R E S E A R C H
Open Access
The global solution of a diffusion equation
with nonlinear gradient term
Huashui Zhan
**Correspondence:
[email protected]; [email protected] School of Applied Mathematics, Xiamen University of Technology, Xiamen, 361024, P.R. China School of Sciences, Jimei University, Xiamen, 361021, P.R. China
Abstract
Consider the viscosity solution to the initial boundary value problem of the diffusion equation
ut= div
(
∇um p–2∇um
)
–uq1m∇ump1 ,withp> 1,m> 0,p1≤2,p> 2p1, its initial valueu(x, 0) =u0(x)∈Lq–1+ 1
m(
), 3 >q> 1 and its boundary valueu(x,t) = 0, (x,t)∈
∂
×(0,∞). Ifp> 1 +m1, by considering the regularized problem and using Moser’s iteration technique, we get the locally uniformly bounded property of the solution and the locally bounded property of the Lp-norm of the gradient. By the compactness theorem, the existence of the viscositysolution of the equation is obtained provided that
mNq1
Nm(p– 1) –N+mq+
p1(m(p– 1) +m– 2) m(p– 1) – 1 < 1.
If 2 <p< 1 +m1, the existence of solution is obtained in a similar way, and the extinction of the solution is proved in this case.
MSC: 35K55; 35K65; 35B40
Keywords: diffusion equation; Moser iteration; viscosity solution; extinction
1 Introduction
The objective of the paper is to study the nonnegative weak solution of the following non-linear parabolic equation:
ut=div∇ump–∇um–umq∇ump inS=×(,∞), (.)
u(x, ) =u(x), x∈, (.)
u(x,t) = , (x,t)∈∂×(,∞), (.)
where⊂RNis a bounded open domain,p> ,m> ,p
≤,p> p,N≥, ≤u(x)∈
Lq–+m(), >q> , and∇is the spatial gradient operator.
The equation of the form (.) was suggested as a mathematical model for a variety of problems in mechanics, physics and biology, which can be seen in [–]etc.It has been widely researched, whether it is linear (i.e.,m= , p= ,p= ,mq= ) or nonlinear, fast diffusion (m(p– ) < ) or slow diffusion (m(p– ) > ). For example, the existence of
a nonnegative solution of (.)-(.) without the damping term –umq|∇um|p, defined in
some weak sense, is well established (see [, ]). For other examples, Bertshet al.[] and Zhouet al.[] discussed the existence and properties of viscosity solution for the equation
ut=uu–γ|∇u|, (.)
whereγ is a positive constant. Zhanget al.[] discussed the existence and properties of the viscosity solution for the equation
ut=u–a(x)|u|q–|∇u|, (.)
wherea(x) is a known function.
The most important characteristic of equation (.) or (.) is in that, generally, the uniqueness of the solutions is not true; one can refer to [–]. Thus, for the equation of the type (.), we should mainly consider the existence of the viscosity solution (see Defi-nition . below) and the related properties such as large time behaviors; one can refer to [–]etc.for some progress on this problem.
Now, we quote the following definition.
Definition . A nonnegative functionu(x,t) is called a weak solution of (.)-(.) ifu
satisfies
(i)
u∈L∞loc,∞;L∞(), (.)
ut∈Lloc,∞;L(), um∈L∞loc,∞;W,p(); (.)
(ii)
S
uϕt–∇um p–
∇um· ∇ϕ–umq∇umpϕdx dt= , ∀ϕ∈C
(S); (.)
(iii)
lim t→
u(x,t) –u(x)dx= . (.)
We will get the solution of (.)-(.) by considering the regularized problem
ut=div
∇um+ k
p–
∇um –umq∇ump
, (.)
with the initial value (.) and the homogeneous boundary value (.). The solutions of the regularized equation (.) are denoted byuk.
