Existence and Nonexistence of Global Solutions of the Quasilinear Parabolic Equations with Inhomogeneous Terms
© Yasumaro Kobayashi. 2010
Received: 20 April 2010
Accepted: 14 October 2010
Published: 18 October 2010
On the other hand, [6–9] and so on. study the inhomogeneous equations (i.e., in (1.1)). Bandle et al.  study the case , , and Zeng  and Zhang  study the case . In this paper, we investigate the critical exponents of (1.1) in the case . Our results are as follows.
We will prove Theorem 1.1(a) and (b) in Sections 3 and 4, respectively. The proof of Theorem 1.2 is included in the proof of Theorem 1.1(a).
In this section, we first give the definition of a solution for Problem (1.1) and then cite the comparison theorem and a known result.
Lemma 2.2 (the comparison theorem).
Lemma 2.3 (the monotonicity property).
3. Proof of Theorem 1.1(a)
It is clear that the positive solution of Problem (3.1) is a sub-solution of Problem (1.1). If every positive solution of Problem (3.1) blows up in finite time, then, by Lemma 2.2, every positive solution of Problem (1.1) also blows up in finite time. Therefore, we only need to consider Problem (3.1).
It is obvious that 0 is a sub-solution of Problem (3.2) and does not satisfy Problem (3.2). Thus, by making use of Lemmas 2.2 and 2.3, the positive solution of Problem (3.1) is monotone increasing to .
Thus, there exists with , such that , which implies that and then blow up in finite time. It contradicts our assumption. Therefore, every positive solution of Problem (3.1) blows up in finite time. Hence, every positive solution of Problem (1.1) blows up in finite time.
4. Proof of Theorem 1.1(b)
where . Thus, if and , then is a supersolution of Problem (1.1). It is obvious that is s sub-solution of Problem (1.1). Therefore, by the iterative process and the comparison theorem, Problem (1.1) admits a global positive solution.
This paper was introduced to the author by Professor Kiyoshi Mochizuki in Chuo University. The author would like to thank him for his proper guidance. The author would also like to thank Ryuichi Suzuki for useful discussions and friendly encouragement during the preparation of this paper.
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