1. Introduction
In this paper, we discuss the following semi-linear parabolic system with nonlinear nonlocal sources and nonlinear boundary conditions:
where
are positive constants,
,
,
and
is a bounded region in
(
is an integer) with a smooth boundary
,
is an outward unit normal vector of
,
are nonnegative continuous functions satisfying some assumptions, and
denotes the Lebesgue measure of
.
Our study is motivated by [
1,
2], in which the existence and blow-up of partial differential equations under nonlinear boundary conditions were researched. As stated in [
1], in certain thermal explosion problems that require long induction times (times prior to ignition), such as the safe storage of energetic materials or nuclear waste, the Dirichlet boundary condition
is no longer applicable because during this time the boundary of the reaction material will be preheated to a temperature significantly higher than the surrounding temperature. As a result, we need the heat-loss boundary conditions in (
1) to describe the temperature distribution of the boundary. In recent years, many authors have investigated the reaction-diffusion system with Dirichlet boundary conditions as follows:
For details, see [
3,
4,
5,
6,
7] and their references. In [
3], the blow-up and global existence criteria of the nonlocal parabolic system
were obtained. In [
4,
7], the authors studied nonlocal nonlinear parabolic systems of the form
where
are constants. They all gave some results on the existence of solutions, the blow-up solutions, and the blow-up rate or profile. In [
5], Li et al. suggested that if one of
or
holds, the solution of (
2) blows up in finite time for sufficiently large initial data, when
and
. Furthermore, they also obtained the blow-up profile of the solution
uniformly on any compact subset of
, if
,
, and
. Thereafter, in [
8], Kong and Wang extended these results to parabolic systems with nonlocal boundaries
by improving the methods used in [
9,
10].
In addition, there are some interesting results on the global well-posed theory related to our work that can be found in [
11,
12,
13,
14].
In this paper, we present the global existence, the uniform blow-up criteria, and the uniform blow-up profiles of system (
1) with nonlinear heat-loss boundary conditions. To this end, we first give the following assumptions.
- (H1)
, are nonnegative, bounded, and , ;
- (H2)
satisfy the local Lipschitz condition;
- (H3)
are increasing functions.
This paper is divided into five sections. In
Section 2, we prove some new comparison principles for a modification of our system and using the obtained principles, we can obtain the local existence of (
1) if our system has sub- and super-solutions as we define. In
Section 3, we give some new definitions of uniform blow-up solutions and global solutions and present the global existence and uniform blow-up criteria of solutions to system (
1), respectively, with the help of the methods used in [
8,
15]. In
Section 4, we generalize several relevant lemmas from [
9,
10] and based on them give the blow-up profile of (
1), which describes the asymptotic behavior of the solution near the blow-up time. In
Section 5, we simulate numerically the global existence and uniform blow-up of solutions.
2. Local Existence
In this section, we intend to prove the local existence of solutions to (
1) using the monotone iterative technique (see [
16,
17,
18]) and the fixed-point theorem. To do this, we need to define the sub- and super-solutions, as well as to generalize the comparison principle from [
2,
8,
15]. Here we define the sub- and super-solutions as follows.
Definition 1. A pair of nonnegative functions is called a super-solution of (1) if , and satisfies Similarly, a sub-solution can be defined with reversed inequalities in (7). Then, we prove the following comparison principle.
Lemma 1. Suppose that and are the continuous and bounded functions on , respectively; and are the nonnegative and bounded functions in ; and and are the nonnegative functions and the positive increasing functions on , respectively. If satisfythen in . Proof. Firstly, we prove that if
satisfy
then
in
.
Let
,
and
. Set
,
, then we have
By boundary conditions and Assumption (H1),
on
can be obtained. Considering initial conditions
,
, by continuity, there exists a
such that
,
for all
. Let
and take
, then we have
. If
, the conclusion is proven, and if
, with no loss of generality we can suppose that there are some
such that
. Therefore,
w takes the minimum on
. According to the first inequality of (
10), we obtain
By the strong maximum principle (Theorem 1 in [
19] Chap. 2),
can be obtained, which is a contradiction. Hence,
; that is
.
Then, we prove that if
satisfy
then
.
