R E S E A R C H
Open Access
Blow-up and global existence for the
non-local reaction diffusion problem with
time dependent coefficient
Iftikhar Ahmed
*, Chunlai Mu, Pan Zheng and Fuchen Zhang
*Correspondence:
[email protected] College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P.R. China
Abstract
Blow-up and global existence for the non-local reaction diffusion problem with time dependent coefficient under the Dirichlet boundary condition are investigated. We derive the conditions on the data of problem (1.1) sufficient to guarantee that blow-up will occur, and obtain an upper bound fort∗. Also we give the condition for global existence of the solution.
Keywords: blow-up; global existence; non-local reaction diffusion problem; Dirichlet boundary condition
1 Introduction
In this work, we consider the following non-local reaction diffusion problem with time dependent coefficient under the Dirichlet boundary condition
⎧ ⎪ ⎨ ⎪ ⎩
ut=u+
u
pdx–k(t)uq, x= (x
,x, . . . ,xN)∈,t∈(,t∗),
u(x,t) = , x∈∂,t∈(,t∗), u(x, ) =u(x)≥, x∈,
(.)
where⊂RN is a bounded domain with a smooth boundary∂,is the Laplace op-erator, and t∗ is the possible blow-up time. By the maximum principle, it follows that u(x,t)≥ in the time interval of existence. The coefficient k(t) is assumed to be non-negative. The particular case ofk= const of problem (.) has already been investigated by many authors, in [, ], they studied the question of blow-up for the solution, and in [–], they derived lower bounds for blow-up time under different boundary conditions. To deal with problem (.) with time dependent coefficient, we make the assumption on the parameterspandq, that is,p=q> .
The motivation of this article comes from the work of Payne and Philippin in [], where they investigated the blow-up phenomena of the solution of the following problem
⎧ ⎪ ⎨ ⎪ ⎩
ut=u+k(t)f(u), x= (x, . . . ,xN)∈,t∈(,t∗),
u(x,t) = , x∈∂, u(x, ) =u(x), x∈,
(.)
whereis a bounded sufficiently smooth domain inRN,N≥, and the coefficientk(t) is
assumed nonnegative or strictly positive depending on the situation.
In next, we employ a method used by Kaplan in [] to obtain a condition, which leads to blow-up at some finite time and also leads to an upper bound for the blow-up time. In Section , we derive the condition on the data of problem (.) sufficient to guarantee the global existence ofu(x,t).
2 Conditions for blow-up in finite timet∗
Letλbe the first eigenvalue, and letφbe the associated eigenfunction of the
Dirichlet-Laplace operator defined as
φ= –λφ, φ> ,x∈; φ= ,x∈∂, (.)
φdx= . (.)
Let the auxiliary functionη(t)
η(t) :=||–k(t) q–
uφdx (.)
defined in (,t∗), whereu(x,t) is the solution of (.) andq> . We assume that for allt∈(,t∗),
||>k(t) > , –k
(t)
||–k(t)≥β, maxx∈φ|| ≤, (.)
for some constantβ, and
γ :=λ–
β
q– . (.)
We deduce from (.) and (.) that
η(t)≥ β q– η(t) +
||–k(t) q–
φ
u+
uqdx–k(t)uq
dx
= –γ η(t) +||–k(t)
q–
uqdx–k(t)
φuqdx
≥–γ η(t) +||–k(t)
q–
maxx∈φ
–k(t)
φuqdx
≥–γ η(t) +||–k(t) q q–
φuqdx. (.)
Furthermore, using (.) and Hölder’s inequality, we get
φu dx≤
φuqdx
q
. (.)
Combining (.) and (.), we get
By integrating (.), we get
η(t) –q≤(t) :=
((η())–q–γ)eγ(q–)t+
γ, γ= ,
(η())–q+ ( –q)t, γ= . (.)
If(T) = for someT> , thenη(t) blows up at timet∗<T. This result is summarized
in the following theorem.
Theorem Let u(x,t)be the solution of problem(.).Then the auxiliary functionη(t) defined in(.)blows up at time t∗<Twith
T:=
⎧ ⎨ ⎩
γ(q–)log(–
γ((η())–q–γ)) if <γ(η()) –q< ,
(q–)(η())q– ifγ ≤.
(.)
