# Blow-Up Analysis for a Reaction-Diffusion Model with Nonlocal and Gradient Terms

Authors: Xuhui Shen; Lun Lan
Journal: Mathematical Problems in Engineering
 (2022)

---

## Abstract

In this paper, we investigate the blow-up phenomena for the following reaction-diffusion model with nonlocal and gradient terms:ut=Δu+aup∫Ωuαdxm−∇uqinΩ×0,t∗∂u/∂ν=huon∂Ω×0,t∗ux,0=u0x≥0inΩ¯. Here Ω⊂ℝNN≥3 is a bounded and convex domain with smooth boundary, and constants m,p,q,α are supposed to be positive. Utilizing the Sobolev inequality and the differential inequality technique, lower bound for blow-up time is derived when blow-up occurs. In addition, we give an example as application to illustrate the abstract results obtained in this paper.


---

## Body

## 1. Introduction

As we all know, reaction-diffusion models can be used to illustrate many natural phenomena such as heat flow, combustion, and gravitational potentials, so they have received extensive attention from many scholars [1, 2]. Since early sixties, lots of papers concerning the problem of blow-up or global existence of solutions to reaction-diffusion models have been published. After that, qualitative properties of reaction-diffusion models were investigated, such as the blow-up set, blow-up rate, blow-up profile, and boundedness of global solutions (see the papers [3–6] and books [7, 8]).Especially, when the solution of reaction-diffusion models blows up, one would like to know in what time the solution blows up. Weissler in [9] firstly studied the blow-up time of reaction-diffusion models, and then much attention has been paid to finding the blow-up time of solutions. However, much work has been done in deriving the upper bound for the blow-up time [10]. In practical situations, lower bound for the blow-up time is more useful than upper bound for the blow-up time in predicting the critical state of the systems. This makes the study of the lower bound for the blow-up time more meaningful. In 2006, Payne and Schaefer introduced the first differential inequality technique to give the lower bound for the blow-up time (see [11]). Based on Payne’s methods, lots of works are devoted to giving the lower bound for the blow-up time when blow-up occurs (refer to [12–17]).In this paper, we investigate the blow-up time of the following reaction-diffusion model with nonlocal and gradient terms:(1)ut=Δu+aup∫Ωuαdxm−∇uqinΩ×0,t∗,∂u∂ν=huon∂Ω×0,t∗,ux,0=u0x≥0inΩ¯,where Ω⊂ℝNN≥3 is a bounded and convex domain with smooth boundary, m,p,q,α are positive constants, and t∗ is the maximal existence time of the solution u. Also, we assume that h is nonnegative C1ℝ+ function, where ℝ+=0,+∞. u0x is assumed to be nonnegative C1ℝ+ function, which is compatible with the boundary conditions. Problem (1) can describe many physical phenomena and biological species theories. For example, in the density of some biological species for population dynamics, nonlocal source term represents the births of the species, and gradient terms can illustrate the natural or the accidental deaths.To complete our research, we focus our attention on the following blow-up phenomena of the reaction-diffusion models (see [18–21]). Marras, et al. in [20] studied(2)ut=Δum+a∫Ωupdx−buq−c∇u2inΩ×0,t∗,∂u∂ν=guon∂Ω×0,t∗,ux,0=u0x≥0inΩ¯,where Ω⊂ℝNN≥1 is a bounded and convex domain with smooth boundary. When p>q, they determined lower bounds for t∗ when blow-up occurs in Ω⊂ℝ2 and Ω⊂ℝ3. In addition, when p<q, a global existence criterion for u was derived on Ω⊂ℝNN≥1.Ding and Shen in [21] investigated the following nonlocal reaction-diffusion model:(3)hut=∇⋅ρ∇u2∇u+axfuinΩ×0,t∗,∂u∂ν+γu=0on∂Ω×0,t∗,ux,0=u0x≥0inΩ¯,where Ω⊂ℝNN≥2 is a bounded and convex domain whose boundary is sufficiently smooth. Assuming that(4)axfu≤a1+a2up∫Ωuldxm,they obtained a lower bound for the blow-up time when Ω⊂ℝ3. Moreover, an upper bound for the blow-up time and global solution were also discussed.Inspired by the research studies mentioned above, we deal with the blow-up phenomena of problem (1). The highlight of this paper is to investigate the model with both nonlocal terms and gradient terms, making the research closer to reality. In addition, there is little research on the blow-up phenomenon of the solution of problem (1) and even less research on the lower bound for the blow-up time. The key to achieving our work is to build suitable auxiliary functions. Since auxiliary functions given in problems (2) and (3) are no longer applicable, we need to establish new auxiliary functions and use the Sobolev inequality to accomplish our research.The paper is organized as follows. In Section2, when Ω⊂ℝNN≥3, we derive a lower bound for the blow-up time when blow-up occurs. In Section 3, an example is given to illustrate the application of the abstract results obtained in this paper.


