# Global Existence and Blow-Up of Solutions to a Parabolic Nonlocal Equation Arising in a Theory of Thermal Explosion

Authors: Wenyuan Ma; Zhixuan Zhao; Baoqiang Yan
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Focusing on the physical context of the thermal explosion model, this paper investigates a semilinear parabolic equationut=Δu+a∫Ωupdx,x,t∈QT,n·∇u+guu=0,x,t∈ST,ux,0=u0x,x∈Ω with nonlocal sources under nonlinear heat-loss boundary conditions, where a,p>0 is constant, QT=Ω×0,T, ST=∂Ω×0,T, and Ω is a bounded region in RN,N≥1 with a smooth boundary ∂Ω. First, we prove a comparison principle for some kinds of semilinear parabolic equations under nonlinear boundary conditions; using it, we show a new theorem of subsupersolutions. Secondly, based on the new method of subsupersolutions, the existence of global solutions and blow-up solutions is presented for different values of p. Finally, the blow-up rate for solutions is estimated also.


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## Body

## 1. Introduction

This paper studies the following semilinear parabolic equations under nonlinear boundary conditions(1)ut=Δu+a∫Ωupdx,x,t∈QT,n·∇u+guu=0,x,t∈ST,ux,0=u0x,x∈Ω,where a,p>0 is constant, QT=Ω×0,T, ST=∂Ω×0,T, and Ω is a bounded region in RN,N≥1 with a smooth boundary ∂Ω, n is outward unit normal vector of ST, initial value u0x is nonnegative continuous function, satisfying assumption (H1) (see below), and Ω denotes Lebusgue measure of Ω.This equation can be used to describe thermal explosion or spontaneous combustion problems (see [1–3]). It differs from the classical Dirichlet boundary conditions discussed in most of the literature (see [3–9]). For examples, in [5, 7], the authors considered the following equation:
(2)ut=fuΔu+a∫Ωudx,x,t∈QT,under Dirichlet boundary conditions, where a is positive constant. And they proved the existence of global solution and showed that all the blow-up solutions are blow up globally if f satisfies ∫0∞1/fsds=∞. Furthermore, authors gave the blow-up rate in special cases as follows:
(3)c1T∗−t−1/p≤maxx∈Ω¯ux,t≤C1T∗−t−1/p,where c1,C1 are positive constants and fu=up,0<p<1. In [8], Li and Xie studied global existence of the following equation:
(4)ut−Δum=aup∫Ωuqdx,x,t∈QT,with Dirichlet boundary conditions, where a>0,m>1,p,q≥0. They obtained that there exists a global positive classical solution if p+q≤m and when p+q>m, and the solution blows up in finite time if the initial value u0 is sufficiently large. Then, the blow-up rate was given as follows:
(5)C1T∗−t−1/p+q+1≤maxx∈Ω¯ux,t≤C2T∗−t−1/p+q+1,where C1,C2 are positive constants and T∗ is the blow-up time of ux,t.In [10], the authors investigated the parabolic superquadratic diffusive Hamilton-Jacobi equations as follows:
(6)ut−Δu=∇up,inΩ×0,∞,with Dirichlet boundary condition, where p>2. They studied the gradient blow-up (GBU) solutions which are defined as
