Article

Blow-Up Rate Estimates for a System of Reaction-Diffusion Equations with Gradient Terms

Abstract

This paper is concerned with the blow-up properties of Cauchy and Dirichlet problems of a coupled system of Reaction-Diffusion equations with gradient terms. The main goal is to study the influence of the gradient terms on the blow-up profile. Namely, under some conditions on this system, we consider the upper blow-up rate estimates for its blow-up solutions and for the gradients.
Research Article
Blow-Up Rate Estimates for a System of Reaction-Diffusion
Equations with Gradient Terms
Maan A. Rasheed , Hassan Abd Salman Al-Dujaly, and Talat Jassim Aldhlki
Department of Mathematics, College of Basic Education, Mustansiriyah University, Baghdad, Iraq
Correspondence should be addressed to Maan A. Rasheed; maan.rasheed.edbs@uomustansiriyah.edu.iq
Received 16 November 2018; Revised 12 January 2019; Accepted 22 January 2019; Published 10 February 2019
Academic Editor: Irena Lasiecka
Copyright © 2019 Maan A. Rasheed et al. is is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
is paper is concerned with the blow-up properties of Cauchy and Dirichlet problems of a coupled system of Reaction-Diusion
equations with gradient terms. e main goal is to study the inuence of the gradient terms on the blow-up prole. Namely, under
some conditions on this system, we consider the upper blow-up rate estimates for its blow-up solutions and for the gradients.
1. Introduction
In this paper, we consider the following problem:
𝑡=−|∇|𝑞1+V𝑝1,(,)∈×(0,)
V𝑡=V|V|𝑞2+𝑝2(,)∈×(0,)
(,0)=0()≥0, ∈
V(,0)=V0()≥0, ∈,
(1)
where 1,2∈(1,);1,2(1,2);
=𝑛or 𝑅(a ball in 𝑛with radius ).
Moreover, for =
𝑅,and Vsatisfy the zero Dirichlet
boundary conditions:
(,)=V(,)=0, ∈,∈(0,);(2)
0,V0∈2()are both nonzero, satisfying the monotonicity
conditions:
0∇0𝑞1+V0𝑝1≥0, ∈ (3)
V0V0𝑞2+0𝑝2≥0, ∈ (4)
Moreover, in case of =
𝑅,(
0,V0)should satisfy
compatibility conditions:
0()=V0()=0, ∈ (5)
As an application to system (1), a single equation of this
system can be considered a simple model in population
dynamics, [1, 2]:
𝑡=−|∇|𝑞+𝑝,in ×{≥0}(6)
where ,>1.
Let the domain represent a territory where a biological
species live on. (,)refers to the spatial density of individ-
uals located near a point ∈at time ≥0.
In fact, the evolution of this density is the result of three
types of mechanisms: displacement, birth, and death. For
more details of deriving the evolution equation satised by
, see [1].
Basically, under dierent assumptions on the mecha-
nisms of accidental death, the corresponding term should
more generally be a nondecreasing function of the density
and its gradient ∇.
Moreover, homogeneous Dirichlet’s conditions can be
added to this model which, for instance, correspond to a
nonviable environment in the boundary zone.
It is expected that, with a large size initial function
(initial distribution of population 0), the density becomes
unbounded in a nite time >0. erefore, Chipot and
Weissler [3] studied the eect of the damping term in this
equation on global existence or nonexistence.
Hindawi
International Journal of Mathematics and Mathematical Sciences
Volume 2019, Article ID 9807876, 7 pages
https://doi.org/10.1155/2019/9807876
2 International Journal of Mathematics and Mathematical Sciences
e blow-up phenomena in Reaction-Diusion equa-
tions have been intensively studied; see, for instance, [4–
7]. One of the studied cases is the Cauchy problem of the
semilinear heat equation:
𝑡=+𝑝,(,)R𝑛×(0,),
(,)=0()≥0, ∈R𝑛(7)
e second studied case is zero Dirichlet problem of the
semilinear heat equation:
𝑡=+𝑝,(,)∈𝑅×(0,),
(,)=0, ∈𝑅
(,)=0()≥0, ∈𝑅
(8)
where >1.
