# Blow-Up Phenomena in Reaction-Diffusion Problems with Nonlocal and Gradient Terms

Authors: Xuhui Shen; Dandan Wu
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

This paper considers the blow-up phenomena for the following reaction-diffusion problem with nonlocal and gradient terms:ut=Δum+∫Ωurdx−∇us,inΩ×0,t∗,∂u/∂ν=gu,on∂Ω×0,t∗,ux,0=u0x≥0,inΩ¯. Here m>1, and Ω⊂ℝNN≥2 is a bounded and convex domain with smooth boundary. Applying a Sobolev inequality and a differential inequality technique, lower bounds for blow-up time when blow-up occurs are given. Moreover, two examples are given as applications to illustrate the abstract results obtained in this paper.


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## Body

## 1. Introduction

Since the 1960s, the blow-up phenomena of reaction-diffusion problems have received considerable interest. From then on, questions concerning finite-time blow-up of solutions to reaction-diffusion problems, as well as other classes of problems, have attracted more attention. A number of studies appeared on the topic of blow-up in time (see [1–4]) or global existence and boundedness of solutions (see [5]). Moreover, qualitative properties were investigated such as the blow-up set, rate, and profile of the blow-up [6].From a practical point of view, apart from considering the above problems, one would like to know whether the solutions blow up, and if so, at what time blow-up occurs. However, when the solution does blow up at some finite time, the blow-up time can seldom be determined explicitly, and much effort has been devoted to the calculations of bounds for the blow-up time. As we know, the studies of many papers are lead to upper bounds for the blow-up time when blow-up does occur (see [7–10]). For the practical situations, lower bounds for the blow-up time are more important, which can be used to predict the unsteady state of the systems more accurately.In 2006, Payne and Schaefer [11] used the differential inequality technique to derive the lower bound for the blow-up time when Ω⊂ℝ3. Based on their work, many scholars tend to seek lower bounds for quantity of reaction-diffusion problems when Ω⊂ℝ3 (see [12] and the reference therein) and when Ω⊂ℝNN≥3 (see [13, 14]). In order to expand the underpinning theory of the mathematical analysis of problem, we aim to derive the results that have been extended to more general reaction-diffusion problems when Ω⊂ℝNN≥2. In addition, aiming to be closer to more realistic models, in this paper, we deal with the following reaction-diffusion problem with nonlocal and gradient terms:(1)ut=Δum+∫Ωurdx−∇usinΩ×0,t∗,∂u∂ν=guon∂Ω×0,t∗,ux,0=u0x≥0inΩ¯,where m>1, Ω⊂ℝℕN≥2 is a bounded and convex domain with smooth boundary, ∂u/∂ν denotes the outward normal derivative on ∂Ω, and t∗ is the blow-up time if blow-up occurs. Throughout this paper, we suppose g are nonnegative Cℝ+¯ functions and u0 are nonnegative C1Ω¯ functions satisfying compatibility conditions. Problem (1) appears in the mathematical models for gas or fluid flow in porous media (see [15]); it can also be used to describe the evolution of some biological population u (cells, bacteria, etc.), which live in a certain domain and whose growth is influenced by the law +∫Ωurdx−∇us. Nonlocal terms +∫Ωurdx represent the births of the species, and −∇us represents the accidental deaths of the species. For other related references, readers can refer to [16, 17].In order to achieve our goal, we mainly pay our attention to the following works [18, 19]. Marras et al. in [18] investigated the following problem:(2)ut=Δu+k1tupdx−k2t∇uqinΩ×0,t∗,∂u∂ν+γu=0on∂Ω×0,t∗,ux,0=u0x≥0inΩ¯,where Ω⊂ℝNN≥2 is a bounded and convex domain with smooth boundary. They obtained A lower bound for the blow-up time when Ω⊂ℝ3. Song in [19] considered the following reaction-diffusion problem:(3)ut=Δu+a∫Ωupdx−kuqinΩ×0,t∗,∂u∂ν=0on∂Ω×0,t∗,ux,0=u0x≥0inΩ¯,where Ω⊂ℝ3 is a bounded and convex domain with smooth boundary. The authors obtained A lower bound for the blow-up time when blow-up occurs.Inspired by the aforementioned research studies, we consider the blow-up phenomena of problem (1). The highlight of this paper is considering both gradient term and nonlocal term sources, which make the problem more closer to the 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 main difficulty in studying (1) is to build suitable auxiliary functions. Since auxiliary functions defined in problems (2) and (3) are no longer applicable for our study, it is necessary to construct new auxiliary functions and use Sobolev inequalities to accomplish our research.The paper is organized as follows. In Section2, when Ω⊂ℝNN≥3, we obtain a criterion for blow-up of the solution of problem (1) and give a lower bound for blow-up time. In Section 3, when Ω⊂ℝ2, a lower bound for blow-up time is derived. In Section 4, we present two examples to illustrate the applications of the abstract results obtained in this paper.


