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Blow-up problems for a compressible reactive gas model
Boundary Value Problems volume 2012, Article number: 101 (2012)
Abstract
This paper investigates a compressible reactive gas model with homogeneous Dirichlet boundary conditions. Under the parameters and the initial data satisfying some conditions, we prove that the solutions have global blow-up, and the blow-up rate is uniform in all compact subsets of the domain. Moreover, the blow-up rates of and are precisely determined.
MSC:35K05, 35K55, 35D55.
1 Introduction and main results
In this paper, we investigate blow-up and the blow-up rate of nonnegative solutions for the following degenerate reaction-diffusion system with nonlocal sources:
where () is a ball centered at the origin with the radius , , exponents , , and is the maximal existence time of a solution, .
The system (1.1) models such as heat propagations in a two-components combustible mixture gases [1]. This problem is worth studying because of the applications to heat and mass transport processes (see [2, 3]). In addition, there exist interesting interactions among the multi-nonlinearities described by these exponents in the problem (1.1).
In the past decades, many physical phenomena have been formulated into nonlocal mathematical models and studied by many authors. Here, we will recall some of those results concerning the first initial boundary value problem.
At first, the global solutions and blow-up problems for a single parabolic equation with nonlocal nonlinearity sources had been studied extensively, see [4–10] and references therein. As a typical example, in [4] Souplet considered the equation with spatial integral term
and the equation with both local and nonlocal terms
These two equations are related to some ignition models for compressible reactive gases. The author introduced a method to investigate the profile of blow-up solutions of (1.2) and (1.3) and observed the asymptotic blow-up behaviors of the solutions. In addition, an important model in the theory of nuclear reactor dynamics can be described by the following equation with the space-time integral term:
The blow-up of its solutions was studied by Pao [5], Guo and Su [6].
In 2003, Li and Xie [7] considered the following problem:
By introducing some transformations , (1.5) takes the form
Then they proved that the solution of (1.6) blows up in finite time for large initial data and obtained the blow-up rate. Recently, Liu et al. in [8] investigated the blow-up rate of solutions to diffusion equation (1.6). Their approach was based on sub- and super-solution methods which were very different from those previously used in the study of the blow-up rate. They proved, by using the maximum principle, that the solutions have global blow-up, and the rate of blow-up is uniform in all compact subsets of the domain. Here the global blow-up means that there exists such that
Secondly, we should point out that in the case of , the system (1.1) becomes a semilinear system. To our knowledge, there do not seem to be any results in the literature on blow-up problems of these types. But other related works of the semilinear case have been deeply investigated by many authors, e.g., see [11, 12], and the authors of this paper in [13] studied the system
where the simultaneous and non-simultaneous blow-up criteria were obtained by using the fundamental solution of the heat equation. On the other hand, there are many known results concerning the global solutions and blow-up problems for the parabolic system with local nonlinearities, localized nonlinearities and nonlinear boundary conditions, see [14–17] and references therein. In particular, Ling and Wang in [18] considered the following degenerate parabolic system:
in a bounded domain Ω, with the help of the super- and sub-solution methods, the critical exponent of the system was determined. Motivated by the above works, under the following conditions:
we consider a more general degenerate parabolic system (1.1) which includes the problems considered in [7, 8] and [17] as special cases. Employing the ideas in [7, 8], we describe the blow-up rate of the radially symmetric solutions to (1.1). Here we discuss the blow-up of radially symmetric solutions as well as derive their blow-up rate. Moreover, we get the accurate coefficient of the blow-up rate. For the related discussion on a radially symmetric solution, we refer the readers to [19] and references therein.
In this paper, we always assume that the initial data (V is defined by (1.8)) and satisfies the following (H1)-(H3) or (H4):
(H1) , .
(H2) in B, , on ∂B.
(H3) , are radially symmetric, for , .
Denote the set of initial data, depending only on the radial variable in the spherical coordinate system of :
where
It is noted that the set V is not empty. For example, for the simplest case and , for any constant exponents m, n and , , , , there exist positive constants , such that with , , .
(H4) Let , , be positive constants (will be given in Section 3), and there exists a constant such that
here , and , , are defined by (2.9) and (2.6).
Then, our main results read as follows in detail.
Theorem 1 Assume that and satisfies (H 1)-(H 3). If , then the positive solution of (1.1) blows up in finite time, where ρ is defined by (2.12).
Theorem 2 Under the assumptions of Theorem 1, if on and satisfies (H 4), then the following statements hold uniformly on any compact subset of B:
where and are defined by (3.1).
Theorem 3 Under the assumptions of Theorem 2, if and , then
uniformly on compact subsets of B, where
This paper is organized as follows. The result pertaining to blow-up of a solution in finite time is presented in Section 2, while results regarding the blow-up rates are established in Section 3. Some discussions are given in Section 4.
2 Proof of Theorem 1
In this section, we will discuss the blow-up of the solution to (1.1) and prove Theorem 1. By a simple computation, we have
Since , , from (2.2), we can derive the inequality
Moreover, by (1.1), (2.1) and (2.3), we have
Thus,
Similarly,
Denote , and
Then , , and , satisfy
Consider now the following problem:
where
Since , satisfy (H1)-(H2), then (2.8) has a unique classical solution (see [20]). In the meantime, by the comparison principle, we observe
Let G be a bounded domain of . Consider the problem
where and are some constants. By the standard method (see [3]), we can show that (2.11) has a unique classical solution and . Denote by the unique positive solution of the linear elliptic problem
Set , then we have:
Lemma 1 If , then the positive solution of (2.11) blows up in finite time.
Proof Set , then
where . Let , then
Since , from Jensen’s inequality, it follows that
That is . In view of , it follows that there exists such that , and hence blows up in finite time. □
Let be the unique positive solution of the following linear elliptic problem:
and
Lemma 2 If , then for the solution of (2.8), there exists a sufficiently small constant such that
for all .
