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

Authors: Maan A. Rasheed; Hassan Abd Salman Al-Dujaly; Talat Jassim Aldhlki
Journal: International Journal of Mathematics and Mathematical Sciences
 (2019)

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## 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.


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## Body

## 1. Introduction

In this paper, we consider the following problem:(1)ut=∆u-∇uq1+vp1,x,t∈Ω×0,Tvt=∆v-∇vq2+up2x,t∈Ω×0,Tux,0=u0x≥0,x∈Ωvx,0=v0x≥0,x∈Ω,where p1,p2∈1,∞;q1,q2∈1,2;Ω = R n or BR (a ball in Rn with radius R).Moreover, forΩ=BR,u and v satisfy the zero Dirichlet boundary conditions: (2)ux,t=vx,t=0,x∈∂Ω,t∈0,T;u0,v0∈C2Ω- are both nonzero, satisfying the monotonicity conditions:(3)∆u0-∇u0q1+v0p1≥0,x∈Ω(4)∆v0-∇v0q2+u0p2≥0,x∈ΩMoreover, in case of Ω=BR,(u0,v0) should satisfy compatibility conditions:(5)u0x=v0x=0,x∈∂ΩAs an application to system (1), a single equation of this system can be considered a simple model in population dynamics, [1, 2]: (6)ut=Δu-∇uq+up,inΩ×t≥0wherep,q>1.Let the domainΩ represent a territory where a biological species live on. u(x,t) refers to the spatial density of individuals located near a point x∈Ω at time t≥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 satisfied byu, see [1].Basically, under different assumptions on the mechanisms of accidental death, the corresponding term should more generally be a nondecreasing function of the densityu and its gradient ∇u.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 populationu0), the density u becomes unbounded in a finite time T>0. Therefore, Chipot and Weissler [3] studied the effect of the damping term in this equation on global existence or nonexistence.The blow-up phenomena in Reaction-Diffusion equations have been intensively studied; see, for instance, [4–7]. One of the studied cases is the Cauchy problem of the semilinear heat equation: (7)ut=Δu+up,x,t∈Rn×0,T,ux,t=u0x≥0,x∈RnThe second studied case is zero Dirichlet problem of the semilinear heat equation: (8)ut=Δu+up,x,t∈BR×0,T,ux,t=0,x∈∂BRux,t=u0x≥0,x∈BRwhere p>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 finite time. In [5, 10], it has been shown that the upper blow-up rate estimate for this equation is as follows: (9)ux,t≤CT-t1/p-1,x,t∈Ω×0,T.The 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 Ω⊆Rn, it is well known that blow-up can only occur if p>q; see [3, 11, 12].Moreover, ifΩ=BR, then blow-up occurs at the center of BR, and this follows from the upper point-wise estimate: (10)ux,t≤cxα,x∈BR∖0,t∈0,T,where α>2/p-1, for 1<q<2p/p+1,whileα>q/p-q, for 2p/p+1≤q<p.It is clear thatq/p-q>2/p-1, where q>2p/p+1Therefore, the profile of blow-up solutions of (6) is similar to that of problem (8), where q<2p/p+1 (see [12]), while if q>2p/p+1, the gradient term causes more effect on the plow-up profile and it becomes more singular.Moreover, it has been proved in [4, 13, 14], that there are positive constants a and b, such that the upper and lower blow-up rate estimates for this equation, where <2p/p+1, take the following form: (11)aT-t1/p-1≤ux,t≤bT-t1/p-1In [15–18], the coupled system of Reaction-Diffusion equations was considered: (12)ut=Δu+vp1,vt=Δv+up2,(13)x,t∈Ω×0,T,where p1,p2>1;Ω=BR or Rn.It was shown that if the initial functions satisfyu0,v0≥0, both being nonzero and large enough, then blow-up occurs in a finite time.For the Cauchy problem associated with (12), it was proved in [16] that if (14)max⁡α,β≥n2,then blow-up occurs in a finite time, where (15)α=p1+1p1p2-1,β=p2+1p1p2-1Later, in [4, 15], it was proved that the upper blow-up rate estimates of this system are as follows: (16)ux,t≤C1T-tα,x,t∈Ω×0,T,vx,t≤C2T-tβ,x,t∈Ω×0,T.for some C1,C2>0.The system (1) has been studied in [19], where Ω∈R3 and is a bounded convex domain and (17)p=p1=p2α=β=11-p;q=q1=q2;p>q>1.It has been shown that if a classical solution of this system blows ups (becomes unbounded) in the W-norm, where (18)Wt=∫Ωu2p+v2pdxthen blow-up time for this problem can be estimated from below as follows: (19)T≥12AW02where W(0)=W0=∫Ω‍(u02p+v02p)dxandA 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].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:(20)ux,t≤C1T-t-α,∇ux,t≤C1T-t-1+2α/2,vx,t≤C2T-t-β,∇vx,t≤C2T-t-1+2β/2,where (x,t)∈Ω×0,TandC1,C2>0,α,β are given in (15).