(7)T<∞⇒limt⟶T−∇u∞=∞,where T is the existence time of the unique maximal classical solution. And it was showed that in the singular region, the normal derivatives uν and uνν, which satisfy uνν~−uνp, play a dominant role.Moreover, some Fujita type results for parabolic inequalities are also studied. In [11], authors studied the quasilinear parabolic inequalities with weights and showed the existence of Fujita type exponents. And in [12], it investigated the nonexistence of nonnegative solutions of a class of quasilinear parabolic inequalities featuring nonlocal terms.There are also some interesting results on the behaviour and stability for perturbed nonlinear impulsive differential systems (see [13–19]). And the stability of stochastic differential equations with impluses is studied in [20, 21].In this paper, we will show the existence of global solution and the blow-up property of problem (1).Now some assumptions are listed below.(H1)u0∈C2+αΩ∩CΩ¯0<α<1,u0x≥0,∂u0/∂n<0(H2)g>0 and satisfies the local Lipschitz conditionIn our paper, we use the method of subsupersolutions (see [22–25]). Since the there exist nonlinear boundary conditions and nonlocal term, we list the definitions of super- and subsolutions for our problem as follows.Definition 1.
u¯x,t∈C2,1Q¯T is called a a supersolution to equation (1) if it satisfies that
(8)u¯t≥Δu¯+a∫Ωu¯pdx,x,t∈QT,n·∇u¯+gu¯u¯≥0,x,t∈ST,u¯x,0≥u0x,x∈Ω.
u¯x,t∈C2,1Q¯T is called a subsolution to equation (1) if it satisfies that
(9)u¯t≤Δu¯+a∫Ωu¯pdx,x,t∈QT,n·∇u¯+gu¯u¯≤0,x,t∈ST,u¯x,0≤u0x,x∈Ω.
Blow-up and global existence solutions are defined as follows.Definition 2.
The solutionu of the problem (1) blows up in finite time if there exists a positive real number T∗<∞, such that
(10)limt⟶T∗−supx∈Ω¯ux,t=+∞.
And the solutionu of the problem (1) exists globally if for any t∈0,+∞,
(11)supx∈Ω¯ux,t<+∞.Theorem3 states the problem of local existence of the solution to equation (1) and is the main conclusion of this paper.Theorem 3.
Supposeu¯,u¯≥0 are the sub- and supersolutions to equation (1), respectively, and u¯≤u¯. If u0x satisfies assumption (H1) and the function g satisfies assumption (H2), then there exists u^∈u¯,u¯∩Wp2,1QT, which is the solution to equation (1).
The following two theorems show that whether the solution to equation (1) exists globally or blows up in finite time is related to constant p.Theorem 4.
Suppose assumptions (H1) and (H2) hold, and the equation (1) satisfies one of the following conditions.
(i)
0<p≤1(ii)
p>1, and the initial value u0x is sufficiently small
Then, the solutionu^ of this equation exists globally.Theorem 5.
Assume (H1) and (H2). Ifp>1 and the initial value u0x is sufficiently large, then the solution u^ to equation (1) blows up in finite time.And the blow-up rate of the equation is given by Theorem6.Theorem 6.
Assume (H1)–(H3) (see below). Then, there exists a solutionux,t blowing up at T∗<∞. Specifically, there exist constants C1,C2 such that
(12)C1T∗−t−1/p−1≤ux,t≤C2T∗−t−1/p−1.Remark 7.
See Definition2 for the description of global existence and blow-up solutions.
This paper is organized as follows. In Section2, the local existence theory of solutions to equation (1) is established and Theorem 3 is proved. In Section 3, the conditions for the global existence of the solution are discussed and Theorem 4 is proved. In Section 4, the conclusions related to the blow-up solution are obtained and Theorem 5 is proved. In Section 5, the blow-up rate of the blow-up solution to equation (1) is further discussed and Theorem 6 is proved.


