R E S E A R C H
Open Access
Extinction properties of solutions for a fast
diffusion equation with nonlocal source
Zhong Bo Fang
*and Mei Wang
*Correspondence:
[email protected] School of Mathematical Sciences, Ocean University of China, Songling Road 238, Qingdao, 266100, P.R. China
Abstract
In this paper, we investigate extinction properties of nonnegative nontrivial solutions for an initial boundary value problem of a fast diffusion equation with a nonlocal source in bounded domain. By using the super- and sub-solution and the energy methods, we obtain some sufficient conditions for extinction and non-extinction of the weak solutions and give corresponding decay estimates which depend on the initial data, coefficients, and domains.
MSC: 35K65; 35K20; 35B40
Keywords: fast diffusion equation; extinction; non-extinction; decay estimate
1 Introduction
We consider the fast diffusion equation with a nonlocal source
ut=um+auq
up(y,t)dy, x∈,t> , (.)
subject to the homogeneous Dirichlet boundary and initial conditions
u(x,t) = , x∈∂,t> , (.)
u(x, ) =u(x), x∈, (.)
where <m< ,p≥,q> ,a> , and⊂RN(N≥) is a bounded domain with smooth
boundary andu∈L∞()∩W,() is a nonnegative function. The symbols · p(p≥)
and||denote theLp-norm and the measure of, respectively.
Equation (.) describes the fast diffusion of concentration of some Newtonian fluids through porous media or the density of some biological species in many physical phe-nomena and biological species theories. It has been known that the nonlocal source term presents a more realistic model for population dynamics, see [–]. In the nonlinear dif-fusion theory, there exist obvious differences among the situations of slow (m> ), fast ( <m< ), and linear (m= ) diffusions. For example, there is a finite speed propagation in the slow and linear diffusion situations, whereas an infinite speed propagation exists in the fast diffusion situation.
Recently, many scholars have been devoted to the study on blow-up and extinction prop-erties of solutions for nonlinear diffusion equations with nonlocal terms, see [–]. Ex-tinction of a function is a phenomenon for which there exists a finite timeT> such that
the solution is nontrivial on (,T) and thenu(x,t)≡ for all (x,t)∈×[T, +∞). In this case,T is called an extinction time. It is also an important property of solutions to non-linear parabolic equations which have been studied by many researchers. For example, Kalashnikov [] studied the Cauchy problem of a semilinear parabolic equation with an absorption term
ut= u–λup, x∈RN,t> ,
and obtained extinctions as well as localization and finite propagation properties of the solutions. Evans and Knerr [] investigated extinction behaviors of the solutions for the Cauchy problem of a semilinear parabolic equation with a fully nonlinear absorption term
ut(x,t) = u(x,t) –βu(x,t), x∈RN,t> .
Ferrieira and Vazquez [] studied extinction phenomena of the solutions for the Cauchy problem of a porous medium equation with an absorption term
ut=umxx–up, x∈R,t> ,
by using the analysis of a self-similar solution. By constructing a suitable comparison func-tion, Li and Wu [] considered the problem of a porous medium equation with a local source term
ut= um+λup, x∈,t> ,
subject to homogeneous Dirichlet boundary condition (.) and initial condition (.). They obtained some conditions for extinction and non-extinction of the solutions to the above equation and decay estimates. On extinctions of solutions to thep-Laplacian equa-tion or the doubly degenerate equaequa-tions, refer to [, ] and the references therein.
For equation (.) withp= andN> , Han and Gao [] showed thatq=mis a critical exponent for occurrence of extinction or non-extinction. Recently, Fang and Xu [] con-sidered equation (.) withp= and a linear absorption term, when the diffusion term was replaced withp-Laplacian operator in the whole dimensional space, and showed that the extinction of the weak solution is determined by competition of source and absorption terms. They also obtained the exponential decay estimates which depend on the initial data, coefficients, and domains. Thereafter, they obtained the same results for a class of nonlocal porous medium equations with strong absorption, see [].