3 Condition for global existence
In this section, our argument makes use of the following Sobolev-type inequality
vdx
/
≤
|∇v|dx
/
, =.
–/
π , (.)
valid inRfor a nonnegative functionvthat vanishes on∂. In this section, our results
restricted toRfor proof of (.), see [].
We consider the auxiliary functionσ(t) defined as
σ(t) :=M–||–k(t) n
un(p–)dx, t∈,t∗ , (.)
with
M:=||–k()n
un(q–)dx, (.)
we assume that for allt∈(,t∗),
||>k(t) > , –k
(t)
||–k(t)<β, (.)
for some constantβ. In (.)-(.),nis subjected to restrictions
n(q– )≥, n>
. (.)
For convenience, we set
v(x,t) =un(q–), (.)
and compute
σ(t) = n –k
(t)
||–k(t)σ(t) + n(q– )M
–||–k(t) n
×
un(q–)–
u+
uqdx–k(t)uq
with
un(q–)–u dx= –n(q– ) – n(q– )
|∇v|dx, (.)
due to (.), we obtain
σ(t)≤nβσ(t) –[n(q– ) – ] n(q– )
||–k(t) nM–
|∇v|dx
+ n(q– )M–||–k(t) n
||
v+ndx–k(t)
v+ndx
= nβσ(t) –[n(q– ) – ] n(q– )
||–k(t) nM–
|∇v|dx
+ n(q– )M–||–k(t) n+
v+ndx. (.)
By using Hölder’s inequality,
v+ndx≤
vdx
(n–)/n
vdx
/n
, (.)
and Sobolev-type inequality (.), we obtain
||–k(t) n+
v+ndx
≤||–k(t)n+
vdx
(n–)/n
|∇v|dx
/n
/n
=/nM(n–)/nσ(n–)/n||–k(t)n+
|∇v|dx
/n
, (.)
whereis defined in (.). Joining (.) and (.), we obtain
σ(t)
≤nβσ(t) –[n(q– ) – ] n(q– ) M
–||–k(t)n
|∇v|dx
+ n(q– )/nM/nσ(n–)/n
M–||–k(t) n
|∇v|dx
/n
= nβσ(t) + n
λ–M–||–k(t) n
|∇v|dx
/n
×
λ/n(q– )/nσ(n–)/nM/n–[n(q– ) – ] n(q– )
×λ
M–||–k(t)nλ–
|∇v|dx
(n–)/n
(.)
with arbitraryλ= . Choosingλ:=λ, the first eigenvalue of problem (.), we have
|∇v|dx≥λ
by the Rayleigh principle. By using (.) in the last factor of (.), we obtain
σ(t)≤nβσ(t) + n
λ–M–
||–k(t) n
|∇v|dx
/n
×
λ/n(q– )/nσ(t)(n–)/nM/n–
[n(q– ) – ] n(q– ) λσ(t)
(n–)/n
= nβσ(t) + nσ(t)/n×σ(t)(n–)/nωσ(t)/n– (μ+β), (.)
with
ω=λ/n(q– )/nM/n, μ=
[n(q– ) – ]
n(q– ) λ–β. (.)
Suppose thatβis small enough to satisfy the condition
μ> , (.)
and that initial data is small enough to satisfy the condition
ω–μ< . (.)
Then eitherω(σ(t))/n–μremains negative for all time, or there exists a first timet such
that
ωσ(t) /n
–μ= . (.)
Then we obtain the differential inequality
σ(t)≤nσ(t)ωσ(t) /n–μ≤, t∈(,t). (.)
Integrating this differential inequality, we obtain
σ(t)≤
–ω μ
eμt+ω
μ
–n
, t> . (.)
This result is summarized in the next theorem.
Theorem Letbe a bounded domain in R,and assume that the data of problem(.)
satisfy conditions(.), (.), (.).Then the auxiliary functionσ(t)defined in(.) sat-isfies(.),and u(x,t)exists for all time t> .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Acknowledgements
The authors would like to express sincere gratitude to the referees for their valuable suggestions and comments on the original manuscript. This work is supported in part by the NSF of PR China (11371384) and in part by the Natural Science Foundation Project of CQ CSTC (2010BB9218).
Received: 4 July 2013 Accepted: 18 September 2013 Published:09 Nov 2013
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