand the semigroupS(t), we find that, for any fixedλ> , the map
Fλσ ≡D σ λS
λ–
defined on compact setS()BσM,+is chaotic. Here the delation operatorDσ
λis defined as
Dσ
λϕ(x) =λ
σ σ(p–)+pϕλ
σ(p–)+px
forϕ∈Lloc(RN).
The rest of this paper is organized as follows. In the next section, we give some def-initions and some propositions of the solutions for the problem (.)-(.). Section is devoted to giving the propagation speed estimate and decay estimate for the solutions of problem (.)-(.). The continuity of the semigroupS(t) is consider in Section . We re-veal the fact that the problem (.)-(.) generates a chaotic dynamical system on certain compact subsets ofC(RN) in Section .
2 Preliminaries
conditions is thatFis transitive; that is, for all non-empty open subsetsUandVofXthere exists a natural numberksuch that
Fk(U)∩V=∅.
In a certain sense, transitivity is an irreducibility condition. The second of Devaney’s con-ditions is that the periodic points ofF form a dense subset ofX. The final condition is called sensitive dependence on initial conditions;Fverifies this property if there is aδ> such that, for every pointx∈Xand every neighborhood ofx, there exist a pointy∈
and a nonnegative integerksuch that
dFk(x),Fk(y)>δ.
This sensitivity condition captures the idea that in chaotic systems minute errors in exper-imental readings eventually lead to large scale divergence. Sensitive dependence on initial conditions is thus widely understood as being the central idea in chaos.
Definition .(Devaney’s definition of chaos []) Let (X,d) be a metric space. A con-tinuous map
F:X→X
is said to be chaotic onXif . Fis transitive;
. the periodic points ofFare dense inX;
. Fhas sensitive dependence on initial conditions.
To discuss chaotic dynamical system in the evolutionp-Laplacian equation, we need to adopt some concepts as that in [, , ]. Forf ∈Lloc(RN), we define
|||f|||r=sup R≥rR
–N(pp–)+– p
{|x|≤R}
f(x)dx
and
(f) = lim r→∞|||f|||r.
The spaceXis given by
X≡ϕ∈LlocRN;|||ϕ|||<∞
with the norm||| · |||. Hence it is a Banach space. The spaceXis defined as
X≡
ϕ∈X;(ϕ) = .
Definition . Foru∈X, a measurable functionu=u(x,t) defined inQT=RN×(,T), T> , is a weak solution of the problem (.)-(.) if
u∈Cloc
(,T);LlocRN∩Llocp ,T;Wloc,pRN
and for every test functionϕ∈C∞(QT), the following identity holds:
QT
uϕt–|∇u|p–∇u· ∇ϕ
dxdt= ,
andu(x,t) satisfies the initial-value equation (.) in the following sense:
u(x,t)→u(x) inLloc
RN
ast→.
The existence and uniqueness of weak solution of the problem (.)-(.) for the initial valueu∈Xis shown in [, ], and these solutions satisfy the following proposition.
Proposition .([, , ]) For every u∈X,there exist a time T=T(u)and a weak solution u(x,t)of the problem(.)-(.)in QT.Moreover,for
<t≤T(u) =C(u)–(p–), (.)
the solutions u(x,t)are Hölder continuous in QT≡(,T)×RN and satisfy the following estimates:
u(·,t)r≤C|||u|||r (.)
and
u(x,t)≤Ct–N(p–)+N pR p p–|||u
|||
p N(p–)+p
r if r≤R and|x| ≤R, (.)
where BRis the closed ball inRN with the radius R.
From the above proposition, the following proposition can easily been proved; see [, ].
Proposition . ([, ]) Let u(x,t)be the nonnegative weak solution of the problem
(.)–(.).Given x∈RN,if
B(x) =sup R>
R–N(pp–)+– p
|x–y|<R
u(y) dy<∞,
then,for all <t≤CB(x)–(p–),
Proposition .([]) If the initial value u∈X,one can easily see that T(u) =∞and thus these solutions are global.Moreover,the evolution p-Laplacian equation generates a bound semigroup in Xgiven by
S(t) :u→u(x,t). (.)