Motivated by the mentioned works above, we study extinction behaviors of the solutions for problem (.)-(.) in the whole dimensional space. The main tools we use are the super- and sub-solution and the energy methods to obtain some sufficient conditions for extinction of the solutions, and we give exponential decay estimates which depend on the initial data, coefficients, and domains. In fact, the energy method has a wide application, especially for the equations that do not satisfy the maximum principle (cf.[]).
2 Preliminaries and main results
Due to the singularity of equation (.), problem (.)-(.) has no classical solutions in general, and hence it is reasonable to find a weak solution of the problem. To this end, we first give the following definition of a weak local solution.
Definition A functionu∈L∞(QT) is called a super-solution of problem (.)-(.) in
QTif the following conditions hold:
(i) u(x, )≥u(x)in,
(ii) u(x,t)≥on∂×(,T),
(iii) for everyt∈(,T)and every test functionξ,
u(x,t)ξ(x,t)dx≥
u(x)ξ(x, )dx
+
t
uξs+umξ+auq
up(y,t)dyξ(x,s)
dx ds,
where≤ξ∈C(QT∩C,(QT),ξ
t,ξ∈L(QT),ξ≥,ξ∂×(,T)= , and
QT=×(,T). A sub-solution can be similarly defined by replacing the inequality sign≥in the above conditions with≤. A function is called a local solution of (.)-(.) if it is both super- and sub-solution for someT.
Letϕ(x) be the unique positive solution of the following linear elliptic problem:
– ϕ(x) = , x∈; ϕ(x) = , x∈∂. (.)
Throughout this paper, the constantsMandμare defined asM=supx∈ϕ(x) andμ=
ϕ
p
m(x)dx. The existence of local solutions can be obtained by utilizing the method of the standard regularization, and the regularities of the solutions can be derived by the ar-gument similar to that in []. Since the regularization procedure is important to showing the uniqueness of the solution to problem (.)-(.) for some special cases, we sketch the outline below.
Consider the regularized problem
ut=um+auq
up(y,t)dy, x∈, <t<T,
u(x,t) =
k, x∈∂, <t<T, (.)
u(x, ) =u(x) +
k, x∈,
whereT> can be chosen sufficiently small so that there exists a solutionukof (.) on
QTfor everyk∈Nanduk∞is bounded for allk. Furthermore,l≤ul≤ukfork<l, and
a super-solution (sub-solution) comparison theory holds for (.) (see []).
Since the sequence{uk}is monotone and bounded, we may defineU(x,t)≡limk→∞uk(x,
of (.)-(.), we then have
(u–uk)ξ(x,t)dx=
t
(u–uk)ξs+
um–umk ξ
+aξ(x,s)
uq
up(y,t)dy–uqk
upk(y,t)dy dx ds
+
k t
∂ξ
∂ndSxds–
k
ξ(x, )dx
≤
t
(u–uk)ξs+um–umkξ
+aξ(x,s)
uq
up(y,t)dy–uqk
upk(y,t)dy dx ds
=
t
(u–uk)ξs+
um–umk ξ+aξ
up(y,t)dyuq–uqk
+aξ(x,s)uqk
up(y,t) –ukp(y,t)dydx ds.
Here, we have used the fact ∂ξ∂n≤ on∂to derive this inequality. Define the functions ,F, andGas
(x,t) =m
θu+ ( –θ)ukm–dθ,
F(x,t) =q
θu+ ( –θ)ukq–dθ,
and
G(x,t) =p
θu+ ( –θ)ukp–dθ,
and letϕu=um, we then have
=
t
ϕu
θu+ ( –θ)uk
dθ≥
ϕu
uk
dθ
≥
x∈,s∈[,t]max{minu+uk(x,t)∈QT}≥p≥l
ϕu(p)≡η,
and we also haveF,G∈L∞. We can choose sequences of smooth functions n,Fn, andGn
such that n→ ,Fn→F, andGn→GinL(Qt) and also find a constantγ for which η< n<γ for alln.