Definition . Ifukis a solution of the initial boundary value problem of (.)-(.)-(.),
limk→∞uk=u, a.e. inS, such thatuis a weak solution of (.)-(.), thenuis said to be a
The main aim of the paper is to show how the damping term –umq|∇ump|affects the
equation, including how the damping term affects the existence of the solution and how the damping term affects the properties such as the extinction of the solution. By consider-ing the solutionukof the regularized problem (.) and using Moser’s iteration technique, we getuk’s local bounded properties and the local bounded properties of theLp-norm of
the gradient∇uk. By the compactness theorem, we get the existence of the viscosity
solu-tion of the diffusion equasolu-tion itself. Apart from the general process of the proof such as in [–, , ]etc., in which the main difficulty is how to prove that
∇umk p–
∇um
k (∗)χ=∇um p–∇
um, weakly star inL∞loc,∞;L p p–(),
in our paper, in addition to overcoming the above difficulty, we have to solve another dif-ficulty lying in how to prove that
–umq
k ∇umk p
(∗)ν= –umq∇ump, weakly star inL∞
loc
,∞;L p p().
Also, we need to overcome the difficulty which comes from the damping term –umq|∇ump|when we prove the uniqueness of the viscosity solutions of (.)-(.).
In order to get the desired results, some important relationships among the exponents
p,q,q,p,m,Nare imposed. We also need the following lemmas.
Lemma .[] (Gagliardo-Nirenberg) If≤l<N, +β≤q, ≤r≤q≤(+β)Nl/(N–l),
suppose that u+β∈W,l(),then
uq≤c/(+β)u–r θu+
βθ/(+β)
,l , (.)
whereθ= (β+ )(r––q–)/(N––l–+ (β+ )r–).
Lemma .[] Let y(t)be a nonnegative function on(,T].If it satisfies
y(t) +Atλθ–y+θ(t)≤Bt–ky(t) +Ct–δ, <t≤T, (.)
where A,θ> ,λθ≥,B,C≥,k≤,then
y(t)≤A–θλ+ BT–k
θt–λ
+ Cλ+BT–k–t–δ, <t≤T. (.)
We will prove the following theorems. As usual, the constantscin what follows may be different from one to another.
Theorem . If≤u(x)and
p> +
m, (.) u(x)∈Lq–+m(), (.)
p≤, p<p, >q> , (.)
mNq
Nm(p– ) –N+mq+
p(m(p– ) +m– )
then(.)-(.)has a weak viscosity solution which satisfies
um∈L∞loc,∞;Lq+–m()∩L∞
loc
,∞;W,p() (.)
and
um(t)∞≤c +t–λ( +t)–/(p––m), t> , (.)
whereλ=N(pq+ (p– –m)N)–.Moreover,if p> ,then
∇ump≤c +t–μ( +t)–σ, t> , (.)
whereμ= +m(mp–)–– ,σ=p(m(q+)–)+mp
(m(p–)–)(p–p) .
The condition (.) is only used to prove (.); ifp= =q, this is a natural condition. We conjecture that this condition can be weakened.
Theorem . Let u be a weak solution of(.)-(.).If p> +m,p+q>p– ,then
suppu(·,s)⊂suppu(·,t) (.)
for all s,t with <s<t.
Theorem . If <p< +m <p+q,
≤u∈Lq–+
m(), >q> , (.)
then(.)-(.)has a weak solution which satisfies(.),and there exists a positive T>
such that
u(x,t)≡, ∀(x,t)∈(x,t)∈×(T,∞). (.)
If the damping term disappears in (.), say, if (.) without –umq|∇ump|by [], then
we know that the extinction of the solution as Theorem . is true. For other related works on equation (.), one can refer to the references [–]etc.We use some ideas in [] and [].
2 TheL∞estimate of the solution
Consider the regularized problem
ut=div∇um+
k p–
∇um –umq∇ump
, (.)
u(x, ) =uk(x), x∈, (.)
where ≤uk(x) is a suitably smooth function such that
u(x)∈L∞(), lim
k→∞ukq–+m =uq–+m.
Clearly,
–umq∇ump=mpumq+p(m–)|∇u|p,
if let
b(x,t,z,p) = –mpzmq+p(m–)|p|p.
