- Research Article
- Open Access
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
We consider the quasilinear parabolic equation with inhomogeneous term , , where , , , , and , . In this paper, we investigate the critical exponents of this equation.
- Ordinary Differential Equation
- Functional Equation
- Stationary Problem
- Global Solution
- Random Selection
where , , , , and .
If , we say that is a global solution; if , we say that blows up in finite time.
if , then every nontrivial solution of (1.1) blows up in finite time;
if , then (1.1) has a global solution for some initial value ,
where for , and for , , for , . This is called the critical exponent.
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.
If , then every nontrivial solution of (1.1) blows up in finite time.
If , , and , then (1.1) has a global solution for some constants and .
Suppose that , , , and . Then every nontrivial solution of (1.1) blows up in finite time.
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 the following, and are two given positive real numbers greater than 1. is a positive constant independent of and , and its value may change from line to line.
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.
for any bounded domain and for all and vanishing on ,
for all .
Lemma 2.2 (the comparison theorem).
Then for all , where is a bounded domain in with smooth boundary or and .
Lemma 2.3 (the monotonicity property).
Let be a nonnegative sub-solution to the stationary problems of Problem (1.1). Then the positive solution with initial data is monotone increasing to .
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 .
We argue by contradiction. Assume that Problem (3.1) has a global positive solution for .
in ; in , in ; , ;
in ; in , in ; .
in ; in , in ;
for any , there exists a positive constant such that .
Thus, let be small enough such that , then .
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.
In this section, we prove that for , there exist some and , such that Problem (1.1) admits a global positive solution.
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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