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

In this section, our aim is to determine a lower bound for blow-up timet∗ when Ω⊂ℝNN≥3. We now assume(4)0<gu≤aul,with constants a>0,l>1. Moreover, we suppose constants r>1, s>2, and(5)s>m>1,m+2l>r+2.Let us define the following auxiliary function:(6)Jt=∫Ωuαdx,where(7)α>max1,3−m,3−m−2m−2s−2,Nl−1.It is known from Corollary 9.14 in [20] 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. In this section, we need to use Sobolev inequality (9). The main result is formulated next.Theorem 1.
Letu be a nonnegative classical solution of problem (1). Suppose (4)–(7) hold. If the solution u blows up in the measure Jt at some finite time t∗, then t∗ is bounded by(10)t∗≥∫J0+∞dηA1+A2η2α+Nm−1−m+2l−3N−2/2α+2N1−l,where(11)A1=A+M4l−1α+m+2l−3α+m+2l−32α+m−1−α+m−1/2l−1Ω,(12)A2=M4α+N1−l2α+Nm−1CNm+2l−3/α+N1−l2α+Nm−1Nm+2l−3−Nm+2l−3/α+N1−l+ε2−Nm+2l−3/α+N1−l,(13)A=maαNl−1ρ0α+m+2l−3Ω+αm+2l−r−2α+m+2l−3Ω2+2αls−2−m−2sα+m+2l−3s2−2/s−2Ω,(14)M=maαNα+m+l−2ρ0α+m+2l−3+maαdα+m+l−22ε1ρ0+αα+r−1α+m+2l−3Ω+αs−2α+m−3+2αm−2sα+m+2l−3s2−2/s−2,(15)ε1=ρ0mα−1+1madα+m+l−2,ε2=αmα−1+12α+Nm−1MNm+2l−3α+m−12,where Ω is the measure of the bounded and convex domain Ω, ρ0=min∂Ωx⋅ν>0, and d=maxΩ¯x.Proof.
We compute by using (4), (7), and the divergence theorem(16)J′t=α∫Ωuα−1utdx=α∫Ωuα−1Δum+∫Ωurdx−|∇u|sdx=α∫Ωuα−1Δumdx+α∫Ωuα−1dx∫Ωurdx−α∫Ωuα−1∇usdx=−mαα−1∫Ωuα+m−3∇u2dx+αm∫∂Ωuα+m−2∂u∂νdS+α∫Ωuα−1dx∫Ωurdx−α∫Ωuα−1∇usdx=−mαα−1∫Ωuα+m−3∇u2dx+αm∫∂Ωuα+m−2gudS+α∫Ωuα−1dx∫Ωurdx−α∫Ωuα−1∇usdx≤−mαα−1∫Ωuα+m−3∇u2dx+aαm∫∂Ωuα+m+l−2dS+α∫Ωuα−1dx∫Ωurdx−α∫Ωuα−1∇usdx.