Then, if|z| ≤M, sincep≤,
|b| ≤c|p|,
by Chapter of [], viewing (.) as a divergent form of a quasilinear parabolic equation, we know that (.)-(.) has a unique nonnegative classical solutionuk. In what follows, in the proof of the related lemmas, we only denoteukasufor simplicity.
Lemma . If p> +
m,ukis the solution of(.)-(.),then u m
k ∈L∞loc(,∞;Lq–+
m())
and
umkq–+
m ≤c( +t)
–
p–– m, t≥, (.)
where >q> .
Proof LetAn= (q– )n–q,Bn= ( –q)n–qand
fn(s) =
⎧ ⎨ ⎩
sq– ifs≥
n, Ans+Bns if ≤s<n.
The condition >q> assures thatf(um) defined above is nonnegative. If we multiply
(.) byfn(um) and integral on, then we have
fnumdiv∇um+
k p–
∇um dx
= –
∇um+
k p–
∇umfnumdx
≤–
∇umpfnumdx= –
∇
um
fn(s)
pds p
dx, (.)
–
fnumumq∇ump
dx≤. (.)
From the above calculation, we have
fn
umutdx+
∇
um
fn(s)pds p
By the Poincare inequality, we have
fn
umutdx+c
um
fn(s)pds p
dx≤. (.)
Letn→ ∞in (.). We can deduce that
d dt
um(q–)+dx+c
um[q–+m+p––m]dx≤. (.)
By the Jessen inequality, from (.) we get
d dtu
mq–+m q–+m +cu
mq–+m+p––m q–+m ≤,
then
umq+–
m ≤c( +t)
–
p–– m.
We get the desired result.
Lemma . If p> +
m,ukis the solution of(.)-(.),then
umk∞≤ct–λ, <t≤, (.) umk∞≤c( +t)–
p–– m, t≥, (.)
whereλ= N (p––m)N+q.
Proof Multiply (.) byum(l–)and integral on, then
um(l–)u
tdx=
div∇um+ k
p–
∇um um(l–)dx
–
umq∇umpum(l–)dx
= –(l– )
∇um+
k p–
∇umum(l–)dx
–
umq∇umpum(l–)dx
≤–(l– )
∇um+
k p–
∇umum(l–)dx,
which deduces that
d dtu
ml–+m l–+m +c
l– +
m
–p
∇um
p+l–+ m–– m
SetL=l– +m. Then
d dtu
mL L+cL
–p
∇um
L+p–– m
p pdx≤, (.)
wherecis a constant independent ofl.
Now, if we chooseL=q– –m,Ln=rLn–– (p– –m),θn=rN( –Ln–L–n)(p+N(r–
))–,μ
n= (Ln+p– –m)θn––Ln,r> + (p– –m)q–,n= , , . . . , by Lemma ., we
have umL
n≤c
p/(Ln+p––m)um–θn Ln–∇u
m(Ln+p––m)/ppθn/(p––
m+Ln)
p . (.)
If we chooseL=Lnin (.), by (.) we have
d dtu
mLn Ln+c
–p/θnL–p n um
Ln+μn
Ln u
mp––m–μn
Ln– ≤, <t≤. (.)
We will prove that there exist two bounded sequences{ξn},{λn}such that
umL n≤ξnt
–λn, <t≤. (.)
Ifn= , by Lemma .,λ= ,ξ=supt≥um(t)q––m makes (.) sure. If (.) is true forn– , then from (.),
d dtu
mLn Ln+c
–p/θnL–p n um
Ln+μn Ln ξ
p––m–μn n– t–(p––
m–μn)λn–≤, <t≤. (.)
We can choose
λn=
λn–
μn–p+ +
m + μ
–
n, ξn=ξn–
cp/θnLp–
n λn
/μn
, n= , , . . . ,
by Lemma . and (.), (.) is true.
Moreover, by Lemma ., asn→ ∞,λn→λ=(p––N
m)N+q
. It is easy to see that{ξn}is
bounded. Thus (.) is true.