Proof From (H1) and (H2) we see that there exists a sufficiently small constant such that
and
Let , , then we have by (2.14)
Thus it follows from (2.13) and (2.15) that is a sub-solution of (2.8). Hence, by the comparison principle. □
Lemma 3 The solution of (2.8) blows up in finite time if and , satisfy (H 1)-(H 3).
Proof In view of , we can choose a smooth sub-ball such that
where and satisfies
On the other hand, there exists a sufficiently small such that
Let , here ε is determined by Lemma 2. Then and
by Lemma 2. Therefore, in satisfies
Now, consider the following system:
Similarly, we can show that there exists a nonnegative classical solution of (2.18) for , where denotes the maximal existence time. The standard comparison principle for a parabolic system implies that and
Therefore, it suffices to show that blows up in finite time, because if so, its upper bound does exist up to a finite time T.
Since the initial data is a sub-solution of (2.18), the standard super-solution and sub-solution methods assert that , , which implies that
Hence for . Thus, satisfies
with the corresponding initial and boundary conditions and .
Since , there exist positive constants , with , and l such that
Let
where is a unique positive solution of (2.11) with
From (2.21) and Lemma 1, we know that blows up in finite time . Moreover, , that is, , since the initial data is a sub-solution of (2.11). In addition, from and Hölder’s inequality, we have
Thus, a series of computations yields
It follows from (2.20), (2.23) and the comparison principle that . Hence blows up in finite time, and so does the solution of (2.8) from (2.19). The proof now is completed. □
Considering Lemma 3 and (2.10), we directly obtain the results of Theorem 1.
3 Proofs of Theorems 2 and 3
In this section, we assume that the solution of (1.1) blows up in finite time T and will prove Theorems 2 and 3. We use c or C to denote the generic constant depending only on the structural data of the problem, and it may be different even in the same formula.
For the problem (1.1), denote
Then we have
Lemma 4 Suppose that , satisfy (H 1)-(H 3), then we have
Proof According to the hypotheses, we know that , , . Let
Then, , are Lipschitz continuous (see [21]) and , . Since is radially symmetric and non-increasing in , is also a radially symmetric and non-increasing function, i.e., with . Thus, and always reach their maxima at , which means that for any , i.e., for any . Therefore, it follows from (2.1) and (1.1) that
Integrating (3.3) over , we obtain
From and , we get
□
Next, we first give some auxiliary lemmas about the solutions of (2.8), which will be used in the proofs of theorems. Similar to (3.2), we let
By (2.8), we see that and satisfy
Let , , then . By Young’s inequality, we have
Integrating the above inequality over , we obtain
where by (1.7).
Lemma 5 Suppose that , satisfy (H 1)-(H 3) and the solution of (2.8) blows up in finite time T. Then, we have
where
Proof Let be as (3.5), then from (2.8), we have
Integrating (3.9) over , we obtain
Similar to the proofs of and , we have
□
Lemma 6 Suppose that , satisfy (H 1)-(H 4). Then, we have
here , .
Proof Set , . Then,
A series of computations yields and
From the condition (1.7), it is easy to calculate that . Then, it entails
By the Hölder inequality, for any , it follows that
Furthermore, by Young’s inequality, for any and satisfying , the following inequality holds:
Now, we take
Therefore, by (3.12) and (3.13), it follows that
where
We can determine a number in the similar way. Let , similar to the above, one has
By the comparison principle of Lemma 1 in [20], we have . This completes the proof. □
Lemma 7 Suppose that , satisfy (H 1)-(H 4), then there exist positive constants c and C such that
Proof It follows from (3.11) that
Combining with (3.6), we can obtain
The direct computation yields , . It follows from (3.16) that
Therefore, combining (3.17) with (3.7) gives
Integrating (3.15) from t to T, we end the proof. □
Lemma 8 Suppose that , satisfy (H 1)-(H 4) and , then
uniformly on compact subsets of B.
Proof Here we consider the first eigenvalue problem
Normalize as in B and . Define
A series of computations yields
where . By (3.10), we know that , which means . Then
Integrating (3.19) from 0 to t yields
That is
Denote . Since , using Lemma 4.5 in [4], we obtain
It follows from (3.21) and (3.10) that
for any . On the other hand, we know from (3.10), (3.14) and that
Therefore,
Noting that
Then
Thus
Similarly,
□
Proof of Theorem 2 According to , it follows from (2.9), (3.1), (3.8) and (3.18) that
On the other hand, from (3.4), we estimate
Combining (3.23) with (3.24), we obtain
Similarly,
This completes the proof of the theorem. □
Proof of Theorem 3 By Theorem 2, we have that, as ,
where the notation means that . Hence, we obtain
A series of computations yields
Combining with Lemma 7, we obtain the results of Theorem 3 immediately. □
4 Discussions
The results in this paper show the interactions among the multi-nonlinearities in the reaction-diffusion system (1.1). Roughly speaking, either large exponents m, n, large coupling exponents , or large constants a, b benefit from the occurrence of the finite blow-up. For example, to make a finite blow-up to the problem (1.1), for fixed m, , , and n, , , , constants a and b should be properly large such that the following inequality
holds.
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Acknowledgements
The authors are supported by National Natural Science Foundation of China and they would like to express their many thanks to the editor and reviewers for their constructive suggestions to improve the previous version of this paper. This work is supported by the NNSF of China (11071100).
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Ling, Z., Wang, Z. Blow-up problems for a compressible reactive gas model. Bound Value Probl 2012, 101 (2012). https://doi.org/10.1186/1687-2770-2012-101
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DOI: https://doi.org/10.1186/1687-2770-2012-101