Volume 2022, Issue 1 4629799

Research Article

Open Access

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

Wenyuan Ma,

Wenyuan Ma

School of Mathematics and Statistics, Shandong Normal University, Jinan, Shandong, 250000, China sdnu.edu.cn

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Zhixuan Zhao,

Zhixuan Zhao

School of Mathematics and Statistics, Shandong Normal University, Jinan, Shandong, 250000, China sdnu.edu.cn

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Baoqiang Yan,

Corresponding Author

Baoqiang Yan

[email protected]

orcid.org/0000-0001-6601-6433

School of Mathematics and Statistics, Shandong Normal University, Jinan, Shandong, 250000, China sdnu.edu.cn

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Wenyuan Ma,

Wenyuan Ma

School of Mathematics and Statistics, Shandong Normal University, Jinan, Shandong, 250000, China sdnu.edu.cn

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Zhixuan Zhao,

Zhixuan Zhao

School of Mathematics and Statistics, Shandong Normal University, Jinan, Shandong, 250000, China sdnu.edu.cn

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Baoqiang Yan,

Corresponding Author

Baoqiang Yan

[email protected]

orcid.org/0000-0001-6601-6433

School of Mathematics and Statistics, Shandong Normal University, Jinan, Shandong, 250000, China sdnu.edu.cn

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First published: 28 June 2022

https://doi.org/10.1155/2022/4629799

Citations: 1

Academic Editor: Azhar Hussain

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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 a, p > 0 is constant, Q_T = Ω × (0, T], S_T = ∂Ω × (0, T], 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 different values of p. Finally, the blow-up rate for solutions is estimated also.

1. Introduction

This paper studies the following semilinear parabolic equations under nonlinear boundary conditions

(1)

where a, p > 0 is constant, Q_T = Ω × (0, T], S_T = ∂Ω × (0, 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 u₀(x) 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)

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

. Furthermore, authors gave the blow-up rate in special cases as follows:

(3)

where c₁, C₁ are positive constants and f(u) = u^p, 0 < p < 1. In [8], Li and Xie studied global existence of the following equation:

(4)

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)

where C₁, C₂ are positive constants and T^∗ is the blow-up time of u(x, t).

In [10], the authors investigated the parabolic superquadratic diffusive Hamilton-Jacobi equations as follows:

(6)

with Dirichlet boundary condition, where p > 2. They studied the gradient blow-up (GBU) solutions which are defined as

(7)

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

, 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) g > 0 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

(8)

is called a subsolution to equation (1) if it satisfies that

(9)

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

(10)

And the solution u of the problem (1) exists globally if for any t ∈ (0, +∞),

(11)

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 u₀(x) satisfies assumption (H1) and the function g 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 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 u₀(x) is sufficiently small

Then, the solution of this equation exists globally.

Theorem 5. Assume (H1) and (H2). If p > 1 and the initial value u₀(x) 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 u(x, t) blowing up at T^∗ < ∞. Specifically, there exist constants C₁, C₂ such that

(12)

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

(13)

where c_i(x, t), i = 1, 2, 3 are continuous and bounded functions in Q_T, c₂, c₃ ≥ 0, d(x, t) ≥ 0, (x, t) ∈ Q_T. Then, w(x, t) ≥ 0, (x, t) ∈ Q_T.

Proof 1. Let and v = e^−λtw, where . Then, the first equation in equation (13) can be deduced to

(14)

Hence,

(15)

Assume by contradiction that v < 0 at some points (x, t) ∈ Q_T, so there must be a negative minimum value of v due to continuity, denoted as v₀ = v(x₀, t₀). The following two cases are discussed.

(i)
If x₀ ∈ ∂Ω, then

(16)

At this point, we have n · ∇v₀ ≥ −g(v₀)v₀ > 0, which is contradictory to n · ∇v₀ < 0.

(ii) If x₀ ∈ Ω^∘, consider the values of each function at (x₀, t₀). Then,

(17)

It yields , contradicting .

Combining (i) and (ii), there is no negative minimum value of v; thus, v is nonnegative. So v = e^−λtw ≥ 0, i.e., w ≥ 0. Lemma 8 is proved.

Suppose that the assumptions of Theorem 3 hold. Consider the following auxiliary problem

(18)

where K and G_k(u) satisfy the following rule. Let G(u) = −g(u)u. We have that G(u) is Lipschitz continuous on the interval

, which implies that for any u₁ ≥ u₂ given, there exists a fixed positive real number K such that

(19)

Thus,

(20)

Let G_k(u) = G(u) + Ku. Then, the function G_k(u) 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

such that v = Tu and construct the following sequences

(21)

It can be proved that operator T is increasing. The proof is as follows. For any

, let z₁ = Ty₁, z₂ = Ty₂, w = z₂ − z₁. And

(22)

Applying Lemma 8, where c₁ = −1, c₂ = c₃ = 0, d = 1, we have w = z₂ − z₁ ≥ 0, i.e., z₂ ≥ z₁. Letting

, the above equation is transformed into

(23)

from which we deduce to

. The same procedure may be easily adapted to obtain

. Thus,

(24)

By mathematical induction on n, the above sequence (21) exhibits the following comparative relationship

(25)

which shows that the sequences {u_n}, {v_n} are increasing and bounded. So limits

(26)

exist. And

. Considering the compactness of the nonlinear operator T and |Ω| < ∞, 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, u(x, t) = 0 satisfies

(27)

Therefore, u(x, 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)

Let ϕ = Mφ where M > 0 is a constant. Obviously, on the boundary, we have

(29)

And the initial value ϕ₀ = ϕ ≥ 0 is

(30)

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

(31)

When p is fixed, ∫_Ωφ^pdx is a constant. Set μ = ∫_Ωφ^pdx.