To prove (.), we setτ=log( +t),t≥,w(τ) = ( +t)
p–– mum(t). By (.), we have
d dτw(τ)
L L+cL
–p∇wL+p–– p mp p≤
L
p– –m w(τ)
L
L, τ ≥log. (.)
By Lemma . in [], we can get (.); we omit details here.
3 TheL∞estimation of the gradient
Lemma . If p>max{, +m},ukis the solution of(.)-(.),then
∇umkp≤ct–(+m(mp–)–– ), <t≤, (.)
∇umkp≤c( +t)–
p(m(q+)–)+mp
Proof If we multiply (.) byum
t and integral on, then
m
um–(ut)dx
=
div∇um+
k p–
∇um umt dx–
umq∇umpum
t dx, (.)
div∇um+
k p–
∇um umt dx
= –
∇um+
k p–
∇um∇umt dx= –
∇um+
k p–
∇umt dx
= – d dt
|∇um|
s+
k p–
ds dx= –
d dtk∇u
m
, (.)
–
umq∇umpum
t dx ≤m
um–(ut)dx+c
umq+
m–
m ∇ump
dx. (.)
By (.)-(.), we have
um–(ut)dx+
m d dtk∇u
m≤c
ump+
m–
m ∇ump
dx. (.)
If we multiply (.) byumand integral on, then
m+
d dtu
m+dx=
div∇um+
k p–
∇um umdx–
umq∇ump
umdx
= –
∇um+
k p–
∇umdx–
umq∇umpumdx
and
k∇um
≤
∇um+
k p–
∇umdx
= –
m+
d dtu
m+dx–
umq∇umpumdx
≤
m+ u
m+
u
m– ut . So, m d dtk∇u
m
+ (m+ )um+–
k∇um
≤c
umq+mm–∇ump
dx. (.)
Setting γ= q+ –m, for∀a∈[, γ], if we notice thatp> p, we have
uma∇ump
dx≤um(t)a∞
um
(γ–a)p p–p dx
p–p
p
∇ump
If γ ≥(p– p)(N+ )/N, leta= (γ – (p– p)( +Nq))+. By Lemma .,
um
(γ–a)p p–p dx
p–p
p
≤cum(t)s(γ–a)(–θ)∇umpp–p, (.)
whereθ= (s–– ( –p
p )(γ–a)–)/(N––p–+s–), ands= (γ–p+ p–a)N/(p– p)
when γ≥(p– p)( +q/N),s=q, when (p– p)( +N–)≤γ≤(p– p)( +q/N). By Lemma . and Lemma ., from (.), we have
uma∇umpdx≤ct–λa∇umpp≤ct–λak∇um
, <t≤. (.)
At the same time, if we chooseq= in Lemma ., we have
um+
m =
um+dx m m+
≤ct–(p––mm+)–
and
um+
=
um+dx≤ct–m(mp–)–+ . (.)
By (.), we have
k(t) +ctm(mp–)–+
k(t)≤ct–
λa
k(t), <t≤. (.)
If γ< (p– p)(N+ )/Nandp– p≤a≤γ, then
uma∇umpdx≤c∇uma(–θ)∇umpaθ+p≤c∇umpp≤ck∇um
,
<t≤. (.)
If γ< (p– p)(N+ )/Nandp– ≥a≥, then
uma∇umdx≤c +∇umpp≤c +k∇um
,
<t≤. (.)
The inequalities (.) and (.) mean that the inequality (.) is still true when γ < (p– p)(N+ )/N. Using Lemma ., we get
k(t)≤ct–(+ m–
m(p–)–), <t≤,
which means (.) is true. Now, we will prove (.). Fort≥, by (.) we obtain
uma∇ump
dx≤c∇umpum(t)γγp/p–p
≤c( +t)–γ/(p––m)∇ump
k∇um
=
|∇um|
s+
k p–
ds≤c∇umpp=c∇ump
p
p
p, (.)
um+
=
um+dx
≤c( +t)–(p––m)–. (.)