Moreover,if≤q≤ ∞and u∈Lq(RN)⊂X,then S(t)is a contraction bounded semi-group in Lq(RN).
For <σ<N,λ> andu∈X, the space-time dilationσλis defined as
σλS(t)u (x) =Dσλ
Sλtu (x) =λ
σ
σ(p–)+puλσ(p–)+ px,λt,
whereS(t) is the semigroup given by (.) and the dilationDσ
λis given by
Dσ
λϕ(x) =λ
σ σ(p–)+pϕλ
σ(p–)+px
forϕ∈Lloc(RN). In this paper, we consider our problem in theL∞weight spaceWσ(RN)≡
{ϕ∈L
loc(RN);|x|
σϕ(x)∈L∞(RN)} with <σ <N. We equip this space with the norm ϕWσ(RN)=| · |σϕ(·)L∞(RN). Hence it is a Banach space. Meanwhile, one easily verifies that, for <σ<N,Wσ(RN)⊂X. The closed convex set
BσM,+≡ϕ∈Wσ
RN;ϕW
σ(RN)≤Mandϕ≥
with the weak-star topology ofWσ(RN) is compact and separable. Thus it can be meter-izable. We use the symboldMσ,∗to denote this metric. So, for allM≥, the metric space (BσM,+,dMσ,∗) is compact, hence complete and separable.
In the rest of this section, we study the relation between the semigroup operatorS(t) and the dilation operatorDσλ. Supposeu(x,t) is a weak solution of the problem (.)-(.) with initial valueu∈X. Let
v(x,t) =λσ
S(t)u (x) =λ
σ
σ(p–)+pSλtu
λ
σ(p–)+px. (.)
So
∂v
∂t= div
|∇v|p–· ∇v.
This mean thatv(x,t) is a weak solution of the following Cauchy problem: ⎧
⎨ ⎩ ∂v
∂t = div(|∇v|
p–· ∇v) inRN×(,∞),
v(x, ) =λ
σ σ(p–)+pu
(λ
σ(p–)+px) =Dσ
λu(x) inRN.
Note that
So,u∈Wσ(RN), hence
v(x,t) =S(t)Dσλu
(x).
Then we get the following commutative relation between the semigroup operatorS(t) and the dilation operatorDσ
λ:
σλS(t)u =Dσλ
Sλtu =S(t)
Dσ
λu . (.)
For any fixedλ> and <σ<N, we now define the mapFσ λ :S()B
σ,+
M →S()B
σ,+ M as
Fλσ ≡D σ λS
λ– =S
–
λ
Dσλ.
3 Some estimates
In this section, we first estimate the propagation speed of solutions for Problem (.)-(.) with the nonnegative initial valueu∈Wσ(RN). For this purpose, we need some concepts. Let
d(x)≡supR;u(y) = a.e. inBR(x)
be the distance fromxto the support ofuand let us introduce the following symbol to
denote the positive set ofu(x,t) at timet:
(t)≡x∈RN;u(x,t) > .
We also define theρ-neighborhood of the set (t) as
ρ(t)≡
x∈RN;dx, (t)<ρ,
whered(x, (t)) is the distance fromxto (t).
Theorem . (Propagation estimate) Suppose < σ < N and the initial value u ∈ W+
σ(RN),i.e.,u≥and u∈Wσ(RN).Let u(x,t)be a nonnegative weak solution of the
problem(.)-(.).For any≤t<t<∞,then
(t)⊂ ρ(t–t)(t),
whereρ(t–t) =C(t–t)
σ(p–)+pu
p–
σ(p–)+p
Wσ(RN).
Proof Without loss of generality, we restrict our consideration to the caset= . Assume x∈RNwithd(x) > . Note first that, ifR<d(x), then
BR(x)
u(y) dy= . (.)