Let
,
,
,
and
where
,
is an arbitrary positive real number. Then, we know
,
. Due to the boundary conditions, we have
which holds in
. Further, in
, we have
Therefore, we know that
, then, letting
,
is obtained. The proof of Lemma 1 is completed. □
We give the following conclusion about the properties of the sub-solutions and super-solutions.
Lemma 2. Suppose Assumptions (H1)–(H3) hold, , are derivable, and the nonnegative , , are the sub- and super-solutions to system (1), respectively. is a solution to problem (1). If Case (i)is satisfied, we have , ; otherwise, if Case (ii)is satisfied, we have , . Proof. Case (i). We assume that (
16) holds. Let
,
, and
,
; obviously,
,
. According to (
1), and using the mean value theorem, we have
where
,
are between
u and
;
,
are between
v and
. Similar to the proof of Lemma 1, let
,
, then, (
18) can be transformed into
By boundary conditions and
,
, we have
,
on
. Since
,
, by continuity, there exists a
such that
,
for all
. Let
and
, then we have
. When
, the conclusion is obvious and we discuss the case
. Supposing that there are some
such that
,
takes the minimum on
. According to the first inequality of (
19), we obtain
By the strong maximum principle (Theorem 1 in [
19] Chap. 2),
can be obtained, which is a contradiction. We have
,
; that is,
w,
.
and
are proved.
Case (ii). The proof of Case (ii) is similar to that of Case (i), and we omit it. □
Based on Definition 1 and Lemma 1, we give the theorem describing the local existence of (
1).
Theorem 1. Suppose Assumptions (H1)–(H3) hold, the nonnegative , , are the sub- and super-solutions to system (1), respectively, and , , then there exists a pair of functions satisfying , , which is a solution to system (1). Proof. According to Assumptions (H1)(H2), we know that for any
given, since the function
is Lipschitz continuous, there exists a fixed real number
such that
thus
Let
and similarly
, which are increasing. Then, we consider the following auxiliary problem
where
,
are two nonnegative functions. The auxiliary problem (
22) has Robin boundary conditions, which means that there exists a unique solution
to it. Define nonlinear operator
and
such that
and
. Then, construct the following sequences
Next, we will show that the operators
and
are increasing. For any
and
, let
,
, and
,
, and take
,
. Then, we have
Applying Lemma 1 with
,
,
,
,
, we know
; that is,
,
. Letting
and
, we can obtain
thus,
and
can be deduced. Similarly,
and
can be deduced, too.
By mathematical induction on
n, we have
which shows that the sequences
,
,
,
are increasing and bounded. Therefore, limits
satisfying
,
and
,
, exist. Considering the compactness of the nonlinear operators
,
and
, we know that
and
are the solutions to problem (
1). The local existence of the solutions to problem (
1) is proved. □
3. Global Existence and Uniform Blow-Up Results
In this section, we will use the sub-and-super solution theorem and combine it with Lemma 2 to give the global existence and uniform blow-up conditions of system (
1). Firstly, we define the uniform blow-up solution.
Definition 2. The nonnegative solution to problem (1) blows up uniformly in finite time if there exists a positive real number such thatFurther, the solution of problem (1) exists globally if for all , Then, we refer to [
8,
15] to give Theorems 2 and 3, which are about global existence and blow-up, respectively. Further, the proofs are given using the sub-and-super solution theorem.
Theorem 2. Suppose Assumptions (H1)–(H3) hold, , are derivable andThen, the solution of this problem exists globally of system (1). Proof. First, according to condition (
30), we have
so there exist two positive constants
such that
Due to the continuity of functions
and
, let
and
. We consider the following eigenvalue problem
The first eigenvalue of problem (
33) is denoted by
. Take
,
and
,
. Then, let
be the unique solution to the Cauchy problem
where
,
such that
exists globally. Set
,
,
. Then, we assert that
is a super-solution to problem (
1). In
, we have
similarly,
Since
on
, we have that
and
hold in
. Considering the initial values, we have
and
Hence,
is a super-solution to problem (
1), and satisfies (
16). Applying Lemma 2, the solution
to problem (
1) satisfies
,
. Since
,
exist globally, Theorem 2 is proven. □
Theorem 3. Suppose Assumptions (H1)–(H3) hold and , are derivable.