By (.), using (.)-(.) yields
k(t) +c( +t)–(p––m)–
k(t)≤c( +t)
γ/(p––m)
k(t)
p
p ,
and using the Young inequality gives
k(t) +c( +t)–(p––m)–
k(t)≤c( +t)
–m(γp+p) (m(p–)–)(p–p)
=c( +t)–
p(m(q+)–)+mp (m(p–)–)(p–p) ,
which means (.) is true.
Lemma . If p> +m,ukis the solution of(.)-(.),then
T
t
umk–(ukt)dx ds≤ct–(+m(mp–)–– )+ct–(λγ+m(mp–)–– ), <t≤T. (.)
Proof From (.), (.) and (.), (.), we have
T
t
um–(ut)dx ds≤k(t) +c
T
t
umq+
m–
m ∇ump
dx ds
≤k(t) +c
T
t
s–λ(q+mm–)k(s)ds
≤ct–(+m(mp–)–– )+ct–(λγ+m(mp–)–– ). (.)
4 The proof of Theorem 1.5
The proof of Theorem. From Lemma ., Lemma ., Lemma . and Lemma ., using the compactness theory (cf.[]), there is a sequence (still denoted as{uk}) of{uk}such
that whenk→ ∞, we have
uk(∗)u, weakly star inL∞loc,∞;Lm(q–)+(), (.)
uktut, weakly inL
,∞;L(), ∇umk um, weakly inLploc,∞;Lp(),
(.)
∇umkp–∇ukxmi(∗)χi, weakly star inL∞loc
,∞;Lpp–(), (.)
umq
k ∇umk p
(∗)ν, weakly inL∞loc,∞;L p
whereχ={χi: ≤i≤N}and everyχiis a function inL∞loc(,T;L p
p–()),ν∈L∞
loc(,∞;
L p
p()). (.) and (.) are clearly true. In what follows, we only need to prove that
χ=∇ump–∇um inL∞loc,∞;Lpp–() (.)
and
ν=umq∇ump
inL∞loc,∞;L p
p(). (.)
It is easy to know that
S
(uϕt–χ· ∇ϕ–νϕ)dx dt= , ∀ϕ∈C∞(S). (.)
So, if we can prove that
S
∇ump–∇um· ∇ϕdx dt=
S
χ· ∇ϕdx dt, ∀ϕ∈C∞(S), (.)
S umq
k ∇umk p
ϕdx dt=
S
νϕdx dt, ∀ϕ∈C∞(S), (.)
then (.),(.) and (.) are true.
First, for anyψ∈C∞(S), ≤ψ≤,vm∈Lp
loc(,T;W ,p
()), we have
S
ψ∇umkp–∇umk –∇vmp–∇vm· ∇ukm–vmdx dt≥. (.)
If we multiply byumkψthe two sides of (.), then we have
S
ψ∇umk+
k p–
∇umkdx dt
=
m+
S
ψtumk+dx dt–
S
umk∇umk+
k p–
∇umk · ∇ψdx dt
–
S um(q+)
k ∇umk p
ψdx dt. (.)
Noticing that when <p< , we have
∇umk≥∇umk+
k p
–
k p
,
∇umk+
k p–
∇umk≤∇umk+
k p–
,
and whenp≥, we get
∇umk+
k p–
∇umk≥∇umkp, ∇umk+
k p–
By (.), (.), we have
m+
S
ψtumk+dx dt–
S
umk∇umk+
k p–
∇umk · ∇ψdx dt
–
S um(q+)
k ∇umk p
ψdx dt+ k p– mes – S
ψ∇umkp–∇umk · ∇vmdx dt
–
S
ψ∇vmp–∇vm· ∇umk –vmdx dt≥. (.)
Since
∇umk+
k p–
∇umk =∇umkp–∇umk +p– k
∇umk+s
k p–
ds∇umk
and
lim k→∞
p– k S
∇umk+ s
k p–
ds∇umk · ∇ψumk dx dt= ,
if we letk→ ∞in (.), we have
m+
S
ψtum+dx dt–
S
νψdx dt
–
S
ψ∇ξ· ∇vmdx dt–
S
ψ∇vmp–∇vm· ∇um–vmdx dt≥. (.)