## 2. Local Existence and Blow-up

Set(21)F1v,∇u=vp1-∇uq1,F2u,∇v=up1-∇vq1Since the system (1) is uniformly parabolic and its equations have the same principle parts and F1,F2∈C1(R×Rn), also the growths of the nonlinearities in F1 and F2 with respect to the gradient terms are subquadratic; u0,v0∈C2Ω-, and satisfying (5), it follows that the local existence and uniqueness of classical solution to the for system (1), where Ω=BR, with zero Dirichlet boundary conditions, are guaranteed by standard parabolic theory (see Theorem 7.1, [22, 23]).i.e., there existsT>0, such that (22)u∈C2,1Ω×0,T∩CΩ-×0,T.Also, the gradient terms are bounded as long as the components of the solution are bounded; see [23].In case ofΩ=Rn, these results can also be extended to the Cauchy problem associated with system (1) (see Theorem 8.1, [22, 24]).Moreover, from the monotonicity assumptions (3) and (4) and since u0,v0 are 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.,ux,t≥0,vx,t≥0,inΩ×(0,T).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 finite time. Therefore, 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.Theorem 1.
Assume thatp1,p2,q1, and q2 satisfy the following two conditions: (i)
max⁡α,β≥n/2,(ii)
1<q1<2α+2/2α+1,1<q2<2β+2/2β+1,
whereα,β are given in (15).
Let(u,v) be a blow-up solution of the Cauchy (Dirichlet) problem of system (1), with the above conditions, which blows up at T<∞. There exist two positive constants C1 and C2 such that upper blow-up rate estimates for (u,v) and (∇u,∇v) are as follows:(23)ux,t≤C1T-t-α,∇ux,t≤C1T-t-1+2α/2,vx,t≤C2T-t-β,∇vx,t≤C2T-t-1+2β/2,in Ω×(0,T).Proof.
Fort∈(0,T), set (24)Mut=supΩ×0,tux,t+∇ux,t2α/1+2α,Mvt=supΩ×0,tvx,t+∇vx,t2β/1+2β.Clearly, each of Mu,Mv is continuous, nondecreasing, and nonnegative function on (0,T). Moreover, Mu→∞ or Mv→∞ as t→T and that follows from (u,v) blowing up at T.
It will be shown later that we can findδ∈(0,1) such that (25)δ≤Mu-1/2αtMv1/2βt≤1δfor T/2<t<T
So that consequently bothMu and Mvdiverge as t→T.
In order to prove this theorem, we will use a rescaling method as in [4] and the proof will have five steps.
Step 1 (rescaling). If Mu diverges as t→T, then we can apply the following procedure.
Lettingt0∈(0,T), we can choose (x1,t1)∈Ω×(0,t0] such that (26)ux1,t1+∇ux1,t12α/1+2α≥12Mut0.Define the new rescaled functions as follows: (27)φ1γy,s=γ2αuγy+x1,γ2s+t1,(28)φ2γy,s=γ2βvγy+x1,γ2s+t1,(29)y,s∈Ωγ×-t1γ2,T-t1γ2,where γ=γ(t0)=Mu-1/2α(t0) is a scaling factor and (30)Ωγ=y∈Rn:γy+x1∈Ω.It is clear that(31)Ωγ≔RnifΩ=Rn,BR/γifΩ=BR.Next, we aim to show that (φ1γ,φ2γ) is a solution of the system: (32)φ1sγ-Δφ1γ=-γμ1∇φ1γq1+φ2γp1,φ2sγ-Δφ2γ=-γμ2∇φ2γq2+φ1γp2,where (33)μ1=2α+2-2α+1q1,μ2=2β+2-2β+1q2.From assumption (ii), we get μ1,μ2>0.
Clearly,(34)φ1sγ=γ2α+2u,∇φ1γ=γ2α+1∇u,Δφ1γ=γ2α+2Δu.From (1) and (34), we get (35)1γ2α+2φ1sγ=1γ2α+2Δφ1γ+1γq12α+1∇φ1γq1+1γ2p1βφ2γp1.Hence, the first equation of the system (32) can be obtained by multiplying the last equation by γ(2α+2). The same way can be used to show that φ2γ satisfies the second equation of the system (32).
Now, we restricts to s∈(-γ-2t1,0] to show that (36)φ1γy,s+∇φ1γy,s2α/1+2α≤1,for (y,s)∈Ωγ×(-γ-2t1,0].