It follows from the Holder inequality that(17)∫Ωuα+m−3∇u2dx≤∫Ωuα−1|∇u|sdx2/s∫Ωuα+m−3+2m−2/s−2dxs−2/s≤s2∫Ωuα−1|∇u|sdx2/ss2−2/s−2∫Ωuα+m−3+2m−2/s−2dxs−2/s≤∫Ωuα−1∇usdx+s−2ss2−2/s−2∫Ωuα+m−3+2m−2/s−2dx,which is equivalent to(18)−α∫Ωuα−1∇usdx≤−α∫Ωuα+m−3∇u2dx+αs−2ss2−2/s−2∫Ωuα+m−3+2m−2/s−2dx.
Inserting (18) into (16), we derive(19)J′t≤−mαα−1∫Ωuα+m−3∇u2dx+maα∫∂Ωuα+m+l−2dS+α∫Ωuα−1dx∫Ωurdx−α∫Ωuα+m−3∇u2dx+αs−2ss2−2/s−2∫Ωuα+m−3+2m−2/s−2dx≤−4αmα−1+1α+m−12∫Ω∇uα+m−1/22dx+maα∫∂Ωuα+m+l−2dS+α∫Ωuα−1dx∫Ωurdx+αs−2ss2−2/s−2∫Ωuα+m−3+2m−2/s−2dx.
In view of the lemma in [21], we have(20)∫∂Ωuα+m+l−2dS≤Nρ0∫Ωuα+m+l−2dx+α+m+l−2dρ0∫Ωuα+m+l−3∇udx.
For the second term on the right side of (20), we apply the Ho..lder inequality and the Young inequality to get(21)∫Ωuα+m+l−3∇udx≤∫Ωuα+m+2l−3dx1/2∫Ωuα+m−3∇u2dx1/2≤12ε1∫Ωuα+m+2l−3dx+ε12∫Ωuα+m−3∇u2dx=12ε1∫Ωuα+m+2l−3dx+2ε1α+m−12∫Ω∇uα+m−1/22dx,where ε1 is given in (15). Substituting (20) and (21) into (19), we derive(22)J′t≤−4αmα−1+1α+m−12∫Ω∇uα+m−1/22dx+maαNρ0∫Ωuα+m+l−2dx+α+m+l−2dρ0∫Ωuα+m+l−3∇udx+α∫Ωuα−1dx∫Ωurdx+αs−2ss2−2/s−2∫Ωuα+m−3+2m−2/s−2dx≤−4αmα−1+1α+m−12∫Ω∇uα+m−1/22dx+maαNρ0∫Ωuα+m+l−2dx,+maαdα+m+l−2ρ012ε1∫Ωuα+m+2l−3dx+2ε1α+m−12∫Ω∇uα+m−1/22dx+α∫Ωuα−1d∫Ωurdx+αs−2ss2−2/s−2∫Ωuα+m−3+2m−2/s−2dx=−2αmα−1+1α+m−12∫Ω∇uα+m−1/22dx+maαNρ0∫Ωuα+m+l−2dx+maαdα+m+l−22ρ0ε1∫Ωuα+m+2l−3dx+α∫Ωuα−1d∫Ωurdx+αs−2ss2−2/s−2∫Ωuα+m−3+2m−2/s−2dx.
It follows from (5)–(7), the Ho..lder inequality, and the Young inequality that(23)∫Ωuα−1dx∫Ωurdx≤∫Ωuα+r−1dxΩ≤∫Ωuα+m+2l−3dxα+r−1/α+m+2l−3Ωm+2l−r−2/α+m+2l−3Ω=α+r−1α+m+2l−3Ω∫Ωuα+m+2l−3dx+m+2l−r−2α+m+2l−3Ω2,(24)∫Ωuα+m−3+2m−2/s−2dx≤∫Ωuα+m+2l−3dxs−2α+m−3+2m−2/s−2α+m+2l−3|Ω|2ls−2−2m−2/s−2α+m+2l−3≤s−2α+m−3+2m−2s−2α+m+2l−3∫Ωuα+m+2l−3dx+2ls−2−2m−2s−2α+m+2l−3Ω,(25)∫Ωuα+m+l−2dx≤∫Ωuα+m+2l−3dxα+m+l−2/α+m+2l−3|Ω|l−1/α+m+2l−3≤α+m+l−2α+m+2l−3∫Ωuα+m+2l−3dx+l−1α+m+2l−3Ω.