For both cases (7) and (8), it has been proved in [8, 9] that
if the initial function is nonnegative and suitably large, then
blow-up occurs in a nite time. In [5, 10], it has been shown
that the upper blow-up rate estimate for this equation is as
follows:
(,)
(−)1/(𝑝−1) ,(,)∈×(0,).(9)
e blow-up properties of semiliear heat equations with
negative sign gradient terms (damping terms) have been
studied by some authors as in [3, 7, 11].
One of these equations is population model (6). For ⊆
𝑛, it is well known that blow-up can only occur if >;see
[3, 11, 12].
Moreover, if =
𝑅,thenblow-upoccursatthe
center of 𝑅, and this follows from the upper point-wise
estimate:
(,)
||𝛼,
𝑅\{0},[0,),(10)
where >2/(−1),for1<<2/(+1),
while >/(−),for2/(+1)<.
It is clear that /()>2/(1),where>2/(+1)
erefore, the prole of blow-up solutions of (6) is
similar to that of problem (8), where  < 2/(+1) (see
[12]), while if > 2/( + 1),thegradienttermcauses
more eect on the plow-up prole and it becomes more
singular.
Moreover, it has been proved in [4, 13, 14], that there are
positive constants and , such that the upper and lower
blow-up rate estimates for this equation, where <2/(+1),
take the following form:
(−)1/(𝑝−1) ≤(,)
(−)1/(𝑝−1) (11)
In [15–18], the coupled system of Reaction-Diusion equa-
tions was considered:
𝑡=+V𝑝1,
V𝑡=V+𝑝2,(12)
(,)∈×(0,),(13)
where 1,2>1;=𝑅or 𝑛.
It was shown that if the initial functions satisfy 0,V0≥0,
both being nonzero and large enough, then blow-up occurs
in a nite time.
For the Cauchy problem associated with (12), it was
proved in [16] that if
max ,
2,(14)
then blow-up occurs in a nite time, where
= 1+1
12−1,
= 2+1
12−1
(15)
Later, in [4, 15], it was proved that the upper blow-up rate
estimates of this system are as follows:
(,)1
(−)𝛼,(,)∈×(0,),
V(,)2
(−)𝛽,(,)∈×(0,).(16)
for some 1,2>0.
e system (1) has been studied in [19], where ∈3and
is a bounded convex domain and
=1=2== 1
1−;
=1=2;>>1. (17)
It has been shown that if a classical solution of this system
blows ups (becomes unbounded) in the W-norm, where
()=Ω2𝑝 +V2𝑝 (18)
then blow-up time for this problem can be estimated from
below as follows:
≥ 1
22
0(19)
where (0)=0=∫
Ω(2𝑝
0+V2𝑝
0)
and is a constant which depends on the data.
For the blow-up times and applications of other parabolic
systems with damping terms (such as Keller-Segel system
with Neumann and Robin boundary conditions), we refer to
[20, 21].
International Journal of Mathematics and Mathematical Sciences 3
In this paper, with some restricted conditions on system
(1), we show that the upper blow-up rate estimates for this
solution and its gradients terms take the following forms:
(,)≤1(−)−𝛼 ,
|∇(,)|≤1(−)−(1+2𝛼)/2 ,
V(,)≤2(−)−𝛽 ,
|V(,)|≤2(−)−(1+2𝛽)/2 ,
(20)
where (,) ∈ ×(0,) and 1,2> 0,,are given in
(15).
2. Local Existence and Blow-up
Set
1(V,∇)=V𝑝1|∇|𝑞1,
2(,V)=𝑝1|V|𝑞1(21)
Since the system (1) is uniformly parabolic and its equations
have the same principle parts and 1,2∈
1( × 𝑛),
also the growths of the nonlinearities in 1and 2with
respect to the gradient terms are subquadratic; 0,V0
2(), and satisfying (5), it follows that the local existence
and uniqueness of classical solution to the for system (1),
where =𝑅, with zero Dirichlet boundary conditions, are
guaranteed by standard parabolic theory (see eorem 7.1,
[22, 23]).
i.e., there exists >0,suchthat
∈2,1 (×(0,))∩×[0,). (22)
Also,thegradienttermsareboundedaslongasthecompo-
nents of the solution are bounded; see [23].
In case of =
𝑛, these results can also be extended to
the Cauchy problem associated with system (1) (see eorem
8.1, [22, 24]).