## 2. Proof of Theorem3

In this section, the local existence of the solution to equation (1) is proved by using the fixed-point theorem and monotone iterative technique (see [26–29]).First, the following lemma is present, which is proved according to [2].Lemma 8.
Suppose that assumptions (H1) and (H2) hold. Letwx,t∈C2+αQT∩CQT¯ and satisfy
(13)wt−dΔw≥c1w+c3∫Ωc2wdx,x,t∈QT,n·∇w+gww≥0,x,t∈ST,wx,0≥0,x∈Ω,where cix,t,i=1,2,3 are continuous and bounded functions in QT, c2,c3≥0,dx,t≥0,x,t∈QT. Then, wx,t≥0,x,t∈QT.Proof 1.
Letc¯i=supQTcix,t,i=1,2,3 and v=e−λtw, where λ=c¯1+c¯2c¯3Ω+1. Then, the first equation in equation (13) can be deduced to
(14)vt+λv−dΔv≥c1v+c3∫Ωc2vdx.
Hence,(15)vt−dΔv+λ−c1v≥c3∫Ωc2vdx.Assume by contradiction thatv<0 at some points x,t∈QT, so there must be a negative minimum value of v due to continuity, denoted as v0=vx0,t0. The following two cases are discussed.
(i)
Ifx0∈∂Ω, then(16)n·∇v0+gv0v0≥0.At this point, we haven·∇v0≥−gv0v0>0, which is contradictory to n·∇v0<0.(ii) Ifx0∈Ω∘, consider the values of each function at x0,t0. Then,
(17)λ−c1v0≥−dΔv0+λ−c1v0≥c3∫Ωc2v0dx≥c3c¯2Ωv0.It yieldsλ−c1≤c3c¯2Ω, contradicting λ≥c1+c3c¯2Ω.Combining (i) and (ii), there is no negative minimum value ofv; thus, v is nonnegative. So v=e−λtw≥0, i.e., w≥0. Lemma 8 is proved.Suppose that the assumptions of Theorem3 hold. Consider the following auxiliary problem
(18)vt−Δv+v=u+a∫Ωupdx,x,t∈QT,n·∇v+Kv=Gku,x,t∈ST,vx,0=u0x,x∈Ω,where K and Gku satisfy the following rule. Let Gu=−guu. We have that Gu is Lipschitz continuous on the interval u¯,u¯, which implies that for any u1≥u2 given, there exists a fixed positive real number K such that
(19)Gu1−Gu2≤Ku1−u2.Thus,(20)Gu1−Gu2≥−Ku1−u2.LetGku=Gu+Ku. Then, the function Gku is increasing under this definition.The auxiliary problem (12) is a third boundary value problem. It is clear that there exists a unique solution v to it, due to Theorem 3.4.7 in [9]. Define the nonlinear operator T:u¯,u¯↦u¯,u¯ such that v=Tu and construct the following sequences
(21)u1=Tu¯,u2=Tu1,⋯,un=Tun−1,⋯,v1=Tu¯,v2=Tv1,⋯,vn=Tvn−1,⋯.It can be proved that operatorT is increasing. The proof is as follows. For any y1,y2∈u¯,u¯, u¯≤y1≤y2≤u¯, let z1=Ty1,z2=Ty2,w=z2−z1. And
(22)wt−Δw+w=a∫Ωy2p−y1pdx+y2−y1≥0,x,t∈QT,n·∇w+Kw=Gky2−Gky1≥0,x,t∈ST,wx,0=0≥0,x∈Ω.Applying Lemma8, where c1=−1,c2=c3=0,d=1, we have w=z2−z1≥0, i.e., z2≥z1. Letting w=v1−u¯, the above equation is transformed into
(23)wt−Δw+w≥a∫Ωu¯p−u¯pdx+u¯−u¯=0,x,t∈QT,n·∇w+Kw≥Gku¯−Gku¯=0,x,t∈ST,wx,0=0≥0,x∈Ω,from which we deduce to v1≥u¯. The same procedure may be easily adapted to obtain u¯≥u1. Thus,
(24)u¯≤v1=Tu¯≤u1=Tu¯≤u¯.By mathematical induction onn, the above sequence (21) exhibits the following comparative relationship
(25)u¯≤v1≤v2≤⋯≤vn≤⋯≤un≤un−1≤⋯≤u1≤u¯,which shows that the sequences un,vn are increasing and bounded. So limits
(26)u^=limn⟶∞unv^=limn⟶∞vnexist. And u^=Tu^,v^=Tv^. Considering the compactness of the nonlinear operator T and Ω<∞, we know that u^,v^∈u¯,u¯∩Wp2,1QT is the solution to the auxiliary problem, so as to the problem (1). The local existence of the solution to equation (1), i.e., Theorem 3, is proved.