For anyr≥, letR=d(x) +r≥d(x). If|x|< R, then
Therefore,
R–N(pp–)+– p
BR(x)
u(y) dy≤CuWσ(RN)R– N(p–)+p
p–
BR()
|y|–σdy
=CuWσ(RN)R
–pp––σ≤Cu
Wσ(RN)d(x)
–pp––σ
. (.)
If|x| ≥R, then, for anyy∈BR(x),
|y| ≥ |x|–R≥R.
So for <σ<N, we have
|y|–σ ≤
R–σ.
Therefore,
R–
N(p–)+p p–
BR(x)
u(y) dy≤CuWσ(RN)R
–N(pp–)+– p–σ
BR(x) dy
=CuWσ(RN)R
–pp––σ
≤CuWσ(RN)d(x)
–pp––σ .
Combining this with (.) and (.), we have
B(x) =sup R>
R–
N(p–)+p p–
BR(x)
u(y) dy≤CuWσ(RN)d(x)
–(p–)p–σ+p
.
Therefore, Proposition . implies
u(x,t) = for all ≤t≤Cu–(Wp–)
σ(RN)d(x)
(p–)σ+p.
This means
(t)⊂ ρ(t)(),
whereρ(t) =Cu
p–
σ(p–)+p
Wσ(RN)t
(p–)σ+p. So we complete the proof of this theorem.
In the rest of this section, we pay our attention on the properties of the semigroupS(t).
Theorem . Letω∈C(RN \ {})be a homogeneous function of degree.Suppose <
σ<N and set u(x) =|x|–σω(x).It follows that S(t)u(x) =t–
σ
σ(p–)+pgt–σ(p–)+ px, (.)
where g(x)∈Cα(RN)and|x|σg(x) –ω(x)→as|x| → ∞.
Proof From (.) and the definition of the initial valueu, we have
σλ
S(s)u(x) =λ
σ
σ(p–)+pSλsu
λ
σ(p–)+px
=S(s)λσ(p–)+σ pu
λσ(p–)+ p· (x) =S(s)u
First notice that, for <σ<N,
ur=sup R≥rR
–N(pp–)+– p BR
u(x)dx≤Cr–σ–
p
p– → asr→ ∞.
So,
u∈X.
Therefore,
S(s)u∈C
α
,α(,∞)×RN
for some <α< ; see [, ]. In particular,
S()u(x)∈Cα
RN.
Now takings= ,λ=t andg(x) =S()u(x) in the expression (.), we obtain
S(t)u(x) =t–
σ
σ(p–)+pgt–σ(p–)+ px
and
g(x)∈CαRN.
The factS(t)u(x)∈C([,∞)×RN\ {, }) [, ] clearly implies that, for|x|= ,
t–σ(p–)+σ pgt–σ(p–)+ px=S(t)u
(x)→ϕ(x) =|x|–σω(x) =ω(x)
ast→. Let
y=t–σ(p–)+ px.
So
|y| → ∞ ast→.
Therefore,
|y|σg(y) –ω(y)→,
as|y| → ∞. We complete the proof of this theorem.
Theorem .(Space-time decay estimates) Given <σ<N and a constant M> ,there exists a constant C such that if u∈BσM,+(RN),then
S(t)u(x)≤C
tN(p–)+ p+|x|–
σ
, (.)
Proof Let
ϕ(x) =M|x|–σ and
g(x) =S()ϕ(x).
It follows from Theorem . that there exists a constantCsuch that
g(x)≤C +|x|–σ.
So by (.),
S(t)ϕ(x)≤Ctσ(p–)+ p+|x|–σ. By the comparison principle [, ], we get
S(t)u(x)≤S(t)ϕ(x)≤C
tσ(p–)+ p+|x|–
σ
.
So the proof of this theorem is complete.