Moreover, from the monotonicity assumptions (3) and
(4) and since 0,V0are nonnegative, it follows by the
maximum principle [6] that in the interval of existence
the solutions of system (1) are nondecreasing in time and
nonnegative.
i.e., (,)0, V(,)0, ×(0,).
On the other hand, since the existence and uniqueness
of system (1) can only be locally guaranteed and according
to known blow-up results to the single equation (6), blow-up
may occur in this problem in a nite time. erefore, some
authors were interested in studying the blow-up properties
and numerical solutions of system (1) ; see for instance [19, 25].
3. Upper Blow-Up Rate Estimates
In the next theorem, we derive the upper blow-up rate
estimates for any blow-up solution of system (1) and its
gradients.
eorem 1. Assume that 1,2,1,and2satisfy the following
two conditions:
(i) max{,}/2,
(ii) 1<1<(2+2)/(2+1), 1<2<(2+2)/(2+1),
where ,are given in (15).
Let (,V)be a blow-up solution of the Cauchy (Dirichlet)
problem of system (1), with the above conditions, which blows
up at <∞.ere exist two positive constants 1and 2such
that upper blow-up rate estimates for (,V)and (∇,V)are as
follows:
(,)≤1(−)−𝛼 ,
|∇(,)|≤1(−)−(1+2𝛼)/2 ,
V(,)≤2(−)−𝛽 ,
|V(,)|≤2(−)−(1+2𝛽)/2 ,
(23)
in ×(0,).
Proof. For ∈(0,),set
𝑢()=sup
Ω×(0,𝑡] (,)+|∇(,)|2𝛼/(1+2𝛼),
V()=sup
Ω×(0,𝑡] V(,)+|V(,)|2𝛽/(1+2𝛽). (24)
Clearly, each of 𝑢,Vis continuous, nondecreasing, and
nonnegative function on (0,). Moreover, 𝑢→ or
V→ as →and that follows from (,V)blowing
up at .
It will be shown later that we can nd ∈(0,1)such that
≤−1/2𝛼
𝑢()1/2𝛽
V()1
(25)
for /2<<
So that consequently both 𝑢and Vdiverge as →.
In order to prove this theorem, we will use a rescaling
method as in [4] and the proof will have ve steps.
Step 1 (rescaling).If𝑢diverges as →, then we can apply
the following procedure.
Letting 0∈(0,),wecanchoose(1,1)∈×(0,
0]
such that
1,1+∇1,12𝛼/(1+2𝛼) 1
2𝑢0. (26)
Dene the new rescaled functions as follows:
𝛾
1,=2𝛼+1,2+1, (27)
𝛾
2,=2𝛽V+1,2+1, (28)
,𝛾×−1
2,−1
2, (29)
4 International Journal of Mathematics and Mathematical Sciences
where =(0)=−1/2𝛼
𝑢(0)is a scaling factor and
𝛾=∈𝑛:+1∈.(30)
It is clear that
𝛾
𝑛  = 𝑛,
𝑅/𝛾   = 𝑅.(31)
Next, we aim to show that (𝛾
1,𝛾
2)is a solution of the system:
𝛾
1𝑠 −𝛾
1=−𝜇1∇𝛾
1𝑞1+𝛾
2𝑝1,
𝛾
2𝑠 −𝛾
2=−𝜇2∇𝛾
2𝑞2+𝛾
1𝑝2,(32)
where
1=2+2(2+1)1,
2=2+2−2+12.(33)
From assumption (ii), we get 1,2>0.
Clearly,
𝛾
1𝑠 =2𝛼+2,
∇𝛾
1=2𝛼+1∇,
𝛾
1=2𝛼+2.
(34)
From (1) and (34), we get
1
(2𝛼+2) 𝛾
1𝑠 =1
(2𝛼+2) 𝛾
1+1
𝑞1(2𝛼+1) ∇𝛾
1𝑞1
+1
2𝑝1𝛽𝛾
2𝑝1.(35)
Hence, the rst equation of the system (32) can be obtained
by multiplying the last equation by (2𝛼+2).e same way can
be used to show that 𝛾
2satises the second equation of the
system (32).