## 2. Lower Bound for Blow-Up Time WhenΩ⊂ℝNN≥3

In this section, we seek the lower bound for the blow-up time whenΩ⊂ℝNN≥3. For this aim, we firstly assume(5)0<hu≤buβ,with constants b>0,β>1. Moreover, we suppose constants m>0, p>1, q>2. Let the auxiliary function be defined as follows:(6)At=∫Ωuαdx,where(7)α>max2,Nβ−1,Np−12,qq−2+1.Owing to Corollary 9.14 in [22], we know that(8)W1,2Ω⟶L2N/N−2Ω,N≥3,which implies(9)∫Ωw2N/N−2dxN−2/2N≤C∫Ωw2dx+∫Ω∇w2dx1/2.Herew∈W1,2Ω and C=CN,Ω is a Sobolev embedding constant depending on N and Ω. Inequality (9) will be used in our proof. The main results are stated next.Theorem 1.
Letu be a nonnegative classical solution of problem (1). Suppose (5)–(7) hold. If the solution u blows up in the measure At at some finite time t∗, then t∗ is bounded by(10)t∗≥∫A0+∞dηD+D1ηα−β−1N−2/α+D2ηα−β−1N−2/α−Nβ−1+D3ηα+p−1/α+m+D4η2αm+1−p−1N−2/2α−Np−1,where(11)D=B1,D1=B2C2Nβ−1/α,D2=B2α−Nβ−1αε2−Nβ−1/α−Nβ−1C2Nβ−1/α−Nβ−1,(12)D3=aαCNp−1/α,D4=aα2α−Np−12αε3−Np−1/2α−Np−1C2Np−1/2α−Np−1,(13)B1=abNβ−1L0α+2β−2|Ω|+αq−22β−1+qqα+2β−2q2−2/q−2Ω,(14)B2=abNα+β−1L0α+2β−2+αα−1q−2−qqα+2β−2q2−2/q−2+bdαα+β−12L0ó1,(15)ε1=α2L0α+β−1d,ε2=αB2Nβ−1,ε3=2aNp−1.where Ω is the measure of the bounded and convex domain Ω, L0=min∂Ωx⋅ν>0, and d=maxΩ¯x.Proof.
Using (5)–(7) and the divergence theorem, we have(16)A′t=α∫Ωuα−1utdx=α∫Ωuα−1Δu+aup∫Ωuαdxm−∇uqdx=−αα−1∫Ωuα−2∇u2dx+α∫∂Ωuα−1hudS+aα∫Ωuα+p−1dx∫Ωuαdxm−α∫Ωuα−1∇uqdx=−αα−1∫Ωuα−2∇u2dx+bα∫∂Ωuα+β−1dS+aα∫Ωuα+p−1dx∫Ωuαdxm−α∫Ωuα−1∇uqdx.
It follows from the Hölder inequality that(17)∫Ωuα−2∇u2dx≤∫Ωuα−1∇uqdx2/q∫Ωuα−1−q/q−2dxq−2/q≤∫Ωuα−1∇uqdx+q−2qq2−2/q−2∫Ωuα−1−q/q−2dx,which is equivalent to(18)−α∫Ωuα−1∇uqdx≤−α∫Ωuα−2∇u2dx+q−2αqq2−2/q−2∫Ωuα−1−q/q−2dx.
Inserting (18) into (16), we derive(19)A′t≤−α2∫Ωuα−2∇u2dx+bα∫∂Ωuα+β−1dS+aα∫Ωuα+p−1dx∫Ωuαdxm+q−2αqq2−2/q−2∫Ωuα−1−q/q−2dx.
Recalling inequality (2) in [20], we have(20)∫∂Ωuα+β−1dS≤NL0∫Ωuα+β−1dx+α+β−1dL0∫Ωuα+β−2∇udx.
We apply the Hölder inequality and the Young inequality to the term∫Ωuα+β−2∇udx to deduce(21)∫Ωuα+β−2∇udx≤ε1∫Ωuα−2∇u2dx1/21ε1∫Ωuα+2β−2dx1/2≤ε12∫Ωuα−2∇u2dx+12ε1∫Ωuα+2β−2dx,where ε1 is given in (15). The substitution of (21) into (20) yields(22)∫∂Ωuα+β−1dS≤NL0∫Ωuα+β−1dx+α+β−1dε12L0∫Ωuα−2∇u2dx+α+β−1d2L0ε1∫Ωuα+2β−2dx.
We insert (22) into (19) to get(23)A′t≤−12α2∫Ωuα−2∇udx+αbNL0∫Ωuα+β−1dx+bdαα+β−12L0ε1∫Ωuα+2β−2dx+aα∫Ωuα+p−1dx∫Ωuαdxm+q−2αqq2−2/q−2∫Ωuα−1−q/q−2dx=−2∫Ω∇uα/22dx+αbNL0∫Ωuα+β−1dx+bdαα+β−12L0ε1∫Ωuα+2β−2dx+aα∫Ωuα+p−1dx∫Ωuαdxm+q−2αqq2−2/q−2∫Ωuα−1−q/q−2dx.
