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Existence and Nonexistence of Global Solutions of the Quasilinear Parabolic Equations with Inhomogeneous Terms
Advances in Difference Equations volume 2010, Article number: 451619 (2010)
We consider the quasilinear parabolic equation with inhomogeneous term , , where , , , , and , . In this paper, we investigate the critical exponents of this equation.
We consider the quasi-linear parabolic equation with inhomogeneous term
where , , , , and .
For the solution of (1.1), let be the maximal existence time, that is,
If , we say that is a global solution; if , we say that blows up in finite time.
For quasi-linear parabolic equations, the authors of [1–5] and so on. study the homogeneous equations (i.e., in (1.1)). Baras and Kersner  proved that (1.1) with and has a global solution, two constants and depending on and exist such that
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.
Suppose that , , , and . Put
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.
A continuous function is called a solution of Problem (1.1) in if the following holds:
for any bounded domain and for all and vanishing on ,
for all .
Lemma 2.2 (the comparison theorem).
Let , , ; , , , ; , and satisfy
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 .
3. Proof of Theorem 1.1(a)
We first consider the following problem:
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).
The stationary problem of Problem (3.1) is as follows:
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 .
Let and be two functions in , and satisfy
in ; in , in ; , ;
in ; in , in ; .
For and , define , and let be a cut-off function, where , . It is easy to check that
where is a positive number to be determined. Then
Since , there exist and such that for :
Hence, by the definition of and , we have
we obtain from (3.3) that
in . Thus, (3.7) becomes
Let be large enough such that and , and let be as follows:
Then, by making use of Young's inequality, we have
where , . Thus, (3.11) becomes
For , since , , and , we have
For , since , , and , we have
For , since , and , we have
Let such that , then
By the integral mean value theorem, there exists such that
Since is a large positive number and a random selection, and is monotone increasing to , there exists a positive number for any fixed such that, for all ,
By the monotone increasing property of , also is increasing to . This, combined with (3.24), yields that the limit exists such that
Since is nonnegative, is monotone increasing to . This, combined with (3.25), yields that exists. Thus, for any small , there exists a large positive constant which still is denoted by , such that, for ,
Hence, by similar argument as that in (3.24), there exists a large positive number such that
in ; in , in ;
for any , there exists a positive constant such that .
Let and be as defined in (3.26) and (3.27). Multiplying (3.1) by and then integrating by parts in , we have
By the definition of , Hölder's inequality, and (3.27), we have
where , , since
Let and . Then, by making use of (3.29) and for , (3.28) becomes
Thus, let be small enough such that , then .
Let . By making use of Hölder's inequality, we obtain that
where . Thus, we obtain that
Since for all , we have
Let , then
Let such that . Since , by solving the differential inequality (3.35) in , we have
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)
In this section, we prove that for , there exist some and , such that Problem (1.1) admits a global positive solution.
We first consider the stationary problem of Problem (1.1) as follows:
Let , where and the positive constant satisfies
Then, we have
Since and , we have
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.
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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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Kobayashi, Y. Existence and Nonexistence of Global Solutions of the Quasilinear Parabolic Equations with Inhomogeneous Terms. Adv Differ Equ 2010, 451619 (2010). https://doi.org/10.1155/2010/451619
- Ordinary Differential Equation
- Functional Equation
- Stationary Problem
- Global Solution
- Random Selection