Now, we chooseϕ=ψumin (.),
m+
S
ψtum+dx dt–
S
νψdx dt–
S
ψ χ·∇ψumdx dt=
S
ψ∇ξ·∇umdx dt.
From this formula and (.), we have
S
ψχ–∇vmp–∇vm· ∇um–vmdx dt≥. (.)
Letvm=um–λϕ,λ≥,ϕ∈C∞
(S). Then
S
ψχi–∇
um–λϕp–um–λϕx
i
ϕxidx dt≥.
Letλ→. We obtain
S
ψχi–∇ump–umxi
ϕxidx dt≥, ∀ϕ∈C∞(S). (.)
Moreover, if we chooseλ≤, we are able to get
S
ψχi–∇ump–umxi
ϕxidx dt≤, ∀ϕ∈C ∞
Now, if we chooseψ such thatsuppϕ⊂suppψ, and onsuppϕ,ψ= , then from (.)-(.), we can get (.). By (.) and (.), we have
S
uϕt–∇ump–∇um· ∇ϕ–νϕ
dx dt= , ∀ϕ∈C∞(S),
which means (.) is true, and so (.) is true. Secondly, we are to prove (.).
For smallr> , denoter={x∈:dist(x,∂)≤r}. For anyη> , let
sgnη(s) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ifs>η,
s
η if|s| ≤η, – ifs< –η.
For any given smallr> and large enoughk,l, we declare that
r
uk(x,t) –ul(x,t)dx≤
r
uk(x, ) –ul(x, )dx+cr(t), (.)
wherecr(t) is independent ofk,l, andlimt→cr(t) = . By (.) we have t
r
ϕ(ukt–ult)dx dτ
+ t
r
∇ϕ∇umk+
k p–
∇umk –∇uml +
l p–
∇uml
dx dτ
+ t r
umq
k ∇u m k
p
–umq
l ∇u m l
p
ϕdx dτ= ,
∀ϕ∈Lp,T;W,p(). (.)
Suppose thatξ(x)∈C(r) such that
≤ξ≤; ξ|r= ,
and chooseϕ=ξsgnη(um
k –uml ) in (.), then
t
r
ξsgnηumk –ulm(ukt–ult)dx dτ
+ t r
∇umk+
k p–
∇umk –
x∇uml +
l p–
∇uml
× ∇ξsgnηumk –uml dx dτ
+ t r
∇umk+
k p–
∇umk –
x∇uml +
l p–
∇uml
× ∇umk –uml ξsgnηumk –uml dx dτ
+ t r
umq
k ∇umk p
–umq
l ∇uml p
If we notice that the third term on the left-hand side of (.) tends to zero whenη→, then we have
lim
η→ t
r
ξsgnηumk –uml (ukt–ult)dx dτ
+lim
η→ t
r
∇umk+
k p–
∇umk –∇uml +
l p–
∇uml
× ∇ξsgnηum k –uml
dx dτ
+lim
η→ t
r
umq
k ∇umk p
–umq
l ∇uml p
ξsgnηumk –uml dx dτ= . (.)
At the same time,
lim
η→ t
r
ξsgnηumk –uml (ukt–ult)dx dτ
= t
r
ξsgnumk –ulm(ukt–ult)dx dτ
= t
r
ξsgn(uk–ul)(ukt–ult)dx dτ
=lim
η→ t
r
ξsgnη(uk–ul)(ukt–ult)dx dτ
=lim
η→ t
r
ξ
uk–ul
sgnη(s)ds
τ
dx dτ
=lim
η→ t
r
ξ uk–ul
sgnη(s)dst
dx
=
r
ξ|uk–ul|dx–
r
ξ|uk–ul|dx. (.)
By (.) and (.), we have
r
ξ|uk–ul|dx
≤
r
|uk–ul|dx+c
t
r
∇umk+
k p–
+∇uml +
l p–
dx dτ
+ t
r
umq
k ∇u m k
p
–umq
l ∇u m l
p
dx dτ. (.)