From (34), we obtain (37)∇φ1γy,s2α/1+2α=γ2α∇u2α/1+2α.Clearly, (38)ux,t+∇ux,t2α/1+2α≤Mut0,x,t∈Ω×0,t1.From (32), (37), and (38), we get (36).
Moreover,(39)φ2γ+∇φ2γ2β/1+2β≤Mu-β/αt0Mvt0,for (y,s)∈Ωγ×(-γ-2t1,0].
On the other hand, from (26), we obtain (40)φ1γ0,0+∇φ1γ0,02α/1+2α≥12If Mv→∞ as t→T, the same procedure can be repeated by changing the roles of u and.
Step 2 (Schauder’s estimates). In this step, we find the interior Schauder’s estimates of the functions φ1,φ2 on the sets (41)SK=y∈Ωγ,y≤k×-k,kN,where k>0,N=0,1. Assuming that φ1 and φ2 satisfy in S2k the condition(42)0≤φ1γ+∇φ1γ2α/1+2α≤A,0≤φ2γ+∇φ2γ2β/1+2β≤AOur claim is as follows: for any positive and small enough values of k,A, and σ, there exists a constant L=L(k,A,σ) such that (43)φ1γC2+σ,1+σ/2Sk≤L,φ2γC2+σ,1+σ/2Sk≤LFrom (42), we deduce that φ1γ,φ2γ,∇φ1γ, and ∇φ2γ are uniformly bounded functions in S2k. So, the functions (φ1γ)p1,(φ2γ)p2,|∇φ1γ|q1, and |∇φ2γ|q2 are uniformly bounded in S2k. Therefore, the right hand side of each equation in (32) is uniformly bounded function in S2k. By applying the interior regularity theory (see [23]), we get locally uniform estimates in C1+σ,1+σ/2-norms. Consequently, on the right hand side of each equation in (32), we can obtain locally uniform estimates in Hölder norms Cσ,σ/2. Therefore, 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 tj, such that tj→T as j→∞, and (44)Mu-1/2αtjMv1/2βtj→0asj→∞Thus, Mu→∞ as tj→T.
Now, for eachtj which plays the same role of t0, as in Step 1, we can scale about the corresponding point (xj∗,tj∗) for each, where tj∗≤tj. We get the corresponding rescaled solution (φ1γj,φ2γj): (45)φ1γjy,s=γj2αuγjy+xj∗,γj2s+tj∗,(46)φ2γjy,s=γj2βvγjy+xj∗,γj2s+tj∗.where γj=γ(tj)=Mu-1/2α(tj) is the scaling factor.
It is clear that(φ1γj,φ2γj) satisfies, as in Step 1, the following problem: (47)φ1sγj-Δφ1γj=-γjμ1∇φ1γjq1+φ2γjp1,φ2sγj-Δφ2γj=-γjμ2∇φ2γjq2+φ1γjp2,with (48)φ1γj0,0+∇φ1γj0,02α/1+2α≥12,0≤φ1γj+∇φ1γj2α/1+2α≤1,φ2γj+∇φ2γj2β/1+2β≤Mu-β/αtjMvtj,for (y,s)∈Ωγj×(-γj-2tj∗,0], where (49)Ωγj≔RnifΩ=Rn,BR/γjifΩ=BR.Clearly, (50)Ωγj→Rnasj→∞.From (44) and (48), we see that (51)φ2γj+∇φ2γj2β/1+2β→0,asj→∞.Thus φ2γj and ∇φ2γjare bounded in Ωγj×(-γj-2tj∗,0] for all j.
By applying Step 2, there isLk independent of j, such that the uniform Schauder’s estimates of (φ1γj,φ2γj) are as follows: (52)φ1γjC2+σ,1+σ/2y∈Ωγj,y≤k×-k,0≤Lk,φ2γjC2+σ,1+σ/2y∈Ωγj,y≤k×-k,0≤LkSince (φ1γj,φ2γj) is defined on a compact set, by the Arzela-Ascoli theorem, there exists a convergent subsequence, and it is denoted by (φ1γj,φ2γj).
Sinceμ1,μ2>0 and ∇φ1γj,∇φ2γj are bounded, the limit point (φ1,φ2) is a solution of the following system: (53)φ1s=Δφ1+φ2p1,φ2s=Δφ2+φ1p2(54)inRn×-∞,0Since φ2γj→0,wherej→∞, it follows that φ2≡0inRn×(-∞,0].
Consequently, from the second equation of (53), we get (55)φ1≡0,inRn×-∞,0.Thus, (56)φ10,0+∇φ10,02α/1+2α=0,which leads to a contradiction with (48), so the lower bound is proved.
If we change the roles ofu and v, the upper bound of (25) can be proved similarly as in the last proof.