From (7), the Hölder inequality, and the Young inequality, we can deduce that(24)∫Ωuα+β−1dx≤∫Ωuα+2β−2dxα+β−1/α+2β−2Ωβ−1/α+2β−2≤α+β−1α+2β−2∫Ωuα+2β−2dx+β−1α+2β−2Ω,(25)∫Ωuα−1−q/q−2dx≤∫Ωuα+2β−2dxα−1q−2−q/α+2β−2q−2Ω2β−1q−2+q/α+2β−2q−2≤α−1q−2−qα+2β−2q−2∫Ωuα+2β−2dx+2β−1q−2+qα+2β−2q−2Ω.
Combining (24) and (25) with (23), we obtain(26)A′t≤−2∫Ω∇uα/22dx+abNβ−1L0α+2β−2|Ω|+α2β−1q−2+qqα+2β−2q2−2/q−2Ω+abNα+β−1L0α+2β−2+αα−1q−2−qqα+2β−2q2−2/q−2+bdαα+β−12L0ε1∫Ωuα+2β−2dx+aα∫Ωuα+p−1dx∫Ωuαdxm≤−2∫Ω∇uα/22dx+B1+B2∫Ωuα+2β−2dx+aα∫Ωuα+p−1dx∫Ωuαdxm,where B1 and B2 are defined in (13) and (14), respectively. Applying (7) and Sobolev inequality (9), we have(27)∫Ωuα+2β−2dx≤∫Ωuαdx1−β−1N−2/α∫Ωuα/22N/N−2dxβ−1N−2/α≤∫Ωuαdx1−β−1N−2/αC2N/N−2∫Ωuαdx+∫Ω∇uα/22dxN/N−2β−1N−2/α=C2Nβ−1/α∫Ωuαdxα−β−1N−2/α∫Ωuαdx+∇uα/22dxNβ−1/α,(28)∫Ωuα+p−1dx∫Ωuαdxm≤∫Ωuαdx2α−p−1N−2/2α∫Ωuα/22N/N−2dxp−1N−2/2α∫Ωuαdxm≤∫Ωuαdx2α−p−1N−2/2α+mC2N/N−2∫Ωuαdx+∫Ω∇uα/22dxN/N−2p−1N−2/2α=CNp−1/α∫Ωuαdx2α−p−1N−2/2α+m∫Ωuαdx+∫Ω∇uα/22dxNp−1/2α.
Thanks to the basic inequality(29)k1+k2l≤k1l+k2l,k1,k2>0,0≤l<1,we rewrite (27) and (28) as(30)∫Ωuα+2β−2dx≤C2Nβ−1/α∫Ωuαdxα−β−1N−2/α+C2Nβ−1/α∫Ωuαdxα−β−1N−2/α∫Ω∇uα/22dxNβ−1/α,(31)∫Ωuα+p−1dx∫Ωuαdxm≤CNp−1/α∫Ωuαdxα+p−1/α+m+CNp−1/α∫Ωuαdx2α−p−1N−2/2α+m∫Ω∇uα/22dxNp−1/2α.
It follows from the Hölder inequality and the Young inequality that(32)C2Nβ−1/α∫Ωuαdxα−β−1N−2/α∫Ω∇uα/22dxNβ−1/α≤ε2−Nβ−1/α−Nβ−1C2Nβ−1/α−Nβ−1∫Ωuαdxα−Nβ−1N−2/α−Nβ−1α−Nβ−1/α⋅ε2∫Ω∇uα/22dxNβ−1/α≤α−Nβ−1αε2−Nβ−1/α−Nβ−1C2Nβ−1/α−Nβ−1⋅∫Ωuαdxα−β−1N−2/α−Nβ−1+Nβ−1αε2∫Ω∇uα/22dx.(33)CNp−1/α∫Ωuαdx2α−p−1N−2/2α+m∫Ω∇uα/22dxNp−1/2α≤ε3−Np−1/2α−Np−1C2Np−1/2α−Np−1∫Ωuαdx2αm+1−p−1N−2/2α−Np−12α−Np−1/2α⋅ε3∫Ω∇uα/22dxNp−1/2α≤2α−Np−12αε3−Np−1/2α−Np−1C2Np−1/2α−Np−1⋅∫Ωuαdx2αm+1−p−1N−2/2α−Np−1+Np−12αε3∫Ω∇uα/22dx,where ε2,ε3 are given in (15). Moreover, combining (33) and (31) and (32) and (30), respectively, we have(34)∫Ωuα+2β−2dx≤C2Nβ−1/α∫Ωuαdxα−β−1N−2/α+α−Nβ−1αε2−Nβ−1/α−Nβ−1C2Nβ−1/α−Nβ−1∫Ωuαdxα−β−1N−2/α−Nβ−1+Nβ−1αε2∫Ω∇uα/22dx,(35)∫Ωuα+p−1dx∫Ωuαdxm≤CNp−1/α∫Ωuαdxα+p−1/α+m+2α−Np−12αε3−Np−1/2α−Np−1C2Np−1/2α−Np−1∫Ωuαdx2αm+1−p−1N−2/2α−Np−1+Np−12αε3∫Ω∇uα/22dx.
Inserting (34) and (35) into (26), we obtain(36)A′t≤B1+B2C2Nβ−1/αAα−β−1N−2/αt+B2α−Nβ−1αε2−Nβ−1/α−Nβ−1C2Nβ−1/α−Nβ−1Aα−β−1N−2/α−Nβ−1t+aαCNβ−1/αAα+p−1/α+mt+a2α−Np−12ε3−Nβ−1/2α−Nβ−1C2Np−1/2α−Np−1A2αm+1−p−1N−2/2α−Np−1t=D+D1Aα−β−1N−2/αt+D2Aα−β−1N−2/α−Nβ−1t+D3Aα+p−1/α+mt+D4A2αm+1−p−1N−2/2α−Np−1t,where D,D1,…,D4 are defined in (11) and (12). Integrating (36) between 0 and t∗, we arrive at(37)t∗≥∫A0+∞dηD+D1ηα−β−1N−2/α+D2ηα−β−1N−2/α−Nβ−1+D3ηα+p−1/α+m+D4η2αm+1−p−1N−2/2α−Np−1.


