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

Focusing on the physical context of the thermal explosion model, this paper investigates a semilinear parabolic equation with nonlocal sources under nonlinear heat-loss boundary conditions, where a , p > 0 is constant, Q T = Ω × ð 0, T (cid:2) , S T = ∂ Ω × ð 0, T (cid:2) , and Ω is a bounded region in R N , 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 di ﬀ erent values of p . Finally, the blow-up rate for solutions is estimated also.


Introduction
This paper studies the following semilinear parabolic equations under nonlinear boundary conditions where a, p > 0 is constant, Q T = Ω × ð0, T, S T = ∂Ω × ð0, T, and Ω is a bounded region in R N , N ≥ 1 with a smooth boundary ∂Ω, n is outward unit normal vector of S T , initial value u 0 ðxÞ is nonnegative continuous function, satisfying assumption (H1) (see below), and jΩj denotes Lebusgue measure of Ω. This equation can be used to describe thermal explosion or spontaneous combustion problems (see [1][2][3]). It differs from the classical Dirichlet boundary conditions discussed in most of the literature (see [3][4][5][6][7][8][9]). For examples, in [5,7], the authors considered the following equation: 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 Ð ∞ 0 1/f ðsÞds = ∞. Furthermore, authors gave the blow-up rate in special cases as follows: where c 1 , C 1 are positive constants and f ðuÞ = u p , 0 < p < 1.
In [8], Li and Xie studied global existence of the following equation: 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 u 0 is sufficiently large. Then, the blow-up rate was given as follows: where C 1 , C 2 are positive constants and T * is the blow-up time of uðx, tÞ.
In [10], the authors investigated the parabolic superquadratic diffusive Hamilton-Jacobi equations as follows: with Dirichlet boundary condition, where p > 2. They studied the gradient blow-up (GBU) solutions which are defined as where T is the existence time of the unique maximal classical solution. And it was showed that in the singular region, the normal derivatives u ν and u νν , which satisfy u νν~− ju ν j p , 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.
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.

Definition 1.
uðx, tÞ ∈ C 2,1 ð Q T Þ is called a a supersolution to equation (1) if it satisfies that uðx, tÞ ∈ C 2,1 ð Q T Þ is called a subsolution to equation (1) if it satisfies that Blow-up and global existence solutions are defined as follows.
Definition 2. The solution u of the problem (1) blows up in finite time if there exists a positive real number T * < ∞, such that And the solution u of the problem (1) exists globally if for any t ∈ ð0,+∞Þ, Theorem 3 states the problem of local existence of the solution to equation (1) and is the main conclusion of this paper.
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.  And the blow-up rate of the equation is given by Theorem 6. Theorem 6. Assume (H1)-(H3) (see below). Then, there exists a solution uðx, tÞ blowing up at T * < ∞. Specifically, there exist constants C 1 , C 2 such that Journal of Function Spaces Remark 7. See Definition 2 for the description of global existence and blow-up solutions. This paper is organized as follows. In Section 2, 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.

Proof of Theorem 3
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][27][28][29]).
First, the following lemma is present, which is proved according to [2]. Lemma 8. Suppose that assumptions (H1) and (H2) hold. Let wðx, tÞ ∈ C 2+α ðQ T Þ ∩ Cð Q T Þ and satisfy Hence, Assume by contradiction that v < 0 at some points ðx, tÞ ∈ Q T , so there must be a negative minimum value of v due to continuity, denoted as v 0 = vðx 0 , t 0 Þ. The following two cases are discussed.
Suppose that the assumptions of Theorem 3 hold. Consider the following auxiliary problem where K and G k ðuÞ satisfy the following rule. Let GðuÞ = −gðuÞu. We have that GðuÞ is Lipschitz continuous on the interval ½u, u, which implies that for any u 1 ≥ u 2 given, there exists a fixed positive real number K such that Thus, Let G k ðuÞ = GðuÞ + Ku. Then, the function G k ðuÞ 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 : ½u, u ↦ ½u, u such that v = Tu and construct the following sequences It can be proved that operator T is increasing. The proof is as follows. For any y 1 , y 2 ∈ ½u, u, u ≤ y 1 ≤ y 2 ≤ u, let z 1 = Ty 1 , z 2 = Ty 2 , w = z 2 − z 1 . And

Journal of Function Spaces
Applying Lemma 8, where c 1 = −1, c 2 = c 3 = 0, d = 1, we have w = z 2 − z 1 ≥ 0, i.e., z 2 ≥ z 1 . Letting w = v 1 − u, the above equation is transformed into from which we deduce to v 1 ≥ u. The same procedure may be easily adapted to obtain u ≥ u 1 . Thus, By mathematical induction on n, the above sequence (21) exhibits the following comparative relationship which shows that the sequences fu n g, fv n g are increasing and bounded. So limitŝ exist. Andû = Tû,v = Tv. Considering the compactness of the nonlinear operator T and jΩj < ∞, we know thatû, v ∈ ½u, u ∩ W 2,1 p ðQ T Þ 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.

Proof of Theorem 4
In this section, the proof of the global results of solution to equation (1) is given. Case 1. Combining assumptions (H1) and (H2) and Definition 1, uðx, tÞ = 0 satisfies Therefore, uðx, tÞ = 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 −Δφ = 1, x ∈ Ω, n · ∇φ = 0, x ∈ ∂Ω: ( ð28Þ Let ϕ = Mφ where M > 0 is a constant. Obviously, on the boundary, we have And the initial value ϕ 0 = ϕ ≥ 0 is Let equation (30) ≥0. Then, ϕ is a supersolution to equation (1) and satisfies ϕ ≥ 0. So, When p is fixed, (1) In case of 0 < p < 1, equation (31) can be transformed into At this time, let N is a sufficiently large constant such that Nφ ≥ u 0 . Then, we take M = a 1/ð1−pÞ μ 1/ð1−pÞ + N, which can guarantee that ϕ is a supersolution to equation (1) and the global existence of the solution u.
Case 2. In case of p = 1, the form of equation (1) is as follows: Journal of Function Spaces Let b > ajΩj, δ > ku 0 k ∞ , and zðtÞ be the solution to the following Cauchy problem where t ∈ ð0, TÞ and the solution is zðtÞ = δe bt . Then, we have This means that when p = 1, for any given a, zðtÞ is a supersolution to equation (1), and zðtÞ exists globally. Thus, the solution to equation (1) exists globally.
Combined with Cases 1 and 2, Theorem 4 is proved.

Proof of Theorem 5
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) and u 0 ðxÞ > max fMφ, δ 0 g, where M, φ are defined in Section 3 and δ 0 > 0 is a fixed constant, let ψ be the solution of the following eigenvalue problem We normalize ψ, i.e., kψk ∞ = 1, and λ denotes the first eigenvalue of the problem. Let hðtÞ be the solution to the Cauchy problem below It can be seen that the solution hðtÞ of this equation blows up in finite time T * under the condition of p > 1. Let Equations below state that v, as defined above, is a subsolution to problem (1).
Δv + a Consider the boundary and initial value conditions 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 where ξ is a nonnegative function between v and u. Applying Lemma 8 with d = 1, c 1 = 0, c 2 = ξ p−1 , c 3 = ap, w ≥ 0, i.e., u ≥ v is obtained. Since hðtÞ 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 u 0 ðxÞ is sufficiently large. Theorem 5 is proved.

Proof of Theorem 6
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 solution u 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 constant γ > 0 such that inf gðuÞ ≥ γ Let function UðtÞ = sup x∈ Ω juðx, tÞj, where uðx, tÞ is a blow-up solution to equation (1). The following lemma is given according to [8,30,31].