(1)
In case of 0 < p < 1, equation (31) can be transformed into
(32)

At this time, let N is a sufficiently large constant such that Nφ ≥ u₀. Then, we take M = a^{1/(1 − p)}μ^{1/(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 of p > 1, equation (31) can be transformed into
(33)

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

(34)

Without loss of generality, we can take M = a^{1/(1 − p)}μ^{1/(1 − p)} such that u₀(x) ≤ Mφ = a^{1/(1 − p)}μ^{1/(1 − p)}φ, that is, when u₀(x) is sufficiently small, the solution u to equation (1) exists globally.

Case 2. In case of p = 1, the form of equation (1) is as follows:

(35)

Let b > a|Ω|, , and z(t) be the solution to the following Cauchy problem

(36)

where t ∈ (0, T) and the solution is z(t) = δe^bt. Then, we have

(37)

This means that when p = 1, for any given a, z(t) is a supersolution to equation (1), and z(t) 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 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 u₀(x) > max{Mφ, 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)

We normalize ψ, i.e., ‖ψ‖_∞ = 1, and λ denotes the first eigenvalue of the problem. Let h(t) be the solution to the Cauchy problem below

(39)

It can be seen that the solution h(t) of this equation blows up in finite time T^∗ under the condition of p > 1. Let

(40)

Equations below state that

, as defined above, is a subsolution to problem (1).

(41)

(42)

Consider the boundary and initial value conditions

(43)

Hence,

is a subsolution to problem (1), when p > 1. According to equations (42) and (43), set

. Considering mean value theorem, we have

(44)

where ξ is a nonnegative function between

and u. Applying Lemma 8 with d = 1, c₁ = 0, c₂ = ξ^p−1, c₃ = ap, w ≥ 0, i.e.,

is obtained. Since h(t) blows up in finite time, so does

. 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 u₀(x) 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 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 infg(u) ≥ γ

Let function , where u(x, 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)

The following provides an upper bound for the solution u(x, t) to equation (1). Let z(x, t) be the solution of the following auxiliary problem

(46)

where u₀ is stated in Theorem 5, q ∈ (0, 1), and c₀ > 0 is a fixed constant. Obviously z(x, 0) ≥ u₀, according to [8] Theorem 3.1 (the equation discussed in it is a subsolution to equation (46)), z(x, t) blows up in a finite time, denoted as T^∗∗.

Let the function J(x, t) = z_t − δz^p+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)

where

. Then, from Holder’s inequality, we have

(48)

Combining equations (47) and (48) yields

(49)

The boundary condition leads to

(50)

And the initial value condition leads to

(51)

For any ε > 0, applying Lemma 8 on Ω × (0, T^∗∗ − ε], where d = z^q, c₁ = 2qδz^p+q−1, c₂ = z^p−1, c₃ = apz^q, we have J(x, t) ≥ 0. Considering the arbitrary of ε,

, i.e., z_t ≥ δz^p+q ≥ 0 can be obtained. Then, there exists a constant τ ∈ (0, T^∗∗) such that z ≥ 1, when t ≥ τ. So we have z^p+q ≥ z^p, i.e., z_t ≥ δz^p. Inegrating this equation over (t, T^∗∗),

(52)

where C₂ = (δp)^{−1/(p − 1)}. Set w = z − u. According to equation (46), there is z(x, 0) ≥ u(x, 0), i.e., w(x, 0) ≥ 0. On the boundary, we have n∇z + γz = n∇u + g(u)u ≥ n∇u + γu, i.e., n∇w + γw ≥ 0. Considering z ≥ 1 and mean value theorem, we obtain

(53)

where η is a nonnegative function between z and u, (x, t) ∈ Ω × (τ, T^∗∗). Applying Lemma 8 with d = 1, c₁ = 0, c₂ = η^p−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)

where T^∗ is the blow-up time of solution u(x, t). Theorem 6 is proved.

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).

Open Research

Data Availability

No data were used to support the study.

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

Abstract

1. Introduction

2. Proof of Theorem 3

3. Proof of Theorem 4

4. Proof of Theorem 5

5. Proof of Theorem 6

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

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References

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

Abstract

1. Introduction

2. Proof of Theorem 3

3. Proof of Theorem 4

4. Proof of Theorem 5

5. Proof of Theorem 6

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

Citing Literature

References

Related

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