## 3. Proof of Theorem4

In this section, the proof of the global results of solution to equation (1) is given.Case 1.
Combining assumptions (H1) and (H2) and Definition1, ux,t=0 satisfies
(27)ut=0=Δu+a∫Ωupdx,x,t∈QT,n·∇u+guu=n∇u≤0,x,t∈ST,ux,0=0≤u0x,x∈Ω.
Therefore,ux,t=0 is a subsolution to equation (1). According to Theorem 3, we need to determine a globally existing supersolution. Set φ as the unique solution of the ellipse problem
(28)−Δφ=1,x∈Ω,n·∇φ=0,x∈∂Ω.
Letϕ=Mφ where M>0 is a constant. Obviously, on the boundary, we have
(29)n·∇ϕ+gϕϕ=Mgϕφ≥0.
And the initial valueϕ0=ϕ≥0 is
(30)ϕt−Δϕ−a∫Ωϕpdx=−MΔφ−aMp∫Ωφpdx=M−aMp∫Ωφpdx.Let equation (30) ≥0. Then, ϕ is a supersolution to equation (1) and satisfies ϕ≥0. So,
(31)M1−p≥a∫Ωφpdx.Whenp is fixed, ∫Ωφpdx is a constant. Set μ=∫Ωφpdx.
(1)
In case of0<p<1, equation (31) can be transformed into
(32)M≥a1/1−pμ1/1−p.At this time, letN is a sufficiently large constant such that Nφ≥u0. Then, we take M=a1/1−pμ1/1−p+N, which can guarantee that ϕ is a supersolution to equation (1) and the global existence of the solution u.
(2)
In case ofp>1, equation (31) can be transformed into
(33)M≤a1/1−pμ1/1−p.To ensure thatϕ is still the supersolution to equation (1), it needs to satisfy
(34)u0x≤ϕ=Mφ,x∈Ω.Without loss of generality, we can takeM=a1/1−pμ1/1−p such that u0x≤Mφ=a1/1−pμ1/1−pφ, that is, when u0x is sufficiently small, the solution u to equation (1) exists globally.Case 2.
In case ofp=1, the form of equation (1) is as follows:
(35)ut=Δu+a∫Ωudx,x,t∈QT,n·∇u+guu=0,x,t∈ST,ux,0=u0x,x∈Ω.
Letb>aΩ, δ>u0∞, and zt be the solution to the following Cauchy problem
(36)dzdt=bz,z0=δ,where t∈0,T and the solution is zt=δebt. Then, we have
(37)zt=bz>aΩz=Δz+∫Ωzdx,x,t∈QT,n·∇z+gzz=gzz≥0,x,t∈ST,z0=δ>u0x,x∈Ω.
This means that whenp=1, for any given a, zt is a supersolution to equation (1), and zt exists globally. Thus, the solution to equation (1) exists globally.
Combined with Cases1 and 2, Theorem 4 is proved.


## 4. Proof of Theorem5

The above theorem states that the classic solution of equation (1) exists globally when 0<p≤1. In this section, we will get the blow-up results of solutions to equation (1) when p>1 and prove Theorem 5.Given assumptions (H1) and (H2) andu0x>maxMφ,δ0, where M,φ are defined in Section 3 and δ0>0 is a fixed constant, let ψ be the solution of the following eigenvalue problem
(38)Δψ+λψ=0,x∈Ω,ψ=0,x∈∂Ω.We normalizeψ, i.e., ψ∞=1, and λ denotes the first eigenvalue of the problem. Let ht be the solution to the Cauchy problem below
(39)dhdt=−λht+ahpt∫Ωψpdx,h0=δ0.It can be seen that the solutionht of this equation blows up in finite time T∗ under the condition of p>1. Let
(40)v¯x,t=htψx.Equations below state thatv¯, as defined above, is a subsolution to problem (1).
(41)v¯tx,t=h′tψx=−λψht+ahpψ∫Ωψpdx,(42)Δv¯+a∫Ωv¯pdx=hΔψ+ahp∫Ωψpdx=−λψh+ahp∫Ωψpdx≥−λψh+ahpψ∫Ωψpdx=v¯t.Consider the boundary and initial value conditions(43)n·v¯+gv¯v¯=0,x,t∈ST,v¯x,0=δ0ψ≤δ0≤u0x,x,t∈Ω.Hence,v¯ is a subsolution to problem (1), when p>1. According to equations (42) and (43), set w=u−v¯. Considering mean value theorem, we have
(44)wt−Δw≥a∫Ωup−v¯pdx=a∫Ωpξp−1wdx,x,t∈QT,n·∇w+gww=n·∇u+guu=0,x,t∈ST,wx,0=u0x−v¯x,0≥u0x−δ0≥0,x∈Ω,where ξ is a nonnegative function between v¯ and u. Applying Lemma 8 with d=1,c1=0,c2=ξp−1,c3=ap, w≥0, i.e., u≥v¯ is obtained. Since ht blows up in finite time, so does v¯. Therefore, when p>1, the solution u to equation (1) blows up in finite time, which means equation (1) has at least one solution that blows up in finite time, when p>1 and u0x is sufficiently large. Theorem 5 is proved.