4 Continuity of the semigroup
In this section, we first present the fact that the semigroup operatorS(t) are continuous from the metric spaceBσM,+to the spaceC(RN), which is the basis of the proof of our main
result.
Theorem . For any fixedτ > ,let <σ <N and M> ,then, for any t>τ, S(t) : (BσM,+,dMσ,∗)→C(RN)is continuous.In particular,S()is a continuous map from the metric space(BσM,+,dMσ,∗)to the space C(RN).
Proof Let{un}n≥⊂BσM,+andu∈BσM,+such that
dMσ,∗(un,u)→ asn→ ∞.
Therefore,
un→u inD
RNasn→ ∞. (.)
Notice also that
unWσ(RN)≤M for alln≥.
For anyt,R> , let
R(t) =R+ +CM
p–
So, for alln≥,
supp( –χR(t))un ⊂
x∈RN;|x|>R+ +CM
p–
σ(p–)+pt
σ(p–)+p,
whereχR(t) is the cut-off function defined on the ballBR(t)+≡ {x∈RN;|x| ≤R(t) + } relative to the ballBR(t)≡ {x∈RN;|x| ≤R(t)},i.e.,
χR(t)(x)∈C
∞
RN, ≤χ
R(t)(x)≤
and
χR(t)(x) =
⎧ ⎨ ⎩
forx∈BR(t), forx∈/BR(t)+.
By Lemma ., we get
suppS(t)( –χR(t))un ⊂
x∈RN;|x| ≥R+ .
This means that the value ofS(t)un(x) inBRis only dependent on the initial valueunin
BR(t). In other words, forx∈BR,
S(t)un(x) =S(t)[χR(t)un](x). (.)
For any> , taking the aboveRlarge enough, the inequality (.) clearly implies that, if
|x| ≥R, then S(t)un(x)<
for alln≥. (.)
By (.) and the hypothesisun,u∈Wσ(RN), we get, for <q<Nσ,
χR(t)un χR(t)u inL
qRN.
So,
S(τ)[χR(t)un]S(τ)[χR(t)u] inL
qRN.
In particular,
S(τ)[χR(t)un]→S(τ)[χR(t)u] inD(R).
From (.), we know that there exists a constantC(τ) such that S(τ)[χR(t)un]L∞(RN)≤C(τ) for alln≥.
Therefore,
S(τ)[χR(t)un]→S(τ)[χR(t)u] weakly-star inL
From the regularity of the semigroup operatorS(t), we obtain, for any fixedt>τ> , S(t)[χR(t)un] –S(t)[χR(t)u]L∞
loc(RN)→
asn→ ∞. This implies, via (.), that
S(t)[un] –S(t)[u]L∞(BR)=S(t)[χR(t)un] –S(t)[χR(t)u]L∞(BR)→.
By (.), there exists an integerNsuch that ifn>N, then
S(t)[un] –S(t)[u]L∞(RN)
≤S(t)[un] –S(t)[u]L∞(BR)+S(t)[un]L∞(RN\B
R)+S(t)[u]L∞(RN\BR) <.
So we complete the proof of this theorem.
5 Chaotic dynamical system
For anyλ> andM> , we recall that the mapFσ λ:S()B
σ,+
M →S()B
σ,+
M is defined as
Fσ λ =D
σ λS
λ– .
The ideas of the following theorem come from []. We also need a lemma which appeared in [].
Lemma . If f :X→X is transitive and has dense periodic points,then F has sensitive dependence on initial conditions.
Theorem . Ifλ> and M> ,then the map Fσ λ :S()B
σ,+
M →S()B
σ,+
M is chaotic.
Proof We first verify that the mapFλσ is well defined. By Theorem ., the uniqueness the-orem for the solution of (.)-(.) withu∈X, we see thatS() is a continuous, injective,
surjective map from the compact Hausdorff spaceBσM,+onto the Hausdorff spaceS()BσM,+. So we see thatS() is a homeomorphism from the compact set BσM,+ to S()BσM,+ by the fact that a continuous, injective, surjective map of the compact Hausdorff space onto the Hausdorff space is a homeomorphism. This means thatS()BσM,+is a compact set. For any
ϕ∈BσM,+, we have
Fλσ
S()ϕ =S()Dσλϕ ∈S()B σ,+ M .