Substituting (23)–(25) into (22) yields(26)J′t≤−2αmα−1+1α+m−12∫Ω∇uα+m−1/22dx+A+M∫Ωuα+m+2l−3dx,where A,M are defined in (13) and (14), respectively. Applying (7) and the Sobolev inequality (9), we have(27)∫Ωuα+m+2l−3dx≤∫Ωuαdx2α+Nm−1−m+2l−3N−2/2α+Nm−1⋅∫Ωuα+m−1/22N/N−2dxm+2l−3N−2/2α+Nm−1≤∫Ωuαdx2α+Nm−1−m+2l−3N−2/2α+Nm−1⋅C2N/N−2∫Ωuα+m−1dx+∫Ω∇uα+m−1/22dxNN−2m+2l−3N−2/2α+Nm−1=C2Nm+2l−3/2α+Nm−1∫Ωuαdx2α+Nm−1−m+2l−3N−2/2α+Nm−1⋅∫Ωuα+m−1dx+∫ΩdxNm+2l−3/2α+Nm−1.≤∫Ωuαdx2α+Nm−1−m+2l−3N−2/2α+Nm−1∫Ωuα+m−1/22N/N−2dxm+2l−3N−2/2α+Nm−1≤∫Ωuαdx2α+Nm−1−m+2l−3N−2/2α+Nm−1C2N/N−2∫Ωuα+m−1dx+∫Ω∇uα+m−1/22dxNN−2m+2l−3N−2/2α+Nm−1=C2Nm+2l−3/2α+Nm−1∫Ωuαdx2α+Nm−1−m+2l−3N−2/2α+Nm−1∫Ωuα+m−1dx+∫ΩdxNm+2l−3/2α+Nm−1.
Using the inequality(28)i1+i2j≤i1j+i2j,i1,i2>0,0≤j<1,we rewrite inequality (27) as(29)∫Ωuα+m+2l−3dx≤C2Nm+2l−3/2α+Nm−1⋅∫Ωuαdx2α+Nm−1−m+2l−3N−2/2α+Nm−1⋅∫Ωuα+m−1dxNm+2l−3/2α+Nm−1+C2Nm+2l−3/2α+Nm−1⋅∫Ωuαdx2α+Nm−1−m+2l−3N−2/2α+Nm−1⋅∫Ω∇uα+m−1/22dxNm+2l−3/2α+Nm−1.
For the first term on the right side of the (29), we make use of the Ho..lder inequality and the Young inequality to get(30)C2Nm+2l−3/2α+Nm−1∫Ωuαdx2α+Nm−1−m+2l−3N−2/2α+Nm−1⋅∫Ωuα+m−1dxNm+2l−3/2α+Nm−1=2C2Nm+2l−32α+Nm−1Nm+2l−3/2α+2N1−l⋅∫Ωuαdx2α+Nm−1−m+2l−3N−2/2α+2N1−l2α+2N1−l/2α+Nm−1⋅2α+Nm−12Nm+2l−3∫Ωuα+m−1dxNm+2l−3/2α+Nm−1≤2α+2N1−l2α+Nm−1C2Nm+2l−32α+Nm−1Nm+2l−3/2α+2N1−l⋅∫Ωuαdx2α+Nm−1−m+2l−3N−2/2α+2N1−l+∫Ωuα+m−1dx.
Moreover, it follows from the Ho..lder inequality and the Young inequality that(31)∫Ωuα+m−1dx≤α+m+2l−32α+m−1∫Ωuα+m+2l−3dxα+m−1/α+m+2l−3α+m+2l−32α+m−1−α+m−1/2l−1Ω2l−1/α+m+2l−3≤12∫Ωuα+m+2l−3dx+2l−1α+m+2l−3α+m+2l−32α+m−1−α+m−1/2l−1Ω.
