#### Abstract

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 is constant, , , and is a bounded region in with a smooth boundary . 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 . Finally, the blow-up rate for solutions is estimated also.

#### 1. Introduction

This paper studies the following semilinear parabolic equations under nonlinear boundary conditions where is constant, , , and is a bounded region in with a smooth boundary , is outward unit normal vector of , initial value is nonnegative continuous function, satisfying assumption (H1) (see below), and denotes Lebusgue measure of .

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: under Dirichlet boundary conditions, where is positive constant. And they proved the existence of global solution and showed that all the blow-up solutions are blow up globally if satisfies . Furthermore, authors gave the blow-up rate in special cases as follows: where are positive constants and . In [8], Li and Xie studied global existence of the following equation: with Dirichlet boundary conditions, where . They obtained that there exists a global positive classical solution if and when , and the solution blows up in finite time if the initial value is sufficiently large. Then, the blow-up rate was given as follows: where are positive constants and is the blow-up time of .

In [10], the authors investigated the parabolic superquadratic diffusive Hamilton-Jacobi equations as follows: with Dirichlet boundary condition, where . They studied the gradient blow-up (GBU) solutions which are defined as where is the existence time of the unique maximal classical solution. And it was showed that in the singular region, the normal derivatives and , which satisfy , 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)

(H2) and satisfies the local Lipschitz condition

In 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. * is called a a supersolution to equation (1) if it satisfies that
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 of the problem (1) blows up in finite time if there exists a positive real number , such that
And the solution of the problem (1) exists globally if for any ,

Theorem 3 states the problem of local existence of the solution to equation (1) and is the main conclusion of this paper.

Theorem 3. *Suppose are the sub- and supersolutions to equation (1), respectively, and . If satisfies assumption (H1) and the function satisfies assumption (H2), then there exists , 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 .*

Theorem 4. *Suppose assumptions (H1) and (H2) hold, and the equation (1) satisfies one of the following conditions.
*(i)*(ii)**, and the initial value is sufficiently small**Then, the solution of this equation exists globally.*

Theorem 5. *Assume (H1) and (H2). If and the initial value is sufficiently large, then the solution to equation (1) blows up in finite time.*

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 blowing up at . Specifically, there exist constants such that
*

*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.

#### 2. 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 [26–29]).

First, the following lemma is present, which is proved according to [2].

Lemma 8. *Suppose that assumptions (H1) and (H2) hold. Let and satisfy
where are continuous and bounded functions in , . Then, .*

*Proof 1. *Let and , where . Then, the first equation in equation (13) can be deduced to
Hence,

Assume by contradiction that at some points , so there must be a negative minimum value of due to continuity, denoted as . The following two cases are discussed. (i)If , then

At this point, we have , which is contradictory to .

(ii) If , consider the values of each function at . Then,

It yields , contradicting .

Combining (i) and (ii), there is no negative minimum value of ; thus, is nonnegative. So , i.e., . Lemma 8 is proved.

Suppose that the assumptions of Theorem 3 hold. Consider the following auxiliary problem where and satisfy the following rule. Let . We have that is Lipschitz continuous on the interval , which implies that for any given, there exists a fixed positive real number such that

Thus,

Let . Then, the function is increasing under this definition.

The auxiliary problem (12) is a third boundary value problem. It is clear that there exists a unique solution to it, due to Theorem 3.4.7 in [9]. Define the nonlinear operator such that and construct the following sequences

It can be proved that operator is increasing. The proof is as follows. For any , , let . And

Applying Lemma 8, where , we have , i.e., . Letting , the above equation is transformed into from which we deduce to . The same procedure may be easily adapted to obtain . Thus,

By mathematical induction on , the above sequence (21) exhibits the following comparative relationship which shows that the sequences are increasing and bounded. So limits exist. And . Considering the compactness of the nonlinear operator and , we know that 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 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, satisfies
Therefore, is a subsolution to equation (1). According to Theorem 3, we need to determine a globally existing supersolution. Set as the unique solution of the ellipse problem
Let where is a constant. Obviously, on the boundary, we have
And the initial value is

Let equation (30) ≥0. Then, is a supersolution to equation (1) and satisfies . So,

When is fixed, is a constant. Set . (1)In case of , equation (31) can be transformed into

At this time, let is a sufficiently large constant such that . Then, we take , which can guarantee that is a supersolution to equation (1) and the global existence of the solution . (2)In case of , equation (31) can be transformed into

To ensure that is still the supersolution to equation (1), it needs to satisfy

Without loss of generality, we can take such that , that is, when is sufficiently small, the solution to equation (1) exists globally.

*Case 2. *In case of , the form of equation (1) is as follows:
Let , , and be the solution to the following Cauchy problem
where and the solution is . Then, we have
This means that when , for any given , is a supersolution to equation (1), and exists globally. Thus, the solution to equation (1) exists globally.

Combined with Cases 1 and 2, Theorem 4 is proved.

#### 4. Proof of Theorem 5

The above theorem states that the classic solution of equation (1) exists globally when . In this section, we will get the blow-up results of solutions to equation (1) when and prove Theorem 5.

Given assumptions (H1) and (H2) and , where are defined in Section 3 and is a fixed constant, let be the solution of the following eigenvalue problem

We normalize , i.e., , and denotes the first eigenvalue of the problem. Let be the solution to the Cauchy problem below

It can be seen that the solution of this equation blows up in finite time under the condition of . Let

Equations below state that , as defined above, is a subsolution to problem (1).

Consider the boundary and initial value conditions

Hence, is a subsolution to problem (1), when . According to equations (42) and (43), set . Considering mean value theorem, we have where is a nonnegative function between and . Applying Lemma 8 with , , i.e., is obtained. Since blows up in finite time, so does . Therefore, when , the solution to equation (1) blows up in finite time, which means equation (1) has at least one solution that blows up in finite time, when and is sufficiently large. Theorem 5 is proved.

#### 5. 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 of equation (1) blows up in finite time and the assumptions (H1) and (H2) hold. We need the following assumption on the boundary condition:

(H3) There exists a constant such that

Let function , where 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 such that
*

The following provides an upper bound for the solution to equation (1). Let be the solution of the following auxiliary problem where is stated in Theorem 5, , and is a fixed constant. Obviously , according to [8] Theorem 3.1 (the equation discussed in it is a subsolution to equation (46)), blows up in a finite time, denoted as .

Let the function , where . We need to prove that . According to [8]. Since , applying Young’s inequality yields where . Then, from Holder’s inequality, we have

Combining equations (47) and (48) yields

The boundary condition leads to

And the initial value condition leads to

For any , applying Lemma 8 on , where , we have . Considering the arbitrary of , , i.e., can be obtained. Then, there exists a constant such that , when . So we have , i.e., . Inegrating this equation over , where . Set . According to equation (46), there is , i.e., . On the boundary, we have , i.e., . Considering and mean value theorem, we obtain where is a nonnegative function between and , . Applying Lemma 8 with , is obtained. So in . Combined with Lemma 9, there exists solution to equation (1) satisfying where is the blow-up time of solution . Theorem 6 is proved.

#### Data Availability

No data were used to support the study.

#### Conflicts of Interest

No conflicts of interest exist.

#### Acknowledgments

The authors thank the National Natural Science Foundation of China (62073203), the Fund of Natural Science of Shandong Province (ZR2018MA022), the National University Student Innovation Training Project (202110445095), and the Shandong University Students Innovation Training Project (S202110445194).