Letξ≡ξndenote the solution of the problem
ξs+ nξ+aξMp||Gn+aMqμFn= , x∈,t∈(,T), ξ(x,t) =χ(x), x∈,
where χ is a smooth function and has compact support in with ≤χ ≤. From
∇ξ(·,t), ξ,∇ξ≤C. With thisξ, we have
(u–uk)(x,t)χ(x,t)dx
≤
t
(u–uk)( – n)ξ+aξMp||(G–Gn) +aMqμ(F–Fn)dx ds.
Sinceu,uk∈L∞(QT) for some constantMandu≤v, we deduce that
(u–uk)(x,t)χ(x,t)≤K
n– ξ+aMp||G–Gnξ
+aMq||μF–Fn
for some constantK. Lettingn→ ∞, we obtain
(u–uk)(x,t)χ(x,t)dx≤.
Choosingχ=sign(u–uk)+, we have
(u–uk)+(x,t)dx≤.
This implies thatu≤ukonQt. Sincet<Tis arbitrary, we haveu≤uk≤UonQT. To establish the uniqueness of solution to problem (.)-(.) for some special cases, it only remains to prove that the reverse inequality is also true. The desired result can be seen in the following proposition.
Proposition If p+q=m and aμMmq < ,whereμand M are constants defined in(.),
then the nonnegative solution of(.)-(.)is unique.Furthermore,if v is a sub-solution of
(.)-(.),we have v≤u.
Proof We only need to show the uniqueness of solution.
Letube an arbitrary solution of (.)-(.), and letukdenote the solution of (.). We then have
(uk–u)ξ(x,t)dx
=
t
(uk–u)ξs+
umk –um ξ
+aξ(x,s)
uqk
upk(y,t)dy–uq
up(y,t)dy dx ds
–
k t
∂ξ
∂ndSxds+
k
Choosing the unique positive solutionϕ(x) of (.) as a testing functionξ(x,t) and noticing thatp+q=m,aμMmq < , and ≤u≤uk≤M, we get
(uk–u)ξ(x,t)dx=
t
–(uk–u) +aμMmqMp+q– p+qdx ds
–
k t
∂ξ
∂ndSxds+
k
ξ(x, )dx
≤ –
k t
∂ξ
∂ndSxds+
k
ξ(x, )dx.
Lettingk→ ∞, we obtain
(U–u)ξ(x,t)dx≤.
The above inequality together with the factU≥uguarantees thatu≡U.
The following comparison principle and lemmas will play a crucial role in what follows, but the proofs of them are simple, and so we omit them here (see [, ]).
Proposition (Comparison principle) Let u and v be nonnegative bounded super- and sub-solution of(.)-(.),respectively,with u≥δfor someδ> .If v(x, )≤u(x, ),then v≤u on QT.
Lemma Suppose that k andαare positive constants,with k< .If y(t)is a nonnegative absolutely continuous function on[,∞)solving the problem
dy dt+αy
k≤, t≥; y()≥,
we then have the decay estimate
y(t)≤y–k() –α( –k)t
–k, t∈[,T∗),
y(t)≡, t∈[T∗, +∞),
where T∗=yα–(–k()k).
Lemma [] Suppose that <k<m≤and y(t)is a nonnegative function solving the problem
dy dt+αy
k≤γym, t≥; y() =y
> ,
whereαandγ are nonnegative constants.Ifγ <αyk–m,there existsη> such that≤
y(t)≤ye–ηtfor all t≥.
3 Extinction and non-extinction
Theorem If p+q=m and aμMmq < ,then for any nonnegative initial data u,the
unique solution u(x,t)of(.)-(.)vanishes in finite time.
Proof We will prove this theorem by constructing a proper super-solution. Letϕ(x) be the unique positive solution of the following elliptic problem:
–ϕ(x) = , x∈; ϕ(x) = , x∈∂. (.)
