R E S E A R C H
Open Access
Profiles of blow-up solution of a weighted
diffusion system
Weili Zeng
1,2*, Chunxue Liu
1, Xiaobo Lu
1,2and Shumin Fei
1,2*Correspondence: [email protected] 1School of Automation, Southeast
University, Nanjing, 210096, China
2Key Laboratory of Measurement
and Control of CSE, Ministry of Education, Southeast University, Nanjing, 210096, China
Abstract
In this paper, we study the blow-up profiles for a coupled diffusion system with a weighted source term involved in a product with local term. We prove that the solutions have a global blow-up and the profile of the blow-up is precisely determined in all compact subsets of the domain.
Keywords: diffusion system; weighted localized source; blow-up profile
1 Introduction
In this paper, we consider the following coupled diffusion system with a weighted nonlin-ear localized sources:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
ut–u=a(x)up(x,t)vα(,t), x∈B, <t<T∗, vt–v=b(x)uβ(,t)vq(x,t), x∈B, <t<T∗, u(x,t) =v(x,t) = , x∈∂B,t> ,
u(x, ) =u(x), v(x, ) =v(x), x∈B,
(.)
whereBis an open ball ofRN,N≥ with radiusR;α,β,p,qare nonnegative constants
and satisfyα+p> andβ+q> .
System (.) is usually used as a model to describe heat propagation in a two-component combustible mixture []. In this caseuandvrepresent the temperatures of the interacting components, thermal conductivity is supposed constant and equal for both substances, a volume energy release given by some powers ofuandvis assumed.
The problem with a nonlinear reaction in a dynamical system taking place only at a single site, of the form
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
ut–u=up(,t)vα(,t), x∈, <t<T∗, vt–v=uβ(,t)vq(,t), x∈, <t<T∗, u(x,t) =v(x,t) = , x∈∂,t> ,
u(x, ) =u(x), v(x, ) =v(x), x∈,
(.)
was studied by Pao and Zheng [] and they obtained the blow-up rates and boundary layer profiles of the solutions.
As for problem (.), it is well known that problem (.) has a classical, maximal in time solution and that the comparison principle is true (using the methods of []). A number of
papers have studied problem (.) from the point of view of blow-up and global existence (see [, ]).
In [], Chen studied the following problem:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
ut–u=upvα, x∈, <t<T∗, vt–v=uβvq, x∈, <t<T∗, u(x,t) =v(x,t) = , x∈∂,t> ,
u(x, ) =u(x), v(x, ) =v(x), x∈,
(.)
assumingp> , orq> , orαβ> ( –p)( –q), he proved that the solution blows up in finite time if the initial datau(x) andv(x) are large enough.
In the case ofa(x) =b(x) = , Li and Wang [] discussed the blow-up properties for this system, and they proved that:
(i) Ifm,q≤, this system possesses uniform blow-up profiles. (ii) Ifm,q> , this system presents single point blow-up patterns.
Recently, Zhang and Yang [] studied the problem of (.), but they only obtained the es-timation of the blow-up rate, which is not precisely determined. In [], the authors proved there are initial data such that simultaneous and non-simultaneous blow-up occur for a diffusion system with weighted localized sources, but they did not study the profile of the blow-up solution. There are many known results concerning blow-up properties for parabolic system equations, of which the reaction terms are of a nonlinear localized type. For more details as regards a parabolic system with localized sources, see [–].
Our present work is partially motivated by [–]. The purpose of this paper is to de-termine the blow-up rate of solutions for a nonlinear parabolic equation system with a weighted localized source. That is, we prove that the solutionsuandvblow up simultane-ously and that the blow-up rate is uniform in all compact subsets of the domain. Moreover, the blow-up profiles of the solutions are precisely determined.
In the following section, we will build the profile of the blow-up solution of (.).
2 Blow-up profile
Throughout this paper, we assume that the functionsa(x),b(x),u(x) andv(x) satisfy the following three conditions:
(A) a(x),b(x),u(x),v(x)∈C(B);a(x),b(x),u(x),v(x) > inBand
a(x) =b(x) =u(x) =v(x) = on∂B.
(A) a(x),b(x),u(x)andv(x)are radially symmetric;a(r),b(r),u(r)andv(r)are non-increasing forr∈(,R](r=|x|).