So,Fσ
λ is well defined. We will divide the rest proof into four steps.
. The map Fσ
λ is a continuous map on the compact set S()B σ,+ M .
For any sequence{vk}k≥⊂BMσ,+andv∈BσM,+, if
S()vk→S()v inC
then
Sλvk=Sλ– S()vk →Sλ– S()v =S
λv inC
RN
ask→ ∞. Here we have used the facts thatλ> andS(t) is a bounded contractive con-tinuous semigroup inL∞(RN) fort> (Proposition .). Therefore,
Fλσ
S()vk –Fλσ
S()v L∞(RN)
=Dσλ
Sλvk –Dσλ
Sλv L∞(RN) =Sλvk–S
λvL∞(RN)→ ask→ ∞. This means that the mapFσ
λ is a continuous map on the compact setS()B σ,+ M . . The periodic points of Fσ
λ are dense in S()B σ,+ M .
For anyk∈Z+,v∈Bσ,+
M andλ> , letvkbe defined as
vk(x) =
+∞
n=–∞
χn(x)λ
nσk
σ(p–)+pvλσ(p–)+nk px,
where
χn(x) =
⎧ ⎨ ⎩
ifx∈An={λ
(n–)k
σ(p–)+p ≤ |y|<λ (n+)k
σ(p–)+p},
ifx∈/An.
Note that, for allk> ,
vkWσ(RN)≤ vWσ(RN)≤M
and
Dσ λ
k vk=vk.
So
vk∈BσM,+
and
Fλσ k
S()vk=Fλσ k–
S()Dσλvk =· · ·=S()
Dσλ k
vk) =S()vk.
Therefore,S()vkis a periodic point ofFσ
λ. Note also that
vk(x) =v(x) ifx∈A.
This means that
hence
vk→v inBσM,+,dσM,∗ask→ ∞.
Using Theorem ., we get
S()vk→S()v inC
RNask→ ∞.
This proves that the periodic points ofFσ
λ are dense inS()B σ,+ M . . The map Fσ
λ is topologically transitive.
For any open subsetsUandVofS()BσM,+, there exist a constantε> and two functions
ϕ,φ∈BσM,+such that
Bε
S()ϕ⊂U (.)
and
Bε
S()φ⊂V. (.)
Now let
U(x) = ∞
n=
Dσλ– n
χn(x)φ(x) +Dσλ– n–
χn–(x)ϕ(x) , (.)
where
λn=λ
n+
and χn(x) is the cut-off function defined on the set An≡ {x∈ RN;λ
–n
σ(p–)+p <|x| <
λ
n
σ(p–)+p}relative to the setAn
–≡ {x∈RN;λ
–n+
σ(p–)+p <|x|<λ n–
σ(p–)+p}. Notice that, for all
n≥,
suppDσλ– n
χn(x)φ(x) ⊂
x∈RN;λ
n+–n
σ(p–)+p <|x|<λ n++n
σ(p–)+p (.)
and
suppDσλ– n–
χn–(x)ϕ(x) ⊂
x∈RN;λ
n–n+
σ(p–)+p <|x|<λ n+n–
σ(p–)+p. (.)
So, forn≥, from (.) and (.), we have
suppDσ λ–n
χn(x)φ(x) ∩suppDσλ– n+
χn+(x)ϕ(x) =∅
and
suppDσλ– n
χn(x)φ(x) ∩suppDσλ– n–
Here we have used the fact that, ifi>j≥, then
i+–i> j++j.
So
U∈BσM,+.