Case (i). If one of the following conditions holdsand are sufficiently large, then the solution of problem (1) blows up uniformly in a finite time. Case (ii). If (40) is satisfied and are sufficiently small, then the solution exists globally.
Proof. Case (i) (a). Assuming that
, we consider the eigenvalue problem (
33) with
, and for convenience, let
. Take
,
such that
and set
Obviously,
. Let
be the unique solution to the following Cauchy problem
where
is sufficiently large such that the solution to (42) blows up in a finite time.
Let
,
be sufficiently large such that
and
,
,
, where
and
are arbitrarily real numbers. Then, we will show that
is a sub-solution to problem (
1). In
, we yield
and
On
, we yield
when
is sufficiently close to 0. Since
we can obtain that
is a sub-solution to problem (
1) and satisfies (
17). Applying Lemma 2, we know that
,
in
. Due to the arbitrary nature of
, letting
,
,
in
can be obtained.
Specifically, we have and on . Hence, the solution blows up globally when and , are sufficiently large.
Case (i) (b). We omit the proof for the comparable case in which .
Case (i) (c). When
,
and
, we know that
Take
and
, then there exist two positive constants
,
such that
which means that
,
. When
,
is used, and the solution
also blows up, as shown in Case (i).
Case (ii). By the continuity of functions
and
, we have
. Let positive constant
be sufficiently small such that
and
be the unique solution to the elliptic problem
Then, there exists a positive constant
M such that
. Set
where
When
, we have
Taking the initial values of problem (
1)
,
are sufficiently small such that
,
, we have that
is a super-solution to (
1), and satisfies (
16). Using Lemma 2, we obtain
,
. Thus,
u and
v exist globally. Theorem 3 is then fully proved. □
4. Uniform Blow-Up Profile of the Solution
In this section, according to the ideas of [
10], we discuss the uniform blow-up profile of the solution to (
1).
Lemma 3. Assume that (H1)-(H3) hold, , , and , can be derived. Let be a solution to problem (1). If blow up at the finite time simultaneously, then we have that for all , there exists such thatin . Proof. We take the first equation in (
1) as an example. Let
. Since
, we obtain
It is easy to see that
w can obtain its maximum and minimum in
.
On the one hand, if there is a point , , such that , then according to the strong maximum principle of heat equations, we know that w is a constant when .
On the other hand, when
,
w cannot obtain its maximum and minimum in
. For
, set
. Obviously,
. A straightforward computation yields
Further, according to
, we deduce
Applying Theorem 6 in [
20] Chap. 3, when
w takes the minimum value, we have that
Since
u,
v,
and by continuity, there exists a constant
such that
And, when
w takes the maximum value, we have that
So we can deduce that
for
. Combining (55) and (57), we have
The proof of the inequality on
is similar to the one on
, which is omitted. □
According to (50), we have
Integrating Equation (59) from
to
t yields the following lemma.
Lemma 4. Under the assumptions of Lemma 3, we have that for all where are constants and . Proof. After performing the integration, we have
Taking
,
are obtained. Since
u,
in
and due to their continuity, there exists a constant
such that
u,
. □
Remark 1. An obvious conclusion deduced from Lemma 4 is thatwhen u, v blow up. Set
where
.
We consider the eigenvalue problem (
33) with
. Set
as the first eigenvalue of (
33), and
is normalized with
. Define
where
. We have
,
.
Lemma 5. Under the assumptions of Lemma 3, letting , and any such that (i) , if ; or (ii) , where is fixed, if ; and is a constant, it holds thatin , for all , where or . Proof. Case (i). Let
,
in (
33) and
. By Green’s formula, we have
Solving this ODE, we have
On the other hand, Lemma 4 implies
when
for all
, which, combined with Equation (67), implies
Further, by Lemma 3, we have
Fixing
, whose center is noted as
x, the function
,
is a subharmonic implied by (70). Applying the mean-value inequality for subharmonic functions, it follows that
We know that
, where
is related to the distance from
, which, together with (69) and (71), can imply that
Similarly, it follows that
Selecting
, Case (i) can be proved.