LetM=supx∈ϕ(x),δ=infx∈ϕ(x), and letμ=
ϕ
p m
(x)dx. We know from the comparison principle of elliptic problems that ϕ(x) <ϕ(x) for x∈, and δ > and
μ<μ. SinceaμM q
m < andϕ(x) is continuous, we can find a domain⊂such that
aμM q m
< . Letg(t) be the positive solution of the following problem:
g(t) = – –aμM q m
M
m
gm(t), g() =A,
whereA> is a constant large enough so thatu≤Aϕ
m
(x) for allx∈. Since <m< , the functiong(t) vanishes in a finite timeT> . Setv(x,t) =g(t)ϕ
m
(x), and then it can be easily seen thatv(x,t) also vanishes from the timeT.
On the other hand, one can directly verify thatv(x,t) is a super-solution of (.)-(.) for any fixedT such that <T<T. There exist two positive constantsCandCsuch that
C≤v(x,t)≤C. By Proposition , we know thatu(x,t)≤v(x,t) for any (x,t)∈×[,T]. SinceT<Tis arbitrary, one can see thatu(x,T)≡ for someT<T. Then it follows from Proposition thatu(x,t)≡ for allt≥T, which implies thatu(x,t) vanishes from
the timeT.
Theorem Suppose that p+q>m and a≤maxϕ
p+q m ||
δp+q–m .Then,for any nonnegative initial
data u,the unique solution u(x,t)of(.)-(.)vanishes in finite time.
Proof Letv(x,t) =g(t)ϕm(x), where ϕ(x) is the function solving (.). Then v(x,t) is a super-solution of (.)-(.) if and only if the following conditions (.), (.), and (.) hold:
gϕm(x)≥gm(t)ϕ+agp+q(t)ϕ p m
ϕmq dx, x∈, <t<T, (.)
g(t)ϕm(x)≥, x∈∂, <t<T, (.)
g()ϕm(x)≥, x∈. (.)
Letg(t) = –gmH(t), whereminx∈ϕ(x) = andH=maxϕm(x). One can see that the following condition is sufficient to guarantee (.):
gm(t)≥–g(t)ϕm(x) +agp+q(t)ϕ p m
ϕmq dx. (.)
Since
g(t)ϕm(x) =g
m(t)
H ϕ
m ≤ g
andv(x,t)≥δ, we may choose a constanta≤maxϕ
p+q m ||
δp+q–m such that
agp+q(t)ϕmp
ϕmq dx≤ g
m(t). (.)
Combining (.) with (.), it can be seen that (.) is true. Hence, ifu≤g()ϕ(x), the functionvis a super-solution of (.)-(.). From the definition ofg(t), we obtain the inte-gral equality
g
g(t)
ds gm =
t
H.
DefineT= Hggmds. We then havev> for ≤t<T andx∈, andv(x,T) = for
x∈. By Proposition , we conclude thatu≤vforx∈and <t<T.
Theorem Assume that p+q<m or p+q=m and aμMmq > .Then,for any nonnegative
initial data u,the maximal solution U(x,t)of(.)-(.)does not vanish in finite time.
Proof We will prove this theorem by constructing a suitable sub-solution. Setv(x,t) =
g(t)ϕm, whereϕ(x) is the unique positive solution of (.). It can be easily verified that
v(x,t) is a sub-solution of (.)-(.) forp+q<m, ifg(t) solves the problem
g(t) =M–m–gm(t) +aμM q mgp+q;
g() = , g(t) > (t> ).
Ifp+q=mandaμMmq > , chooseg(t) to be the solution of the problem
g(t) =M–maμM q
m – gm(t);
g() = , g(t) > (t> ).
Then v(x,t) is also a sub-solution of (.)-(.). Hence, it follows from the sub-solution
comparison principle thatU(x,t) > in×(,∞).
Remark It can be seen from Theorems - that when p+q<m or p+q=m and
aμMmq > , the maximal solutionU(x,t) is positive for allt> , which means that the effect of the source term is, in some sense, strong and the diffusion term cannot dominate
the source term. However, whenp+q>manda≤maxϕ
p+q m ||
δp+q–m orp+q=mandaμM q m < ,
the effect of the nonlocal source is a little weak, and the diffusion term may cause the non-negative solution of (.)-(.) to vanish in finite time, provided that the initial data are sufficiently small.