Note also that
DσλnU=φ inAn– and
Dσλn+U=ϕ inAn forn≥. So
DσλnU
n→∞
−−−→φ weakly-star inWσ
RN
and
Dσ λn+U
n→∞
−−−→ϕ weakly-star inWσ
RN.
It follows from Theorem . that
S()Dσ λnU
n→∞
−−−→S()φ inC
RN
and
S()Dσλn+U
n→∞
−−−→S()ϕ inC
RN.
Then we conclude from the definition ofFσ λ that
Fσ λ
n–
S()U =S()
Dσ λ
n U
n→∞
−−−→S()φ inC
RN (.)
and
Fλσ n
S()U =S()
Dσλ n+
U −−−→n→∞ S()ϕ inC
RN. (.)
So for the aboveε> , there existsN∈Nsuch that ifn≥N, then
Fλσ n–
S()U ∈Bε
S()φ⊂V
and
Fλσ n
S()U ∈Bε
These results mean that
FλσU∩V=∅.
So we complete the proof of thatFσ
λ is topologically transitive. So the mapFσ
λ is chaotic by Devaney’s definition of chaos and Lemma ., and the proof
of this theorem is complete.
6 Conclusion
In this paper, our concern here is the properties of the solutions to the Cauchy problem of the evolutionp-Laplacian equation with the initial valueubelonging to a weightedL∞
spaces, and we get the following results:
I. The propagation estimate and the decay estimate for the solutions of the problem of (.)-(.) have been established.
II. The semigroupS(t)generated by the evolutionp-Laplacian equation is continuous from the compact setBσM,+to the spaceC(RN).
III. The mapFσ
λ generated by the evolutionp-Laplacian equation is chaotic on the compact subsetS()BσM,+ofC(RN).
In future work, we hope to continue to study the properties of solutions for the Cauchy problem of the evolutionp-Laplacian equation with the initial valueubelonging to other
Banach spaces, especially the unbounded spaces.
Acknowledgements
This research was supported by National Natural Science Foundation of China (11071099 and 11371153), Chongqing Fundamental and Frontier Research Project (cstc2016jcyjA0596) and the Chongqing Municipal Commission of Education (KJ1401003, KJ1601009), Innovation Team Building at Institutions of Higher Education in Chongqing (CXTDX201601035), Chongqing Municipal Key Laboratory of Institutions of Higher Education (No. C16).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The main ideal of continuity of the semigroupS(t) and chaos was proposed by LW and JY. The propagation estimate and space-time decay estimates were proved by JC. All authors read and approved the final manuscript.
Author details
1School of Mathematics and Statistics, Chongqing Three Gorges University, No. 666, Tian Xing Road, Wanzhou District,
Chongqing, 404100, China. 2School of Mathematical Sciences, South China Normal University, No. 55, West of Zhong
Shan Road, Tianhe District, Guangzhou, 510631, China. 3School of Mathematics, and Research Center for Complex
Systems and Network Sciences, Southeast University, No. 2, Southeast University Road, Jiangning District, Nanjing, 210996, China.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 8 May 2017 Accepted: 10 July 2017
References
1. Ladyzhenskaya, OA: The mathematical theory of viscous incompressible flow. Phys. Today17(2), 57-58 (1964) 2. Hutter, K: Mathematical foundation of ice sheet and ice shelf dynamics. A physicist’s view. In: Free Boundary
Problems: Theory and Applications, pp. 192-203. Chapman & Hall/CRC, Boca Raton (1999) 3. Dibenedetto, E: Degenerate Parabolic Equations. Springer, New York (1993)
4. Bénilan, P: Equations d’évolution dans un espace de Banach quelconque et applications. PhD thesis, Univ. Orsay (1972) (in French)
5. Véron, L: Effets régularisants de semi-groupes non linéaires dans des espaces de banach. Ann. Fac. Sci. Toulouse1(2), 171-200 (1971) (in French)
