Discrete Dynamics in Nature and Society
Volume 2014, Issue 1 821503
Research Article
Open Access

Blow-Up and Global Existence for a Quasilinear Parabolic System

Chunchen Wu, 

Corresponding Author

Chunchen Wu

Zhicheng College, Fuzhou University, Fuzhou, Fujian 350002, China fzu.edu.cn

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First published: 13 February 2014
https://doi.org/10.1155/2014/821503
Academic Editor: Elena Braverman

Abstract

The problem of solutions to a class of quasilinear coupling parabolic system was studied. By constructing weak upper-solutions and weak lower-solutions, we obtain the global existence and blow-up of solutions under appropriate conditions.

1. Introduction and Main Result

In this paper, we consider global existence of nonnegative solutions for a class of nonlocal degenerate quasilinear parabolic system as follows:
(1)
where Ω ∈ RN is bounded region; Ω has smooth boundary ∂Ω; the parameters m, n, h > 1,  α, β, γ ≥ 1,  p1, q1, and r1 ≥ 0,  p2, q2, r2 > 0; the initial functions u0, v0, and w0 are nonnegative and bounded, and .

Quasilinear parabolic system is the model for many problems in the scientific field, for example, gas flow model in some seepage medium, some biological population growth model. In recent years, there are many papers to investigate the nonlinear parabolic equation and many excellent results are obtained (see [1–10] and the references cited therein). In this paper, we expand the equation of [10] into 3 and discuss the global existence and blow-up of the solutions for problem (1), and the main results of this paper are the following.

Theorem 1. If one of the following conditions holds, system (1) has global solutions:

  • (1)

    m > p1, n > q1, h > r1, and  p2q2r2 < (m − p1)(n − q1)(h − r1);

  • (2)

    m > p1, n > q1, h > r1, and p2q2r2 = (m − p1)(n − q1)(h − r1), and the magnitude of the region Ω is sufficiently small;

  • (3)

    If m ≤ p1,  n ≤ q1, or h ≤ r1, or if m > p1, n > q1,   h > r1, and p2q2r2 > (m − p1)(n − q1)(h − r1), and the initial data u0, v0, and w0 are sufficiently small.

Theorem 2. If one of the following conditions holds, the solution of system (1) blows up in finite time:

  • (1)

    m > p1, n > q1,  h > r1, and  p2q2r2 > (m − p1)(n − q1)(h − r1), and the initial data u0, v0, and w0 are sufficiently large;

  • (2)

    m ≤ p1,   n ≤ q1, or h ≤ r1, and the initial data u0, v0, and w0 are sufficiently large.

2. Proof of Global Existence

As we know, nonlocal degenerate quasilinear parabolic system may not have classical solutions. Similar to the proof in [10] (see page 388-389), we can obtain that system (1) has nonnegative weak upper-solutions and nonnegative weak lower-solutions, and the following comparison principle holds.

Lemma 3 (comparison principle). Suppose ,   are the nonnegative weak upper-solutions and nonnegative weak lower-solutions of system (1), if

(2)
one has for every (x, t) ∈ Ω × (0, T).

Therefore, in order to prove Theorem 1, we only show that, for all T > 0, there exists a positive bounded weak upper-solutions. Let φ(x) be the unique positive solution of linear elliptic equation as follows:
(3)
Denote C = max⁡x∈Ωφ(x), then 0 ≤ φ(x) ≤ C. Define as follows:
(4)
where l1, l2, l3 < 1 and satisfy ml1, nl2, hl3 < 1. k is a positive constant to be determined suitably. For all T > 0, , , and are bounded, and , and . By deducing, we have
(5)
Similarly
(6)
(7)
Denote
(8)
(1) If m > p1, n > q1, h > r1, and p2q2r2 < (m − p1)(n − q1)(h − r1), there exist positive constants l1, l3 such that
(9)
and ml1, nl2, hl3 < 1. And there exists positive constant l2 such that
(10)
Thus,
(11)
From (11), we can choose k sufficiently large, such that k > max⁡{C1, C2, C3} and
(12)
By (4)–(12) we obtain that is the weak upper-solutions of system (1).
(2) If m > p1, n > q1, h > r1, and p2q2r2 = (m − p1)(n − q1)(h − r1), there exist positive constants l1, l3 such that
(13)
And there exists positive constant l2 such that
(14)
Therefore,
(15)
Without loss of generality, we assume that the domain we discuss is contained in a sufficiently large ball B; denote φB(x) is the unique positive solution of the following linear elliptic equation:
(16)
Let C0 = max⁡x∈BφB(x); hence C ≤ C0. Suppose that |Ω| is sufficiently small such that
(17)
In addition, we choose k large enough, such that k satisfies (12); then from (5), (6), and (12)–(17) we obtain that is the weak upper-solutions of system (1).
(3) At last, if m ≤ p1,  n ≤ q1, or h ≤ r1, or if m > p1, n > q1, h > r1, and p2q2r2 > (m − p1)(n − q1)(h − r1), there exist positive constants l1, l2, and l3 such that
(18)
Therefore, from (18), we can choose k small enough,such that k < min⁡{C1, C2, C3}. Furthermore, if u0, v0, and w0 are sufficiently small to satisfy (12), then by (5), (6), (12), and (18) we obtain that is the weak upper-solutions of system (1).

This completes the proof of Theorem 1.

3. Proof of Blow-Up

In this section we will prove Theorem 2, so we construct blow-up positive weak lower-solutions of system (1). Let λ1, ϕ(x) be the first eigenvalue and corresponding eigenfunction of the eigenvalue problem as follows:
(19)
Then λ1 > 0. Standardized ϕ(x), such that ϕ(x)|Ω > 0 and max⁡x∈Ωϕ(x) = 1, then (∂ϕ/∂n)|∂Ω < 0, where n is the outer normal direction of ∂Ω, suppose Ω1 ⊂ ⊂Ω is a compact subset of Ω, if any solution (u, v, w) blows up in Ω1, also (u, v, w) blows up in Ω.
Define functions
(20)
where l1, l2, and l3 satisfy ml1, nl2, hl3 > 1, and s(t) is the solution of the initial value problem as follows:
(21)
Here k, b > 0,   d > 1 is to be determined suitably. It is clear that s(t) ≥ b and s(t) are unbounded in finite time, and
(22)
where ; by computing we obtain that
(3)
Similarly
(24)
where
(25)
(1) If m > p1, n > q1, h > r1, and p2q2r2 > (m − p1)(n − q1)(h − r1), there exist positive constants l1, l2, and l3 satisfying ml1, nl2, hl3 > 1 and
(26)
Choose
(27)
From (26)-(27) and m, n, h > 1, we have k > 0,  d > 1, and b > 1. Suppose that u0, v0, and w0 are sufficiently large to satisfy
(28)
From (20)–(28),we obtain that () is positive weak lower-solutions of system (1) in Ω1.

(2) If m ≤ p1, n ≤ q1, or h ≤ r1, (26) still holds, the results are obtained by the same methods.

Thus completes the proof of Theorem 2.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to thank the referees for their helpful comments and suggestions which greatly improve the presentation of the paper. This work was supported by the Program for Science and Technology Development Foundation of Fujian Education Bureau (JA12374).

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