By Lemma . and Lemma ., if <t≤, t
r
umq
k ∇u m k
p
–umq
k ∇u m k
p
dx dτ≤c
t
r
t–dx dτ,
where
= mNq
Nm(p– ) –N+mq+
p(m(p– ) +m– )
m(p– ) – < ,
Now, for any given smallr, ifk,lare large enough, by (.), we have
r
u(x,t) –u(x)dx≤
r
u(x,t) –uk(x,t)dx+
r
uk(x) –ul(x)dx
+
r
ul(x,t) –ul(x)dx+
r
ul(x) –u(x)dx.
Lettingt→, we get (.).
5 The uniqueness of the viscosity solution
As we have said in the introduction, the uniqueness of the solutions of (.)-(.) is not true generally. But we are able to prove the uniqueness of the viscosity solution.
Theorem . If u(x)∈L∞(),in addition,|∇u|<c, ≥p≥,then the viscosity solution
of(.)-(.)is unique.
Proof Letu,vbe two viscosity solutions of (.)-(.). Then there are two sequences{uk}
and{vl}, which are the solutions of (.)-(.)-(.), such that
lim
k→∞uk=u, l→∞limvl=v, a.e. inS. (.)
Clearly, sinceu(x)∈L∞(),
uk∞≤c, vl∞≤c. (.)
Let
w=uk–vl, w=umk –vml .
Then
wt=ail(x,t)wxl
xi+b(x,t,w,∇w), (x,t)∈×(,∞), (.) w(x, ) =uk(x) –vl(x), x∈, (.)
w(x,t) = , (x,t)∈∂×(,∞), (.)
where
ail(x,t) =
s∇um+ ( –s)∇vmp–ds·δil
+
(p– )s∇um+ ( –s)∇vmp–sumxi+ ( –s)vmxisumxl+ ( –s)vmxlds,
and sincep≥, using the convexity of the functionsp, by (.), we have
By Chapter of [], we know that uk(x,t) –vl(x,t)∞≤cuk–vl.
Letk,l→ ∞, we know that the uniqueness of the viscosity solution (.)-(.) is true.
Suppose that the viscosity solution of (.)-(.) is unique in what follows. Then, by con-sidering the regularized problem (.) with (.)-(.), we easily get the following lemma.
Lemma . Let u be a weak solution of(.)-(.).If v satisfies
vt≥div∇vmp–∇vm–vmq∇vmp in S=×(,∞), (.)
v(x, )≥u(x), x∈, (.)
v(x,t) = , (x,t)∈∂×(,∞), (.)
then
u(x,t)≥v(x,t), ∀(x,t)∈S. (.)
Now, we will prove Theorem .. Let
v(x,t) =ukr(x,t) =ruk
x,rm(p–)–t, r∈(, ).
Then
vt(x,t) =divDvmp–Dvm–rm(p––q–p)vmqDvmp, (x,t)∈×(,∞), (.)
v(x, ) =ruk(x, ), x∈, (.)
v(x,t) = , (x,t)∈∂×(,∞). (.)
Noticing that we supposed
p+q>p– , <r< ,
which implies that
rm(p––q–p)> ,
and using the argument similar to that in the proof Lemma . of [], we can prove
uk≥ukr.
It follows that
uk(x,rm(p–)–t) –uk(x,t) (rm(p–)–– )t ≥
r– ( –rm(p–)–)tuk
Lettingr→, we get
ukt≥– uk
(m(p– ) – )t. (.)
Hence, we have proved Theorem ..
6 The proof of Theorem 1.7
If <p< +m, from the process of the proof of Lemma ., we also have (.),i.e.,
d dt
umk(q–)+dx+c
um[q–+
m+p––m]
k dx≤. (.)
But, since <p< + m, the Jessen inequality is invalid now, and (.) may not be true. However, in this case, (.) implies that
d dt
umk(q–)+dx≤, (.)
which gives the information ofum∈L∞
loc(,∞;Lq–+
m()) provided thatu∈Lq–+m().
Lemma . Suppose that p< +m and
q+p> +
m. (.)