7. Yin, J, Wang, L, Huang, R: Complexity of asymptotic behavior of the porous medium equation inRN. J. Evol. Equ.
11(2), 429-455 (2011)
8. Wang, L, Yin, J, Jin, C:ω-Limit sets for porous medium equation with initial data in some weighted spaces. Discrete Contin. Dyn. Syst., Ser. B1(1), 223-236 (2013)
9. Wang, L, Yin, J: Critical exponent for the asymptotic behavior of rescaled solutions to the porous medium equation. Electron. J. Differ. Equ.2017, Article ID 10 (2017)
10. Wang, L, Yin, J: Proper spaces for the asymptotic convergence of solutions of porous medium equation. Nonlinear Anal., Real World Appl.38, 261-270 (2017)
11. Cazenave, T, Dickstein, F, Weissler, FB: Universal solutions of a nonlinear heat equation onRN. Ann. Sc. Norm. Super.
Pisa, Cl. Sci.2(1), 77-117 (2003)
12. Wang, L: Relation between solutions and initial values for evolutionp-Laplacian equation. Appl. Math. Lett.69, 55-60 (2017)
13. Cazenave, T, Dickstein, F, Weissler, FB: Multi-scale multi-profile global solutions of parabolic equations inRN. Discrete
Contin. Dyn. Syst., Ser. S5, 449-472 (2012)
14. Carrillo, JA, Vázquez, JL: Asymptotic complexity in filtration equations. J. Evol. Equ.7, 471-495 (2007) 15. Li, Y, McLaughlin, DW, Shatah, J, Wiggins, S: Persistent homoclinic orbits for a perturbed nonlinear Schrödinger
equation. Commun. Pure Appl. Math.49(11), 1175-1255 (1996) 16. Li, Y: Chaos in partial differential equations. Physics983(1), 93-115 (2009)
17. Li, Y: Chaos in PDEs and Lax pairs of Euler equations. Acta Appl. Math.77(2), 181-214 (2003)
18. Cazenave, T, Dickstein, F, Weissler, FB: Universal solutions of the heat equation onRn. Discrete Contin. Dyn. Syst., Ser.
A9(5), 1105-1132 (2012)
19. Battelli, F, Feˇckan, M: Chaos in the beam equation. J. Differ. Equ.209(1), 172-227 (2005)
20. Lan, Y, Li, YC: On the dynamics of Navier-Stokes and Euler equations. J. Stat. Phys.132(1), 35-76 (2008) 21. Li, Y, Wiggins, S: Homoclinic orbits and chaos in discretized perturbed NLS systems: part I. Homoclinic orbits.
J. Nonlinear Sci.7(3), 211-269 (1997)
22. Li, Y, Wiggins, S: Homoclinic orbits and chaos in discretized perturbed NLS systems: part II. Symbolic dynamics. J. Nonlinear Sci.7(4), 315-370 (1997)
23. Li, Y: Persistent homoclinic orbits for nonlinear Schrödinger equation under singular perturbation. Dyn. Partial Differ. Equ.1, 87-123 (2004)
24. Cazenave, T, Dickstein, F, Weissler, FB: Chaotic behavior of solutions of the Navier-Stokes system inRN. Adv. Differ.
Equ.10(4), 361-398 (2005)
25. Devaney, RL: An introduction to chaotic dynamical systems. Bull. Am. Math. Soc.16, 313-315 (1987)
26. DiBenedetto, E, Herrero, MA: On the Cauchy problem and initial traces for a degenerate parabolic equation. Trans. Am. Math. Soc.314(1), 187-224 (1989)
27. Vázquez, JL: Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type. Oxford University Press, Oxford (2006)
28. Kamin, S, Vázquez, JL: Fundamental solutions and asymptotic behaviour for thep-Laplacian equation. Rev. Mat. Iberoam.4(2), 339-354 (1988)
29. Zhao, JN, Yuan, HJ: Lipschitz continuity of solutions and interfaces of the evolutionp-Laplacian equation. Northeast. Math. J.8(1), 21-37 (1992)