Again making use of the Ho..lder inequality and the Young inequality to the second term on the right side of (29), we obtain(32)C2Nm+2l−3/2α+Nm−1∫Ωuαdx2α+Nm−1−m+2l−3N−2/2α+Nm−1⋅∫Ω∇uα+m−1/22dxNm+2l−3/2α+Nm−1≤C2ε2−1Nm+2l−3/2α+2N1−l⋅∫Ωuαdx2α+Nm−1−m+2l−3N−2/2α+2Nm−12α+2N1−l/2α+Nm−1⋅ε2∫Ω∇uα+m−1/22dxNm+2l−3/2α+Nm−1≤2α+2N1−l2α+Nm−1C2ε2−1Nm+2l−3/2α+2N1−l⋅∫Ωuαdx2α+Nm−1−m+2l−3N−2/2α+2N1−l+Nm+2l−32α+Nm−1ε2∫Ω∇uα+m−1/22dx.
Inserting (31)–(33) into (29), we have(33)∫Ωuα+m+2l−3dx≤4α+N1−l2α+Nm−1CNm+2l−3/α+N1−l⋅2α+Nm−1Nm+2l−3−Nm+2l−3/2α+2N1−l+ε2−Nm+2l−3/2α+2N1−l⋅∫Ωuαdx2α+Nm−1−m+2l−3N−2/2α+2N1−l+4l−1α+m+2l−3α+m+2l−32α+m−1−α+m−1/21−lΩ+2Nm+2l−32α+Nm−1ε2∫Ω∇uα+m−1/22dx.
Combining (33) and (26), we derive(34)J′t≤A1+A2∫Ωuαdx2α+Nm−1−m+2l−3N−2/2α+2N1−l,where A1,A2 are given in (11) and (12). Integrating between 0 and t∗, we arrive at(35)t∗≥∫J0+∞dηA1+A2η2α+Nm−1−m+2l−3N−2/2α+2N1−l.


## 3. Lower Bound for Blow-Up Time WhenΩ⊂ℝ2

In this section, we will give a lower bound for the blow-up time whenΩ⊂ℝ2. Here we still suppose that conditions (4) and (5) hold. Since the embedding theorem in (9) is no longer available when N=2, before proving our main theorem, we note that the following Sobolev embedding:(36)W1,2Ω⟶LδΩ,N=2,δ>2,implies that(37)∫Ωwδdx1/δ≤C˜∫Ωw2dx+∫Ω∇w2dx1/2,where C˜=C˜Ω is the embedding constant depending on Ω. In addition, assume that(38)Ht=∫Ωuβdx,where(39)β>max1,3−m,3−m−2m−2s−2,2δl−1δ−2.Our main results are stated next.Theorem 2.
Letu be a nonnegative classical solution of problem (1). Assume that (4), (5), and (39) hold. If the solution u blows up in the measure Ht at some finite time t∗, a lower bound for t∗ is(40)t∗≥∫H0+∞dηA˜1+A˜2ηβδ−2+δm−1−2m+2l−3/βδ−2−2δl−1,where(41)A˜1=A˜+M˜4l−1β+m+2l−3β+m+2l−32β+m−1−β+m−1/2l−1Ω,(42)A˜2=M˜2βδ−2−4δl−1βδ−2+δm−1C˜2δm+2l−3/βδ−2−2δl−1ε˜2−δm+2l−3/βδ−2−2δl−1+βδ−2+δm−12δm+2l−3−δm+2l−3/βδ−2−2δl−1,(43)A˜=2maβl−1ρ0β+m+2l−3Ω+βm+2l−r−2β+m+2l−3Ω2+2βls−2−m−2sβ+m+2l−3s2−2/s−2Ω,(44)M˜=2maββ+m+l−2ρ0β+m+2l−3+maβdβ+m+l−22ε˜1ρ0+ββ+r−1β+m+2l−3Ω+βs−2β+m−3+2βm−2sβ+m+2l−3s2−2/s−2,(45)ε˜1=ρ0mβ−1+1madβ+m+l−2,ε˜2=βmβ−1+1βδ−2+δm−1M˜δm+2l−3δ+m−12,where Ω is the measure of the bounded and convex domain Ω, ρ0=min∂Ωx⋅ν>0, and d=maxΩ¯x.Proof.