If ukis the solution of(.)-(.),then
umk∞≤ct–λ, <t≤, (.) umkL≤c( +t)–Lθ, t≥, (.)
where L= –m(p– ) +m,θ=(–θ)[mmL(L+θp–)–]– and <θ< .
Proof Similarly as in the proof of Lemma ., we multiply (.) byum(l–)and integral on, and then we get the following inequality (.), which is just the same as (.).
d dtu
mL L+cL
–p
∇um
L+p–– m
p pdx≤. (.)
Let{Ln},{λn}be two sequences just the same as those in the proof of Lemma .. Since
(.) implies thatLn+p––m > andλn> , we can deduce the conclusions (.) similarly
as in Lemma ..
To prove (.), we also setτ=log( +t),t≥,w(τ) = ( +t)
p–– mum(t). By (.), we have
d dτw(τ)
L L–
L p– –
m
w(τ)LL+cL–p∇w L+p–– m
p p
p≤, τ≥log,
which implies
d dτw(τ)
L L+cL
–p( +t)–pm(–(L+mp(–)–p–))++–mm(p–)∇
w L+p–– m
p p
By Gagliardo-Nirenberg Lemma ., let +β=L+p––
m p . Then
∇w+βpp≥c–+βw qw–r θ
(+β)p
θ .
If we chooser=q=L, then from the above inequality, we have
∇w L+p–– m
p p
p≥c p θw(–θ)
L+p–– m
θ
L . (.)
By (.), (.), we have
d dτw(τ)
L L+c
–pθL–p( +t)–pm(–(L+mp(–)–p–))++–mm(p–)
w(–θ) L+p–– m
θ
L ≤,
τ≥log. (.)
Now, we choose the constantl= –m(p– ),i.e.,
L=l– +
m= –m(p– ) +
m,
then
–m(L+p– ) –
p( –m(p– )) + +
m
–m(p– )= .
By (.), we have
d dτw(τ)
L L+c
–pθL–pwL(–θ)[mmLθ(L+p–)–
L ≤, τ≥log. (.)
Let
θ=
( –θ)[m(L+p– ) – ]
mLθ – .
Since <θ< ,θ> , by Lemma ., we have w(τ)L≤cτ–
Lθ,
which implies that um(t)L≤c( +t)–
Lθ.
If <p≤ +m, which implies thatm> , then we can get the conclusions of Lemma . in a similar way. As in the proof of Theorem ., we get the existence of the solution for the system (.)-(.) in this case.
Proposition . Let u be a weak solution of(.)-(.).If p< +m,then there exists a finite time T such that
u(x,t)≡ (.)
To prove this proposition, we use the idea of the proof of Theorem . in [], in which the extinction of the solution for the equation
ut=div∇ump–∇um
was studied. In detail, we define an auxiliary function
v(x,t) =k(T–t)
+/m–p
+ log(l+x+x+· · ·+xN)
m, (.)
where
k=
(p– )(m+ –mp)Np
(l)p(log(l))/m
+/m–p
,
T=
max|u| klog
+/m–p
,
l= sup
(x,x,...,xN)∈
|x|,|x|, . . . ,|xN|+ .
Then we have
∂v
∂t≥div∇v mp–
∇vm≥div∇vmp–∇vm–umq∇ump
,
on account of the non-positivity of the damping term –umq|∇um|p.
If we notice that
v(x, )≥u(x), ∀x∈, v(x,t)≥, ∀(x,t)∈∂×(,∞), (.)
applying Lemma ., by (.)-(.), we have
u(x,t)≤v(x,t),
for all (x,t)∈S. By the definition ofv(x,t), we have
u(x,t)≤v(x,t) = , ∀(x,t)∈×(T,∞).
The proof of the proposition is complete.
Theorem . is a direct corollary of the proposition.
Competing interests
The author declares that they have no competing interests.
Acknowledgements
The paper is supported by NSF (no. 2012J01011) of Fujian Province, supported by SF of Xiamen University of Technology, China.
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doi:10.1186/1029-242X-2013-125