4 Decay estimates
Theorem Suppose <m< and m=p+q. ()When N= or,if||< [mγ–(N,)
a(m+) ]
–m or a< mγ
– (N,)
||–m(m+),then for any nonnegative initial data u, the unique solution u(x,t)of (.)-(.)vanishes in finite time,with the
following decay estimates:
u(·,t)≤
u–m–C
–m+
t
–m+
, t∈[,T);
u(·,t)≡, t∈[T, +∞),
whereγ> is an embedding constant,and Cand Twill be determined later. ()When N> ,
(a)ifNN–+≤m< ,and||< (γ– a )
+ N
or a< γ–
||+ N,then for any nonnegative initial data u,the unique solution u(x,t)of(.)-(.)vanishes in finite time,with the following decay
estimates:
u(·,t)++mm≤
u–+mm–C
– m
+m t
– +mm
, t∈[,T);
u(·,t)++mm≡, t∈[T, +∞),
whereγ> is an embedding constant,and Cand Twill be determined later;
(b)if <m<NN–+,and||< [msγ–(N,) a(m+s) ]
+ N
or a< msγ–(N,)
||+
N(m+s),then for any nonnegative initial data u, the unique solution u(x,t)of (.)-(.)vanishes in finite time,with the
following decay estimates:
u(·,t)++ss≤
u–+ms –C
–m+s +s t
–m++ss
, t∈[,T);
u(·,t)++ss≡, t∈[T, +∞),
where s=N( –m) – andγ> is an embedding constant,and Cand Twill be
deter-mined later.
Proof We first consider the caseN= or . Multiplying both sides of equation (.) byus
(s≥) and integrating the result over, we obtain
s+
d dt
us+dx+ ms (m+s)
∇um+s
dx=a
updx
uq+sdx. (.)
By Hölder’s inequality, we get the inequality
updx
uq+sdx≤ s–p–q–s
s up+q+s s ,
wheres≥ will be determined later. Choosings=s+ ≥, we get the inequality
s+
d dt
us+dx+ ms (m+s)
∇um+sdx≤a||+ss+–mum+s
By using the Sobolev embedding inequality, one can show that there exists an embedding constantγ(N,) such that
um+s s≤γ
(N,)∇u m+s
,
wheres≥ will be determined later,i.e.,
us(+mm+s)s
≤γ(N,)∇um+s
.
LetC=
mγ–(N,) (m+) –a||
–m
,s=
+m, and lets= . Thens≥, since <m< . From the
above inequality with the constantsands, we obtain the inequalitydtdu+Cum+≤ , ifC> ,i.e.,||< [
mγ–(N,)
a(m+) ]
–m. By Lemma , we then obtain its decay estimates,
withT=
u–m
C(–m+).
Secondly, we consider the caseN> .
(a) Formsuch thatNN–+≤m< , multiplying both sides of equation of (.) byumand
integrating the result over, we obtain
m+
d dt
um+dx+
∇umdx=a
updx
uq+mdx. (.)
By the Sobolev embedding and Hölder’s inequalities, one can show that
m+
d dtu
(+m)
+m +γ–|| N–
N –+mmum
+m≤a|| +mum
+m,
and hence we get
d dtu
+m
+m+Cu+mm≤,
providedC= (m+ )(γ–|| N–
N –+mm –a||+m) > ,i.e.,||< (γ
– a )
+ N
. By Lemma , we
then obtain its decay estimates, withT= u–+mm C(–+mm)
.
(b) Formsuch that <m<NN–+, multiplying both sides of equation (.) byus(s=N( –
m) – ) and integrating the result over, we obtain
s+
d dt
us+dx+ ms (m+s)
∇um+sdx=a
updx
uq+sdx. (.)
