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: where is bounded region; has smooth boundary ; the parameters , , , , and , ; the initial functions , , and 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), , , and ;(2), , , and , and the magnitude of the region is sufficiently small;(3)If , , or , or if , , and , and the initial data , , and are sufficiently small.
Theorem 2. If one of the following conditions holds, the solution of system (1) blows up in finite time:(1), , , and , and the initial data , , and are sufficiently large;(2), , or , and the initial data , , and 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 one has for every .
Therefore, in order to prove Theorem 1, we only show that, for all , there exists a positive bounded weak upper-solutions. Let be the unique positive solution of linear elliptic equation as follows:
Denote , then . Define as follows: where and satisfy . is a positive constant to be determined suitably. For all , , , and are bounded, and , and . By deducing, we have Similarly
Denote (1) If , , , and , there exist positive constants , such that and . And there exists positive constant such that Thus, From (11), we can choose sufficiently large, such that and By (4)–(12) we obtain that is the weak upper-solutions of system (1).
(2) If , , , and , there exist positive constants , such that And there exists positive constant such that Therefore,
Without loss of generality, we assume that the domain we discuss is contained in a sufficiently large ball ; denote is the unique positive solution of the following linear elliptic equation: Let ; hence . Suppose that is sufficiently small such that In addition, we choose large enough, such that satisfies (12); then from (5), (6), and (12)–(17) we obtain that is the weak upper-solutions of system (1).
(3) At last, if , , or , or if , , , and , there exist positive constants , , and such that Therefore, from (18), we can choose small enough,such that . Furthermore, if , , and 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 , be the first eigenvalue and corresponding eigenfunction of the eigenvalue problem as follows: Then . Standardized , such that and , then , where is the outer normal direction of , suppose is a compact subset of , if any solution blows up in , also blows up in .
Define functions where , , and satisfy , and is the solution of the initial value problem as follows: Here , is to be determined suitably. It is clear that and are unbounded in finite time, and where ; by computing we obtain that Similarly where (1) If , , , and , there exist positive constants , , and satisfying and Choose From (26)-(27) and , we have , , and . Suppose that , , and are sufficiently large to satisfy From (20)–(28),we obtain that () is positive weak lower-solutions of system (1) in .
(2) If , , or , (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).