Step 4 (estimates on doubling M u ). Since Mu→∞ as t→T, Mu is a continuous function. For any t0∈(0,T), the point t0+ can be defined as follows: (57)t0+=max⁡t∈t0,T:Mut=2Mut0.Clearly, (58)ux,t+∇ux,t2α/1+2α≤2Mut0,x,t∈Ω×0,t0+.Take γ=γ(t0)=Mu-1/2α(t0).
We claim that there is0<A which is independent of t0 such that (59)t0+-t0γ2t0≤A,t0∈T2,T,By supposing that this claim is not true, there is a sequence tj→T, as j→∞ such that (60)tj+-tjγj2tj→∞,where (61)tj+=max⁡t∈tj,T:Mut=2Mutj.For each tj, where T/2<tj<tj+<T,∀j, we can choose 0<tj∗≤tj.
As in Step 3, we scale about the corresponding point(xj∗,tj∗), and we can get the corresponding rescaled functions (φ1γj,φ2γj) with the scaling factor: γj=γ(tj)=Mu-1/2α(tj), which satisfies (47) with the following conditions:(62)φ1γj0,0+∇φ1γj0,02α/1+2α≥12,0≤φ1γj+∇φ1γj2α/1+2α≤2,φ2γj+∇φ2γj2β/1+2β≤Mu-β/αtjMvtj+(63)y,s∈Ωγj×-tj∗γj2,tj+-tj∗γj2.From (61) and (62), it follows that (64)φ2γj+∇φ2γj2β/1+2β≤2β/αMu-β/αtj+Mvtj+.From (25), we conclude that(65)Mvt≤δ-2βMuβ/αt,t∈T2,T.Therefore, (64) becomes (66)φ2γj+∇φ2γj2β/1+2β≤2β/αδ2βBy applying Step 2, we use the Schauder estimates for (φ1γj,φ2γj), and we can get a convergent subsequence in Cloc2+σ,1+σ/2(Rn×R) to the solution of system (53) in Rn×R.
Thus, we get a contradiction because under the assumption (i), all nontrival solutions of system (53) blow up in a finite time; see [17].
So, there is0<A such that (67)γ-2t0t0+-t0≤A,t0∈T2,T.
Step 5 (rate estimates).
For anyt0∈(T/2,T), as in Step 4,
defineMu(t1)=2Mu(t0), where t1=t0+∈(t0,T).
By (67), we have (68)t1-t0≤AMu-1/αt0.We can get t2∈(t,T) by using t1 as a new value of t0 such that (69)Mut2=2Mut1=4Mut0.Thus, (70)t2-t1≤AMu-1/αt1=2-1/αAMu-1/αt0.So, for any j≥0, we have (71)tj+1-tj≤2-j/αAMu-1/αt0,where the sequence tj→T as j→∞.
By adding the above inequalities, it follows that(72)T-t0≤∑j≥02-j/αAMu-1/αt0.Thus, (T-t0)≤(1-2-1/α)-1AMu-1/α(t0).
Using (25) results in (73)Mvt0≤δ-2βMuβ/αt0,t0∈T2,T.Thus, (74)Mvt0≤δ-2β1-2-1/α-βAβT-t0-β,t0∈T2,T.From above, there exist positive two constants a and b such that(75)Mut0≤aT-t0-α,t0∈T2,T,Mvt0≤bT-t0-β,t0∈T2,T.From the last two inequalities and the definitions of MuandMv, it follows that there are C1andC2 such that (76)ux,t+∇ux,t2α/1+2α≤C1T-t-α,vx,t+∇vx,t2β/1+2β≤C2T-t-β.Or, we can split the last estimates as follows: (77)ux,t≤C1T-t-α,∇ux,t≤C1T-t-1+2α/2,vx,t≤C2T-t-β,∇vx,t≤C2T-t-1+2β/2where (x,t)∈Ω×(0,T).


## 4. Conclusions and Future Studies

From Theorem1 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)
The gradient terms are bounded for anyt<T.(iii)
The upper blow-up rate estimates for both of systems (1) and (12) take the same forms. This means that, with the two conditions of Theorem 1, the gradient terms in the system (1) do not effect or make any changes on the profile of blow-up solutions.Next, we point out some possible future research directions:(1)
One may try to derive the blow-up rate estimates for problem (1), in case one or both assumptions (i) and (ii) of Theorem 1 are not satisfied.(2)
For the semilinear system (12) defined in a ball, and under some restricted assumptions on (u0,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).



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