Now, we restrict to ∈(−21,0]to show that
𝛾
1,+∇𝛾
1,2𝛼/(1+2𝛼) ≤1, (36)
for (,)𝛾×(−21,0].
From (34), we obtain
∇𝛾
1,2𝛼/(1+2𝛼) =2𝛼 |∇|2𝛼/(1+2𝛼) .(37)
Clearly,
(,)+|∇(,)|2𝛼/(1+2𝛼) ≤𝑢0,
(,)∈×0,1. (38)
From (32), (37), and (38), we get (36).
Moreover,
𝛾
2+∇𝛾
22𝛽/(1+2𝛽) ≤−𝛽/𝛼
𝑢0V0,(39)
for (,)𝛾×(−21,0].
On the other hand, from (26), we obtain
𝛾
1(0,0)+∇𝛾
1(0,0)2𝛼/(1+2𝛼) 1
2(40)
If V→ ∞ as →, the same procedure can be repeated
by changing the roles of and.
Step 2 (Schauder’s estimates). In this step, we nd the interior
Schauder’s estimates of the functions 1,2on the sets
𝐾=∈𝛾,≤×[−,],(41)
where >0, =0,1.Assuming that 1and 2satisfy in 2𝑘
the condition
0≤𝛾
1+∇𝛾
12𝛼/(1+2𝛼) ≤,
0≤𝛾
2+∇𝛾
22𝛽/(1+2𝛽) ≤ (42)
Our claim is as follows: for any positive and small enough
values of ,,and,thereexistsaconstant = (,,)
such that
𝛾
1
𝐶2+𝜎,1+𝜎/2(𝑆𝑘)≤,
𝛾
2
𝐶2+𝜎,1+𝜎/2(𝑆𝑘)≤ (43)
From (42), we deduce that 𝛾
1,𝛾
2,∇𝛾
1,and∇𝛾
2are
uniformly bounded functions in 2𝑘.So, the functions
(𝛾
1)𝑝1,(𝛾
2)𝑝2,|𝛾
1|𝑞1,and|∇𝛾
2|𝑞2are uniformly bounded in
2𝑘.erefore, the right hand side of each equation in (32) is
uniformly bounded function in 2𝑘.By applying the interior
regularity theory (see [23]), we get locally uniform estimates
in 1+𝜎,(1+𝜎)/2-norms. Consequently, on the right hand side of
each equation in (32), we can obtain locally uniform estimates
in H¨
older norms 𝜎,𝜎/2. erefore, the parabolic interior
Schauder’s estimates (43) are held; see [23].
Step 3 (the proof of (25)). Suppose that the lower bound of
(25) is not held. So, there is a sequence 𝑗,suchthat𝑗→ 
as →∞,and
−1/2𝛼
𝑢𝑗1/2𝛽
V𝑗→0 →∞ (44)
us, 𝑢→ ∞ as 𝑗→ .
Now, for each 𝑗whichplaysthesameroleof0,asin
Step 1, we can scale about the corresponding point (
𝑗,
𝑗)
for each, where
𝑗≤
𝑗.We get the corresponding rescaled
solution (𝛾𝑗
1,𝛾𝑗
2):
International Journal of Mathematics and Mathematical Sciences 5
𝛾𝑗
1,=2𝛼
𝑗𝑗+
𝑗,2
𝑗+
𝑗, (45)
𝛾𝑗
2,=2𝛽
𝑗V𝑗+
𝑗,2
𝑗+
𝑗. (46)
where 𝑗=(𝑗)=−1/2𝛼
𝑢(𝑗)is the scaling factor.
It is clear that (𝛾𝑗
1,𝛾𝑗
2)satises, as in Step 1, the following
problem:
𝛾𝑗
1𝑠 −𝛾𝑗
1=−𝜇1
𝑗∇𝛾𝑗
1𝑞1+𝛾𝑗
2𝑝1,
𝛾𝑗
2𝑠 −𝛾𝑗
2=−𝜇2
𝑗∇𝛾𝑗
2𝑞2+𝛾𝑗
1𝑝2,(47)
with
𝛾𝑗
1(0,0)+∇𝛾𝑗
1(0,0)2𝛼/(1+2𝛼) 1
2,
0≤𝛾𝑗
1+∇𝛾𝑗
12𝛼/(1+2𝛼) ≤1,
𝛾𝑗
2+∇𝛾𝑗
22𝛽/(1+2𝛽) ≤−𝛽/𝛼
𝑢𝑗V𝑗,
(48)
for (,)𝛾𝑗×(−2
𝑗
𝑗,0],where
𝛾𝑗
𝑛  = 𝑛,
𝑅/𝛾𝑗  = 𝑅.(49)
Clearly,
𝛾𝑗→ 𝑛 ∞. (50)
From (44) and (48), we see that
𝛾𝑗
2+∇𝛾𝑗
22𝛽/(1+2𝛽) → 0,   → ∞. (51)
us 𝛾𝑗
2and ∇𝛾𝑗
2are bounded in 𝛾𝑗×(−2
𝑗
𝑗,0]for all .
