Existence and Nonexistence of Global Solutions of the Quasilinear Parabolic Equations with Inhomogeneous Terms

Abstract

We consider the quasilinear parabolic equation with inhomogeneous term , , where , , , , and , . In this paper, we investigate the critical exponents of this equation.

1. Introduction

We consider the quasi-linear parabolic equation with inhomogeneous term

(1.1)

where , , , , and .

For the solution of (1.1), let be the maximal existence time, that is,

(1.2)

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 [1] proved that (1.1) with and has a global solution, two constants and depending on and exist such that

(1.3)

Mochizuki and Mukai [2] and Qi [4] study the case , , Pinsky [3] studies the case , , and Suzuki [5] studies the case , . The following two results are proved by them:

• (1) if , then every nontrivial solution of (1.1) blows up in finite time;

• (2) 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. [6] study the case , , and Zeng [8] and Zhang [9] study the case . In this paper, we investigate the critical exponents of (1.1) in the case . Our results are as follows.

Theorem 1.1.

Suppose that , , , and . Put

(1.4)

• (a) If , then every nontrivial solution of (1.1) blows up in finite time.

• (b) If , , and , then (1.1) has a global solution for some constants and .

Theorem 1.2.

Suppose that , , , and . Then every nontrivial solution of (1.1) blows up in finite time.

Remark 1.3.

Theorems 1.1 and 1.2 are the extension of the results of [8]. If we put in these theorems, the same results as Theorem  1 in [8] are obtained.

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.

2. Preliminaries

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.

Definition 2.1.

A continuous function is called a solution of Problem (1.1) in if the following holds:

• (i) ;

• (ii) for any bounded domain and for all and vanishing on ,

(2.1)

for all .

Lemma 2.2 (the comparison theorem).

Let , , ; , , , ; , and satisfy

(2.2)

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:

(3.1)

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:

(3.2)

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

• (i) in ; in , in ; , ;

• (ii) in ; in , in ; .

For and , define , and let be a cut-off function, where , . It is easy to check that

(3.3)

Let

(3.4)

where is a positive number to be determined. Then

(3.5)

Since , there exist and such that for :

(3.6)

Hence, by the definition of and , we have

(3.7)

Since and

(3.8)

we obtain from (3.3) that

(3.9)

in and

(3.10)

in . Thus, (3.7) becomes

(3.11)

Let be large enough such that and , and let be as follows:

(3.12)

Then, by making use of Young's inequality, we have

(3.13)

where and

(3.14)

where , . Thus, (3.11) becomes

(3.15)

For , since , , and , we have

(3.16)

For , since , , and , we have

(3.17)

For , since , and , we have

(3.18)

Let such that , then

(3.19)

that is,

(3.20)

Thus

(3.21)

By the integral mean value theorem, there exists such that

(3.22)

that is,

(3.23)

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 ,

(3.24)

By the monotone increasing property of , also is increasing to . This, combined with (3.24), yields that the limit exists such that

(3.25)

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 ,

(3.26)

Hence, by similar argument as that in (3.24), there exists a large positive number such that

(3.27)

On the other hand, we argue as in [6, 10]. Let be a positive function satisfying.

• (i) in ; in , in ;

• (ii) on ;

• (iii) 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

(3.28)

By the definition of , Hölder's inequality, and (3.27), we have

(3.29)

where , , since

(3.30)

Let and . Then, by making use of (3.29) and for , (3.28) becomes

(3.31)

Thus, let be small enough such that , then .

Let . By making use of Hölder's inequality, we obtain that

(3.32)

where . Thus, we obtain that

(3.33)

Since for all , we have

(3.34)

Let , then

(3.35)

Let such that . Since , by solving the differential inequality (3.35) in , we have

(3.36)

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:

(4.1)

Let , where and the positive constant satisfies

(4.2)

Then, we have

(4.3)

Since and , we have

(4.4)

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.