(A) u(x)andv(x)satisfyu(x) +a(x)up(x)vα(,t)≥and v(x) +b(x)uβ(x)vq(,t)≥inB, respectively.
Theorem . Assume(A), (A),and(A)hold.Let(u,v)be the blow-up solution of(.),
non-decreasing in time,and let the following limits hold uniformly in all compact subsets of B:
(i) Ifp< ,q< andαβ> ( –p)( –q),then
lim
t→T∗u(x,t)
T∗–tθ=a(x)/(–p)Cθθ(σ/θ)β/αβ–(–p)(–q),
lim
t→T∗v(x,t)
where
θ= (α+ –q)/αβ– ( –p)( –q), σ= (β+ –p)/αβ– ( –p)( –q),
C=
a()b()
β
(–p)(–q)–αβb() βθ
–q, C
=
a()b()(–p)(–q)–α αβa()–qαθ.
(ii) Ifp< andq= ,then
lim
t→T∗u(x,t)
T∗–t/β=a(x)/(–p)a()/p–
αb() +β–p
/β
(/β)/β,
lim
t→T∗v(x,t)
T∗–t (+p–β)b(x)
αβb() =a() –a(x)
αb()(/β) (+β–p)b(x)
αβb()
+β–p αb()
(–p)b(x)
αβb()
.
(iii) Ifp= andq= ,then
lim
t→T∗u(x,t)
T∗–t a(x)
βa()=
αb()
a(x)
βa()
,
lim
t→T∗v(x,t)
T∗–t b(x)
αb()=
βb()
b(x)
αb()
.
(iv) Ifp= andq< ,then
lim
t→T∗u(x,t)
T∗–t (+α–q)b(x)
αβa() =b() –a(x)
βa()(/α) (+α–q)b(x)
αβa()
+α–q βa()
(–q)a(x)
αβa()
,
lim
t→T∗v(x,t)
T∗–t/α=b(x)/(–q)b()/q–
βa() +α–q
/α
(/α)/β.
Throughout this section, we denote
g(t) =uβ(,t), G(t) = t
g(s)ds, g(t) =vα(,t), G(t) = t
g(s)ds.
Lemma . Assume that (u,v) is the positive solution of(.), which blow up in finite time T∗.Let p≤and q≤,then
lim
t→T∗g(t) =tlim→T∗G(t) =∞, t→limT∗g(t) =t→limT∗G(t) =∞.
Proof First we claim thatlimt→T∗G(t) =∞. Sinceu(,t) =maxu(x,t), we have
ut(,t)≤a()up(,t)g(t).
By integrating the above inequality over (,t), we get
u–p(,t)≤( –p)a() t
g(s)ds+u–p(), ifp< ,
From limt→T∗u(,t) = ∞, it follows that limt→T∗G(t) =∞. Applying similar
argu-ments as above to the equation ofvin system (.), it is reasonable thatlimt→T∗g(t) =
limt→T∗G(t) =∞.
The following lemma will play a key role in proving Theorem ., which will give the relationships amongu,v,G(t), andG(t).
Lemma . Under the conditions of Theorem.,the following statements hold uniformly in any compact subsets of B:
(i) p< andq< ,then
lim
t→T∗
u–p(x,t)
G(t) = ( –p)a(x), tlim→T∗
v–q(x,t)
G(t) = ( –q)b(x).
(ii) p= andq< ,then
lim
t→T∗
lnu(x,t)
G(t) =a(x), t→limT∗
v–q(x,t)
G(t) = ( –q)b(x).
(iii) p= andq= ,then
lim
t→T∗
lnu(x,t)
G(t) =a(x), t→limT∗
lnv(x,t)
G(t) =b(x).
(iv) p< andq= ,then
lim
t→T∗
u–p(x,t)
G(t) = ( –p)a(x), tlim→T∗
lnv(x,t)
G(t)
=b(x).
Proof (i) Whenp< andq< . A simple computation shows that
du–p dt =u
–p+p( –p)u––p|∇u|+ ( –p)a(x)g(t), x∈, <t<T∗, (.)
dv–p dt =v
–q+q( –q)v––q|∇u|+ ( –q)b(x)g
(t), x∈, <t<T∗, (.)
and the initial and boundary conditions are given by
u–p(x,t) =v–q(x,t) = , x∈∂B,t> , u–p(x, ) =u–p
(x), v–q(x, ) =v –q
(x), x∈B.