Case (ii). Letting
in (
33), considering the mean value theorem, (66) and (67) are transformed into
and
where
, respectively. Similarly, since
and
,
is a subharmonic function. Lemma 3 deduces that
is bounded on
for all
, so it follows that
too. Choosing sufficiently large
, Lemma 5 is proved completely. □
Lemma 6. Under the assumptions of Lemma 3, we have thatif and only ifFurther, if (77) or (78) is fulfilled, thenuniformly on . Proof. (77) ⇒ (78) is a simple corollary of Lemma 4. Assuming (78) holds, we use the method in [
10]. According to Lemma 5 Case (i), for any fixed
, combining Lemma 4 and
yields
Since
,
are increasing, it follows that for all
,
which, together with
,
implies that
.
on
is deduced.
Integrating the first two equations in (
1) from 0 to
t, we have
Since (82), we know that
On the other hand, according to Lemma 4, it follows that
,
; that is,
Applying the maximum principle, we know that the solution
can take the minimum in
. For all
, combining Lemmas 4 and 5 Case (ii) and
, there exists a fixed
such that
where
,
. Since
,
and (84), (85) can imply (79) uniformly on
. Further, we obtain
The proof of Lemma 6 is completed. □
Using the above lemmas and corollaries, we can obtain the following theorems describing blow-up solutions by the same method as in [
8,
21]. Theorems 4 and 5 give the necessary and sufficient conditions that
u,
v blow up simultaneously. Further, Theorem 6 describes the blow-up profile of (
1) when
u,
v blow up simultaneously.
Theorem 4. Assuming (H1)–(H3) hold, , , , are derivable. Let be a classical solution to problem 1. If u and v blow up in a finite time simultaneously, then (i) and , or (ii) and must be satisfied. Theorem 5. Under the assumption of Theorem 4. Let be a classical blow-up solution to problem 1. If and , then u and v blow up simultaneously and uniformly. Theorem 6. Under the assumption of Theorem 4, the following results hold uniformly on :
- (i)
if (a) , or (b) , , then we havewhere - (ii)
if and , then we have - (iii)
if and , then we have - (iv)
if and , then we have
The proofs of Theorems 4–6 are very similar to [
8]; we only need to pay attention to the constants
,
, so we only give the proof by taking Theorem 6 Case (i) as an example, and omit the others.
Proof of Theorem 6 Case (i). When
and
, integrating (89) from 0 to
t, yields
. That is,
While
and
, since
,
, integrating (89) from
t to
, we have
similarly. By (87) and (90), we find
Since
and
, integrating (92) from
t to
, we obtain
According to (93) and Lemma 6, it follows that uniformly on
,
that is,
holds uniformly on
.
Combining (88) and (91), and applying the same proofs of
and
v, we obtain that
holds uniformly on
. □
5. Numerical Simulations
In this section, we give numerical simulations for several specific cases to illustrate Theorems 2 and 3. All calculations are performed through Mathematica.
Firstly, letting
,
,
and
, we obtain problem (
1) of the following form
where
. Because of the symmetry, we only need to simulate the values of its solution on
as shown below.
In this case,
,
satisfies (H2) and (H3); since
,
Further, (94) satisfies (H1) and
,
,
. Therefore, (94) satisfies the conditions of Theorem 2, so its solution should exist globally, which can be illustrated by
Figure 1.
Next, we present a numerical simulation to illustrate the blow-up profiles using Mathematica. As stated in Theorem 3, we consider problem (
1) in the following form satisfying (40):
where
and
is the blow-up time that will be given by the numerical simulation. In this case, the initial values satisfy
and
According to Theorem 3, we know that the solution to (95) blows up in a finite time. Without loss of generality, we take as an example the value of function
u when
,
,
, as in
Figure 2.
As expected, Since (95) satisfies
and
, the solution
blows up at approximately
, simultaneously. We take
m = 1.2,
n = 0.5,
p = 1.3,
q = 0.3 and replace the initial value conditions with
which does not satisfy that
are sufficiently large in Theorem 3. We resolve the solution
of problem (95) (see
Figure 3).
Figure 3 shows that when the initial values are very small, the solution
does not blow up even if (40) is satisfied. Then, we change the values of
m,
n,
p and
q in (95) and perform numerical simulations for the rest of the cases mentioned in Theorem 6. This is shown in
Figure 4,
Figure 5,
Figure 6 and
Figure 7.