Repeating the calculations in (16)–(25), we have(46)H′t≤−2βmβ−1+1β+m−12∫Ω∇uβ+m−1/22dx+A˜+M˜∫Ωuβ+m+2l−3dx,where A˜,M˜ are defined in (43) and (44). Now, we pay our attention to the term ∫Ωuβ+m+2l−3dx. It follows from the Ho..lder inequality and Sobolev inequality (37) that(47)∫Ωuβ+m+2l−3dx≤∫Ωuβdx1−2m+2l−3/βδ−2+δm−1∫Ωuβ+m−1/2δdx2m+2l−3/βδ−2+δm−1≤∫Ωuβdx1−2m+2l−3/βδ−2+δm−1C˜δ∫Ωuβ+m−1dx+∫Ω∇uβ+m−1/22dxδ/22m+2l−3/βδ−2+δm−1=C˜2m+2l−3/βδ−2+δm−1∫Ωuβdx1−2m+2l−3/βδ−2+δm−1∫Ωuβ+m−1dx+∫Ω∇uβ+m−1/22dx2m+2l−3/βδ−2+δm−1.
Again using inequality (28), we rewrite (38) as(48)∫Ωuβ+m+2l−3dx≤C˜2δm+2l−3/βδ−2+δm−1∫Ωuβdx1−2m+2l−3/βδ−2+δm−1∫Ωuβ+m−1dxδm+2l−3/βδ−2+δm−1+C˜2δm+2l−3/βδ−2+δm−1∫Ωuβdx1−2m+2l−3/βδ−2+δm−1∫Ω∇uβ+m−1/22dxδm+2l−3/βδ−2+δm−1.
For the first term on the right side of inequality (48), we make use of the Ho..lder inequality and the Young inequality to obtain(49)C˜2δm+2l−3/βδ−2+δm−1∫Ωuβdx1−2m+2l−3/βδ−2+δm−1⋅∫Ωuβ+m−1dxδm+2l−3/βδ−2+δm−1≤2C˜2δm+2l−3βδ−2+δm−1δm+2l−3/βδ−2−2δl−1⋅∫Ωuβdxβδ−2+δm−1−2m+2l−3/βδ−2−2δl−1βδ−2−2δl−1/βδ−2+δm−1⋅βδ−2+δm−12δm+2l−3∫Ωuβ+m−1dxδm+2l−3/βδ−2+δm−1≤βδ−2−2δl−1βδ−2+δm−12C˜2δm+2l−3βδ−2+δm−1δm+2l−3/βδ−2−2δl−1⋅∫Ωuβdxβδ−2+δm−1−2m+2l−3/βδ−2−2δl−1+∫Ωuβ+m−1dx.
Moreover,(50)∫Ωuβ+m−1dx≤∫Ωuβ+m+2l−3dxβ+m−1/β+m+2l−3Ω2l−1/β+m+2l−3=β+m+2l−32β+m−1∫Ωuβ+m+2l−3dxβ+m−1/β+m+2l−3β+m+2l−32β+m−1−β+m−1/2l−1Ω2l−1/β+m+2l−3≤12∫Ωuβ+m+2l−3dx+2l−1β+m+2l−3β+m+2l−32β+m−1−β+m−1/2l−1Ω.