By applying Step 2, there is 𝑘independent of ,suchthat
the uniform Schauder’s estimates of (𝛾𝑗
1,𝛾𝑗
2)are as follows:
𝛾𝑗
1
𝐶2+𝜎,1+𝜎/2({𝑦∈Ω𝛾𝑗,|𝑦|≤𝑘}×[−𝑘,0]) ≤𝑘,
𝛾𝑗
2
𝐶2+𝜎,1+𝜎/2({𝑦∈Ω𝛾𝑗,|𝑦|≤𝑘}×[−𝑘,0]) ≤𝑘
(52)
Since (𝛾𝑗
1,𝛾𝑗
2)is dened on a compact set, by the Arzela-
Ascoli theorem, there exists a convergent subsequence, and
it is denoted by (𝛾𝑗
1,𝛾𝑗
2).
Since 1,2>0and ∇𝛾𝑗
1,∇𝛾𝑗
2are bounded, the limit
point (1,2)is a solution of the following system:
1𝑠 =1+𝑝1
2,
2𝑠 =2+𝑝2
1
(53)
𝑛×(−∞,0](54)
Since 𝛾𝑗
2→ 0, where ,itfollowsthat2
0𝑛×(,0].
Consequently, from the second equation of (53), we get
1≡0, 𝑛×(−∞,0].(55)
us,
1(0,0)+∇1(0,0)2𝛼/(1+2𝛼) =0, (56)
which leads to a contradiction with (48), so the lower bound
is proved.
If we change the roles of and V, the upper bound of (25)
can be proved similarly as in the last proof.
Step 4 (estimates on doubling 𝑢).Since𝑢→ ∞ as 
,𝑢is a continuous function. For any 0∈(0,),thepoint
+
0can be dened as follows:
+
0=max 0,:𝑢()=2𝑢0. (57)
Clearly,
(,)+|∇(,)|2𝛼/(1+2𝛼) ≤2𝑢0,
(,)∈×0,+
0. (58)
Take =(0)=−1/2𝛼
𝑢(0).
We claim that there is 0<which is independent of 0
such that +
0−0
20≤, 0
2,,(59)
By supposing that this claim is not true, there is a sequence
𝑗→ ,as→∞such that
+
𝑗−𝑗
2
𝑗𝑗→ ∞, (60)
where
+
𝑗=max 𝑗,:𝑢()=2𝑢𝑗. (61)
For each 𝑗,where/2 < 𝑗<
+
𝑗<,,wecanchoose
0<
𝑗≤𝑗.
AsinStep3,wescaleaboutthecorrespondingpoint
(
𝑗,
𝑗), and we can get the corresponding rescaled functions
(𝛾𝑗
1,𝛾𝑗
2)with the scaling factor: 𝑗=(
𝑗)=
−1/2𝛼
𝑢(𝑗),
which satises (47) with the following conditions:
𝛾𝑗
1(0,0)+∇𝛾𝑗
1(0,0)2𝛼/(1+2𝛼) 1
2,
0≤𝛾𝑗
1+∇𝛾𝑗
12𝛼/(1+2𝛼) ≤2,
𝛾𝑗
2+∇𝛾𝑗
22𝛽/(1+2𝛽) ≤−𝛽/𝛼
𝑢𝑗V+
𝑗
(62)
,∈𝛾𝑗×−
𝑗
2
𝑗,+
𝑗−
𝑗
2
𝑗.(63)
6 International Journal of Mathematics and Mathematical Sciences
From (61) and (62), it follows that
𝛾𝑗
2+∇𝛾𝑗
22𝛽/(1+2𝛽) ≤2𝛽/𝛼−𝛽/𝛼
𝑢+
𝑗V+
𝑗.(64)
From (25), we conclude that
V()≤−2𝛽𝛽/𝛼
𝑢(),
2,. (65)
erefore, (64) becomes
𝛾𝑗
2+∇𝛾𝑗
22𝛽/(1+2𝛽) 2𝛽/𝛼
2𝛽 (66)
By applying Step 2, we use the Schauder estimates for
(𝛾𝑗
1,𝛾𝑗
2), and we can get a convergent subsequence in
2+𝜎,1+𝜎/2
𝑙𝑜𝑐 (𝑛×)to the solution of system (53) in 𝑛×.