## 3. Application

In what follows, an example is given as application to illustrate the abstract results in Theorem1.Example 1.
Letu″ be a nonnegative classical solution of the following problem:(38)ut=Δu+23u2∫Ωu6dx−∇u3,inΩ×0,t∗,∂u∂ν=1100u2,on∂Ω×0,t∗,ux,0=19992000+120x2,inΩ¯,where Ω=x=x1,x2,x3|x2=x12+x22+x32<1/100 is a spherical region in ℝ3. Here we choose N=3,m=1,p=2,q=3,a=2/3,b=1/100,α=6,β=2. It is easy to verify that conditions (5)–(7) are established. Moreover, form [23], we can calculate that the embedding constant C≈7⋅5931. With simple calculations, we have L0=d=1/10,Ω=4π/3000,B1≈2.8973×10−3,B2≈0.4381,ϵ1=36/7,ϵ2≈4.5652,ϵ3=1,D1≈3.3265,D2≈2.7664,D3≈11.0223,D4≈11.5896, and(39)A0=∫Ω19992000+120|x|26dx≈41.838×10−3.
It follows from Theorem1 that(40)t∗≥∫A0+∞dηD+D1η5/6+D2η5/3+D3η7/6+D4η23/9≈∫4⋅1838×10−3+∞dη2⋅8973×10−3+3⋅3265η5/6+2⋅7664η5/3+11⋅0223η7/6+11⋅5896η23/9≈0.4168.



---

*Source: 4150728-2022-03-28.xml*