By the Sobolev embedding inequality and a suitables, one can show that
u
s+m
s+ =
us+m–NN– ≤γ
∇u s+m
. (.)
Using Hölder’s inequality, we have
updx
uq+sdx≤ –ms++sum+s
From inequalities (.) and (.), we get the inequality
s+
d dtu
(+s) +s +
ms
(m+s)γ –
us++ms ≤a||–
m+s
+sus+m +s.
Furthermore, we can have dtdu++ss+Cus++ms ≤, providedC = (s+ )(γ–(mms+s) – a||–m++ss) > ,i.e.,||< (msγ
– a(m+s))
+ N
. By Lemma , we then obtain its decay estimates,
withT=
u–m
+s
C–+ms
.
Theorem Suppose that <m< and p+q>m. ()When N= or,if
||<
mγ– (N,)
a(ue–αT)p+q–m(m+ )
–p–q
or
a< mγ
– (N,)
||–p–q(u
e–αT)p+q–m(m+ )
and usmall enough,the unique solution u(x,t)of(.)-(.)vanishes in finite time,with
the following decay estimates:
u(·,t)≤ue–
αt, t∈[,T
);
u(·,t)≤
u–m–C
–m+
t
–m+
, t∈[T,T);
u(·,t)≡, t∈[T, +∞),
whereγ> is an embedding constant,α is a suitable constant,and C,T,Twill be
determined later. ()When N> ,
(a)for m such that NN–+≤m< ,if
||<
γ–
(au++mme–αt)
p+q–m
+m
N+
N + m–p–q
+m
or
a< γ
–
||NN++ m–p–q
+m (u+m +me–αt)
p+q–m
+m
and usmall enough,the unique solution u(x,t)of(.)-(.)vanishes in finite time,with
the following decay estimates:
u(·,t)++mm≤u++mme–αt
, t∈[,T);
u(·,t)++mm≤
u–+mm–C
– m
+m t
– m
u(·,t)++mm≡, t∈[T, +∞),
whereγ> is an embedding constant,αis a suitable constant,and C,T,Twill be
determined later;
(b)for m such that <m<NN–+,if
||<
msγ–
a(m+s)(u
++sse–αt)
p+q–s
+s
–m++pm+q
or
a< msγ
–
||–m++pm+q(m+s)(u+s +se–αt)
p+q–s
+s
and usmall enough,the unique solution u(x,t)of(.)-(.)vanishes in finite time,with
the following decay estimates:
u(·,t)++ss≤u++sse–
αt, t∈[,T
);
u(·,t)++ss≤
u–+ms –C
–s+m +s t
–s++ms
, t∈[T,T);
u(·,t)++ss≡, t∈[T, +∞),
whereγ> is an embedding constant,αis a suitable constant,and C,T,Twill be
determined later.
Proof Multiplying both sides of equation (.) byus (s≥) and integrating the result
over, we obtain
s+
d dt
us+dx+ ms (m+s)
∇um+sdx=a
updx
uq+sdx. (.)
We first consider the casep+q≤.
() WhenN= or , settings= in (.), we get
d dt
udx+ m (m+ )
∇um+
dx=a
updx
uq+dx.
Using Hölder’s inequality and the Sobolev embedding inequality, one can show that
d dtu
+
mγ– (m+ )u
m+ ≤a||
–p–q
up+q+
.
By Lemma , one can see that there exists a constantα> such that ≤ u(·,t)≤
ue–αt, providedu≤[
mγ–
a(m+)||–p–q
]p+q–m. Hence, there exists a constantT >
such that
mγ–
(m+ ) –a||
–p–q
up+q–m
≥
mγ–
(m+ )–a||
–p–q
u
e–
αTp+q–m
for allt∈[T, +∞), and so dtdu+Cum+≤. By Lemma we can obtain its decay estimate, withT=
u–m
C(–m+)+T. () WhenN> ,
(a) formsuch thatNN–+<m< , we chooses=min (.). We then have the inequality
m+
d dtu
m+
m++γ–|| N–
N –+mmum+s
s+ ≤a||– p+q+m
m+ up+q+m
m+ (.)