Denoteλ, the first eigenvalue of –inH(B) and byϕ(x) > andφ(x) > the corre-sponding eigenfunction, normalized byBa(x)ϕ(x)dx= andBb(x)φ(x)dx= .
Multiplying both sides of (.) and (.) byϕ andφ, respectively, and integrating over
B×(,t), we have, for <t<T∗
B
u–pϕdx–
B
u–pϕdx= –λ t
B
u–pϕdx ds
+ t
B
B
v–pφdx–
B
v–pφdx= –λ t
B
v–pφdx ds
+ t
B
p( –p)v–p–|∇v|φdx ds+ ( –p)G(t).
We claim thatlimt→T∗u–p(,t)/g(t) = andlimt→T∗v–q(,t)/g(t) = . In fact, we have ut(,t)≤up(,t)vα(,t), for <t<T∗that is,
lim
t→T∗sup
u–p(,t) G(t)
≤( –p)a(). (.)
Sinceg(t) is non-decreasing, it follows that for allε> ,
≤G(t)
g(t) ≤ T∗–ε
g(s)ds
g(t) +ε,
and usinglimt→T∗g(t) =∞, we deduce thatlimt→T∗G(t)/g(t) = , so that (.) implies
limt→T∗u–p(,t)/g(t) = . By a process analogous to above, we arrive atlimt→T∗v–p(,t)/ g(t) = .
Analogous to the proof of Theorem . in Ref. [], it can be inferred that
lim
t→T∗
u
–pϕdx
G(t) = ( –p), tlim→T∗
v
–qϕdx
G(t)
= ( –q). (.)
From (.) and (.), we know (u–p,v–q) is a sub-solution of the following problem:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
dw
dt =w+ ( –p)a(x)g(t), x∈B, <t<T∗, dz
dt =z+ ( –q)b(x)g(t), x∈B, <t<T∗, w(x,t) =z(x,t) = , x∈∂B,t> ,
w(x, ) =u–p(x), z(x, ) =v–q(x), x∈B,
(.)
Equation (.) and Lemma . assert that
lim
t→T∗ w(x,t)
G(t) = ( –p)a(x), tlim→T∗ z(x,t)
G(t) = ( –q)b(x), (.)
uniformly in all compact subsets ofB.
The rest of the proof of case (i) is similar to Lemma .(i). Cases (ii), (iii), and (iv) can be treated similarly. Now we prove Theorem . by using Lemma ..
Proof of Theorem. (i) Ifp< andq< . By Lemma .(i), we know that for choosing positive constantsδ< <τ, there existst<T∗such that
δ( –p)a()G(t) β/(–p)
≤G(t)≤τ( –p)a()G(t) β/(–p)
, t∈t,T∗
,
δ( –q)b()G(t)α/(–q)≤G(t)≤τ( –q)b()G(t)α/(–q), t∈t,T∗.
Therefore,
(δ( –p)a()G(t))β/(–p) (τ( –q)b()G(t))α/(–q) ≤
dG(t)
dG(t)≤
(τ( –p)a()G(t))β/(–p)
From the right-hand side of (.),
δ( –q)b()G(t) α/(–q)
dG(t)≤
τ( –p)a()G(t) β/(–p)
dG(t), t∈
t,T∗
.
Integrating the above inequality over [,t) yields
( –q)(δ( –q)b())α/(–q)
+α–q G
(+α–q)/(–q) (t)
t
t
≤( –p)(τ( –p)b())β/(–p)
+β–p G
(+β–p)/(–p) (t)
t
t
≤( –p)(τ( –p)b())β/(–p)
+β–p G
(+β–p)/(–p)
(t). (.)
Sincelimt→T∗G(t) =∞andq< , for any constant <ε< , there exists¯t:t≤ ¯t≤T∗ such thatG(+ α–q)/(–q)(t)≤( –ε)G(+
α–q)/(–q)
(t) fort∈[t¯,T∗). Hence, from (.) it can be deduced that fort∈[¯t,T∗),
εδb()α/(–q)( +β–p)( –q)G(t)
(+α–q)/(–q)
≤τa()β/(–p)( +∂–q)( –p)G(t)
(+β–p)/(–p)
. (.)