us, we get a contradiction because under the assump-
tion (i), all nontrival solutions of system (53) blow up in a
nite time; see [17].
So, there is 0<such that
−2 0+
0−0≤, 0∈
2,. (67)
Step 5 (rate estimates).
For any 0(/2,),asinStep4,
dene 𝑢(1)=2𝑢(0),where1=+
0∈(0,).
By (67), we have
1−0≤−1/𝛼
𝑢0. (68)
We can g e t 2∈(,)by using 1as a new value of 0such
that 𝑢2=2𝑢1=4𝑢0. (69)
us,
2−1≤−1/𝛼
𝑢1=2−1/𝛼−1/𝛼
𝑢0.(70)
So, for any ≥0,wehave
𝑗+1 −𝑗≤2−𝑗/𝛼−1/𝛼
𝑢0, (71)
where the sequence 𝑗→ as →∞.
By adding the above inequalities, it follows that
0≤
𝑗≥02−𝑗/𝛼−1/𝛼
𝑢0. (72)
us, (0)≤(1−2−1/𝛼)−1−1/𝛼
𝑢(0).
Using (25) results in
V0≤−2𝛽𝛽/𝛼
𝑢0, 0∈
2,. (73)
us,
V0≤−2𝛽 12−1/𝛼−𝛽 𝛽0−𝛽 ,
0∈
2,. (74)
From above, there exist positive two constants and such
that
𝑢0≤−0−𝛼 ,
0∈
2,,
V0≤−0−𝛽 ,
0∈
2,. (75)
From the last two inequalities and the denitions of
𝑢and V, it follows that there are 1and 2such that
(,)+|∇(,)|2𝛼/(1+2𝛼) ≤1(−)−𝛼 ,
V(,)+|V(,)|2𝛽/(1+2𝛽) ≤2(−)−𝛽 .(76)
Or, we can split the last estimates as follows:
(,)≤1(−)−𝛼 ,
|∇(,)|≤1(−)−(1+2𝛼)/2 ,
V(,)≤2(−)−𝛽 ,
|V(,)|≤2(−)−(1+2𝛽)/2
(77)
where (,)×(0,).
4. Conclusions and Future Studies
From eorem 1 and its proof, we can point out the following
conclusions:
(i) By (25), it follows that the blow-up in system (1) can
only occur simultaneously.
(ii) e gradient terms are bounded for any <.
(iii) e upper blow-up rate estimates for both of systems
(1) and (12) take the same forms. is means that, with
the two conditions of eorem 1, the gradient terms
in the system (1) do not eect ormake any changes on
the prole of blow-up solutions.
Next,wepointoutsomepossiblefutureresearchdirections:
(1) One may try to derive the blow-up rate estimates for
problem (1), in case one or both assumptions (i) and
(ii) of eorem 1 are not satised.
(2) For the semilinear system (12) dened in a ball,
and under some restricted assumptions on (0,V0)
(nonnegative and radial decreasing functions), it is
well known that the blow-up can only occur at
the center point (see [18]). However, it is unknown
whether and under which condition this result can be
extended to the system (1).
Data Availability
https://www.dr opbox.com/home?preview=Data+Availabil-
ity+Statement.pdf.
International Journal of Mathematics and Mathematical Sciences 7
Conflicts of Interest
e authors declare that they have no conicts of interest.
Acknowledgments
e authors would like to thank Mustansiriyah University
(http://www.uomustansiriyah.edu.iq) Baghdad-Iraq for its
support in the present work.
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