## 5. Proof of Theorem6

In this section, we show the blow-up rate of the blow-up solution to equation (1) near its blow-up time.Suppose that the solutionu of equation (1) blows up in finite time T∗ and the assumptions (H1) and (H2) hold. We need the following assumption on the boundary condition:(H3) There exists a constantγ>0 such that infgu≥γLet functionUt=supx∈Ω¯ux,t, where ux,t is a blow-up solution to equation (1). The following lemma is given according to [8, 30, 31].Lemma 9.
Suppose equation (1) satisfies the assumptions (H1) and (H2), and there exists a positive real number C1 such that
(45)Ut≥C1T∗−t−1/p−1,t∈0,T∗.The following provides an upper bound for the solutionux,t to equation (1). Let zx,t be the solution of the following auxiliary problem
(46)zt=zqΔz+a∫Ωzpdx,x,t∈Ω×0,T∗∗,n·∇z+γz=0,x,t∈∂Ω×0,T∗∗,zx,0=maxc0,u0,x∈Ω,where u0 is stated in Theorem 5, q∈0,1, and c0>0 is a fixed constant. Obviously zx,0≥u0, according to [8] Theorem 3.1 (the equation discussed in it is a subsolution to equation (46)), zx,t blows up in a finite time, denoted as T∗∗.Let the functionJx,t=zt−δzp+q, where δ>0. We need to prove that J≥0. According to [8]. Since p/2p+q−1+p+q−1/2p+q−1=1, applying Young’s inequality yields
(47)zp+q−1∫Ωz2p+q−1dxp/2p+q−1≤p+q−12p+q−1θzp+q−12p+q−1/p+q−1+p2p+q−1θ−2p+q−1/p∫Ωz2p+q−1dx,where θ=p+q/2p+q−1p/2p+q−1Ωpp+q−1/2p+q−12. Then, from Holder’s inequality, we have
(48)∫Ωzpdx≤Ωp+q−1/2p+q−1∫Ωz2p+q−1dxp/2p+q−1.Combining equations (47) and (48) yields
(49)Jt−zqΔJ−2qδzp+q−1J−apzq∫Ωzp−1Jdx=qz−1J2+δp+qp+q−1zp+2q−2∇z2+qδ2z2p+2q−1+apδzq∫Ωz2p+q−1dx−aδp+qzp+2q−1∫Ωzpdx≥qδ2z2p+2q−1+apδzq∫Ωz2p+q−1dx−aδp+qzp+2q−1∫Ωzpdx≥δqδ−ap+q−1θ2p+q−12/pp+q−1z2p+2q−1=qδδ−δ1z2p+2q−1≥0.The boundary condition leads to(50)n·∇J+γJ=n·∇zt+γzt−δn·∇zp+q+γzp+q=−δn·∇zp+q+γzp+q=−δγzp+q1−p−q≥0.And the initial value condition leads to(51)Jx,0=ztx,0+δzp+qx,0=δzp+qx,0≥δc0p+q≥0.For anyε>0, applying Lemma 8 on Ω×0,T∗∗−ε, where d=zq,c1=2qδzp+q−1,c2=zp−1,c3=apzq, we have Jx,t≥0. Considering the arbitrary of ε, limε⟶0Jx,t≥0, i.e., zt≥δzp+q≥0 can be obtained. Then, there exists a constant τ∈0,T∗∗ such that z≥1, when t≥τ. So we have zp+q≥zp, i.e., zt≥δzp. Inegrating this equation over t,T∗∗,
(52)zx,t≤C2T∗∗−t−1/p−1,where C2=δp−1/p−1. Set w=z−u. According to equation (46), there is zx,0≥ux,0, i.e., wx,0≥0. On the boundary, we have n∇z+γz=n∇u+guu≥n∇u+γu, i.e., n∇w+γw≥0. Considering z≥1 and mean value theorem, we obtain
(53)wt+Δw≥a∫Ωzp−updx=a∫Ωpηp−1wdx,where η is a nonnegative function between z and u, x,t∈Ω×τ,T∗∗. Applying Lemma 8 with d=1,c1=0,c2=ηp−1,c3=ap, w≥0 is obtained. So z≥u in τ,T∗∗. Combined with Lemma 9, there exists solution u to equation (1) satisfying
(54)C1T∗−t−1/p−1≤ux,t≤C2T∗−t−1/p−1,where T∗ is the blow-up time of solution ux,t. Theorem 6 is proved.



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