By an argument similar to above, there exists˜t<T∗such that˜t<t<T∗,
εδa()β/(–p)( +∂–q)( –p)G(t)
(+α–q)/(–q)
≤τb()α/(–q)( +β–p)( –q)G(t)(+α–q)/(–q). (.)
Sett∗=max{¯t,˜t}, then (.) and (.) hold simultaneously for allt∈[t∗,T∗). Next we choose{δi}∞i=,{εi}∞i=,{τi}∞i=, satisfying <δi,εi< andτi> withδi,εi,τi→ asi→ ∞.
Lett∗<T∗such that (.) and (.) hold fort∗i ≤t<T∗. From Lemma .(i), it follows that for such sequences{δi}∞i=, and {τi}∞i=, there exists{ti}∞i=:ti<T∗ with ti→T∗, as i→ ∞such that
δi( –p)a()G(t)
β/(–p)
≤G(t)≤τi( –p)a()G(t)
β/(–p)
, t∈ti,T∗
. (.)
TakingTi=max{ti∗,ti}, in terms of (.), (.), and (.), we deduce that forT≤t<T∗
G(t)≥δi( –p)a()G(t)
β/(–p)
≥δib()
βθ σ(–q)
δib() τia()
β
(–p)(+β–p)
(εiσ/θ)
β
+β–p( –q)G(t)
βθ
σ(–q), (.)
G(t)≤τib()
βθ σ(–q)
τib() δia()
β
(–p)(+β–p)
(σ/εiθ)
β
+β–p( –q)G
(t) βθ
σ(–q), (.)
Since –βθ/(σ( –q)) = –/(σ( –q)) < andlimt→T∗G(t) =∞, integrating (.) and
(.) over (t,T∗) we have, forT≤t<T∗,
D–i σ(σ/θ)–
β
+β–p≤T∗–t( –q)G(t)/σ(–q)≤d– i σ(σ/θ)
–+ββ–p
, (.)
where
di=
a()
b()
β/(–p)
δib()
βθ σ(–q)
δib() τia()
β
(–p)(+β–p)
(εi)
β
+β–p,
Di=
a()
b()
β/(–p)
τib()
βθ σ(–q)
τib() δia()
β
(–p)(+β–p)
(εi)
β
+β–p.
Clearly,
di,Di→
a()
b()
β/(–p)
b()
βθ σ(–q)
b()
a()
β
(–p)(+β–p)
, asεi,∈?,τi→.
By pluggingi→ ∞into (.) we get
( –q)G(t) /(–q)
∼Cσσ(θ/σ)β/αβ–(–p)(–q)
T∗–t–σ, (.)
whereC= (a())
β
(–p)(–q)–αβ(b()) βθ
–q+
β
(–p)(–q)–αβ.
Applying a similar proof to the one above, we can conclude that
( –q)G(t)/(–p)∼Cθθ(σ/θ)β/αβ–(–p)(–q)
T∗–t–θ, (.)
whereC= (b())
α
(–p)(–q)–αβ(a()) αθ
–q+(–p)(–q)–α αβ.
According to Lemma .(i), (.), and (.), it follows that uniformly in all compact subsets ofB
lim
t→T∗u(x,t)
T∗–tθ=a(x)/(–p)Cθθ(σ/θ)β/αβ–(–p)(–q),
lim
t→T∗v(x,t)
T∗–tσ =b(x)/(–p)Cσσ(θ/σ)α/αβ–(–p)(–q).
The arguments of cases (ii), (iii), and (iv) are very similar to the above, we omit the details. Therefore, we have completed the proof of Theorem ..
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All the authors typed, read, and approved the final manuscript.
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant 61374194, the China Postdoctoral Science Foundation Founded Project under Grant 2013M540405, the National Key Technologies R&D Program of China under Grant 2009BAG13A06, the National High-tech R&D Program of China (863 Program) under Grant 2008AA040202, and the Natural Science Foundation of Jiangsu Province under grant BK20140638.
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doi:10.1186/s13661-014-0143-1