Similar to the computation process in (50), we get(51)C˜2δm+2l−3/βδ−2+δm−1∫Ωuβdx1−2m+2l−3/βδ−2+δm−1⋅∫Ω∇uβ+m−1/22dxδm+2l−3/βδ−2+δm−1≤C˜2ε2˜−1δm+2l−3/βδ−2−2δl−1⋅∫Ωuβdxβδ−2+δm−1−2m+2l−3/βδ−2−2δl−1βδ−2−2δl−1/βδ−2+δm−1⋅ε˜2∫Ω∇uβ+m−1/22dx2δm+2l−3/βδ−2+δm−1≤βδ−2−2δl−1βδ−2+δm−1C˜2ε˜2−1δm+2l−3/βδ−2−2δl−1⋅∫Ωuβdxβδ−2+δm−1−2m+2l−3/βδ−2−2δl−1+δm+2l−3βδ−2+δm−1ε˜2∫Ω∇uβ+m−1/22dx,where ε2˜ is given in (45). It follows from (48)–(51) that(52)∫Ωuβ+m+2l−3dx≤2βδ−2−4δl−1βδ−2+δm−1C˜2δm+2l−3/βδ−2−2δl−1⋅βδ−2+δm−12δm+2l−3−δm+2l−3/βδ−2−2δl−1+ε˜2−δm+2l−3/βδ−2−2δl−1⋅∫Ωuβdxβδ−2+δm−1−2m+2l−3/9βδ−2−2δl−1+4l−1β+m+2l−3β+m+2l−32β+m−1−β+m−1/2l−1Ω+ε˜22δm+2l−3βδ−2+δm−1∫Ω∇uβ+m−1/22dx.
Substituting (52) into (46), we have(53)H′t≤A˜1+A˜2Hβδ−2+δm−1−2m+2l−3/βδ−2−2δl−1t,where A˜1,A˜2 are defined in (41) and (42). Integrating (53) between 0 and t∗, we deduce that(54)t∗≥∫H0+∞dηA˜1+A˜2ηβδ−2+δm−1−2m+2l−3/βδ−2−2δl−1.


## 4. Applications

In this section, two examples are given to illustrate the abstract results of Theorems1 and 2.Example 1.
Letux,t be the nonnegative classical solution of the following problem:(55)ut=Δu2+∫Ωu2dx−∇u3in∂Ω×0,t∗,∂u∂ν=1100u2on∂Ω×0,t∗,ux,0=9.9995×10−2+5×10−4x2inΩ¯,where Ω=x=x1,x2,x3|x2=x12+x22+x32<1/100 is a ball in ℝ3. Here we choose N=3,m=2,l=2,α=6,a=1/100,r=2,s=3,ρ0=d=1/10,Ω=4π/3000, and u0x=9.9995×10−2+5×10−4x2. It is easy to check that conditions (4)–(6) hold. Moreover, from Theorems 2.1 and 3.2 in [22], we have C≈7.5931. By inserting the above constants into (12)–(16), we get ε1≈68.7500,ε2≈0.6034,M≈3.7204,A≈3.3538×10−3,A1≈3.5870×10−2,andA2≈3.3855×103. It follows from Theorem 1 that(56)t∗≥∫J0+∞dηA1+A2η2≈∫4.1883×10−9+∞dη3.5870×10−2+3.3855×103η2≈0.1425,with(57)J0=∫Ωu0αdx=∫Ω9.9995×10−2+5×10−4x26dx≈4.1883×10−9.Example 2.
Letux,t be the nonnegative classical solution of following problem:(58)ut=Δu2+∫Ωu2dx−∇u3in∂Ω×0,t∗,∂u∂ν=1100u2on∂Ω×0,t∗,ux,0=9.9995×10−2+5×10−4x2inΩ¯.
HereΩ=x=x1,x2‖x|2=x12+x22<1/100 is a circular domain in ℝ2. Select δ=4,m=2,l=2,β=7,r=2,s=3,a=1/100, ρ0=d=1/10, Ω=π/100, and u0x=9.9995×10−2+5×10−4x2. Moreover, combining (41)–(45), we can compute A1˜≈2.9720×10−1,A2˜≈1.1905×103,A˜≈2.3210×10−2,ε1˜≈72.2223,ε2˜≈1.6412,andM˜≈3.3269. It is easy to verify that (4), (5), and (39) hold. According Theorem 2, we have the lower bound for the blow-up time:(59)t∗≥∫H0+∞dηA1˜+A2˜τ2≈∫4.1882×10−10+∞dη2.9720×10−1+1.1905×103η2≈8.3509×10−2,where(60)H0=∫Ωu0βdx=∫Ω9.9995×10−2+5×10−4x27dx≈4.1882×10−10.



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*Source: 8364982-2022-04-19.xml*