by the Sobolev embedding and Hölder’s inequalities. By Lemma , one can see that there exists a constantα> such that ≤ u(·,t)mm++≤ umm++e–αt, provided umm++ ≤ [ (m+)γ–
a||+ Nmm–p+–q]
p+q–m. Then there exists a constantT
> such that
( +m)γ–||NN––+mm –a||– m+p+q
+m up+q–m +m
≤( +m)γ–||NN––+mm–a||– m+p+q
+m um+
m+e– αT
p+q–m
+m ≡C (> )
for allt∈[T, +∞). Hence, we get d
dtu m+
m++Cu+mm≤. By Lemma we can obtain its decay estimate,
withT= um–+m C(–mm+)
+T;
(b) formsuch that <m<NN+–, we chooses=N( –m) – in (.). From the Sobolev embedding inequality, we get
u
m+s
s =u
m+s
N
N– ≤γ
∇um+s
. (.)
By Hölder’s inequality, we obtain the inequality
s+
d dtu
s+
s++
γ–ms
(m+s)u
m+s
s+ ≤a||– p+q+s
s+ up+q+s
s+ . (.)
One can see that there exists a constantα> such that ≤ u(·,t)ss++≤ uss++e–αtby Lemma , provideduss++≤[
msγ–
a(m+)||–s+sp++q]
p+q–s. Then there exists a constantT >
such that
( +m)
smγ–
(m+s) –a||
–s++p+squp+q–s
+s
≤( +m)smγ –
(m+s) –a||
–s++p+squ ss++e–
αTp++q–ss≡C (> )
for allt∈[T, +∞). Hence, dtdus+
s++Cums++s≤. By Lemma we can obtain its decay estimate, with
T=
us+ –m
C(–ms++s)+T.
Secondly, we consider the casep+q> .
It can be easily verified thatkϕ(x) is a super-solution of (.)-(.) for sufficiently small
k> , whereϕ(x) is the unique positive solution of (.). We then have
u(x,t)≤kϕ
m
(x)≤kM
m , t>
by Proposition , ifu≤kϕ
m
The above inequality and (.) yield the inequality
s+
d dt
us+dx+ ms (m+s)
∇um+sdx≤akp+q–M
p+q–
m ||
us+dx (.)
by applying Hölder’s inequality to the right-hand side of (.).
Formsuch that NN–+ <m< , we chooses=min (.). It follows from the Sobolev embedding and Hölder’s inequalities that
m+
d dtu
m+
m++γ–|| N–
N –+mmum
m+≤akp+q–M p+q–
m
u+m+m.
By Lemma , there exists a constantα> such that ≤ u(·,t)mm++≤ umm++e–αt, pro-videdumm++≤[
||N –– +mmγ–
akp+q–M
p+q–
m
]–m. Hence, one can find a constantT> such that
( +m)γ–||NN––+mm –akp+q–M p+q–
m
u+–mm
≤( +m)γ–||NN––+mm–akp+q–M p+q–
m
umm++e–
αT–+mm≡C (> )
for allt∈[T, +∞), from which we get
d dtu
m+
m++Cu+mm≤.
By Lemma , we then obtain its decay estimates, withT= um–+m C(–mm+)
+T, as follows:
u(·,t)++mm≤u++mme–
αt, t∈[,T
);
u(·,t)++mm≤
u–+mm–C
– m
+m t
– +mm
, t∈[T,T);
u(·,t)++mm≡, t∈[T, +∞).
Formsuch that <m<NN–, we chooses=N( –m) > in (.), and then we can obtain the extinction results by using a similar argument as the one used for the casep+q≤,
and so the details are omitted.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.
Acknowledgements
This work is supported by the Natural Science Foundation of Shandong Province of China (ZR2012AM018) and the Fundamental Research Funds for the Central Universities (No. 201362032). The authors would like to deeply thank all the reviewers for their insightful and constructive comments.
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