ABSTRACT

In the paper, we study the global existence of weak solution of the fully nonlinear parabolic problem (1.1)-(1.3) with nonlinear boundary conditions for the situation without strong absorption terms. Also, we consider the blow up of global solution of the problem (1.1)-(1.3) by using the convexity method.

1. Introduction

In this paper, we consider the following fully nonlinear parabolic problem:

(1.1)

(1.2)

(1.3)

where is a bounded open domain with smooth boundary, is differentiation in the direction of the outward unit normal to, and .

Denote, and by, , respectively. Also, we need the following conditions:

(D1) and are local Lipschiz continuous with respect to;

(D2) and are positive for all s;

(D3) and with

The problem (1.1)-(1.3) appears in mathematical models of a number of areas of science such as gas dynamics, fluid flow, porous media and biological populations, one can see [1-9]. As for the case of semi-linear or degenerate equations with a nonlinear boundary condition which can be taken as the special case of the problem (1.1)- (1.3), the behavior properties of the above mentioned such as existence and uniqueness, blow up of some special problems, have been established by [2,10-17] and so on.

In this paper, we study the conditions for global existence and blow up of the problem (1.1)-(1.3). The remaining parts of the paper are organized as follows. In Section 2, we give the global solvability condition for the situations with and without strong absorption terms. Finally, we obtain the condition of blowing up of global solution by the convexity method in [18,19].

2. Global Existence

Firstly, we give the definition of weak solution as follows:

Definition 2.1. Given, if

satisfies

(2.1)

for any test function

with, then is called by a weak solution of the problem (1.1), (1.2).

The local existence and uniqueness of weak solution of the problem (1.1)-(1.3), one can see [20]. For the global existence of weak solution, we have the following result:

Theorem 2.1. Assume that there exist strictly non-decreasing positive functions and such that

, (2.2)

(2.3)

where

(2.4)

and satisfies

(2.5)

Then the solution of the problem (1.1)-(1.3) is global.

Proof. Let where is the solution of

(2.6)

and satisfies

(2.7)

From (2.2), (2.3) and (2.6), (2.7), it follows that and are well posed, positive and increasing for all

Thus, there holds

(2.8)

Using (2.5)-(2.7) and (2.3), we have

(2.9)

Using (2.2), (2.5) and (2.6), we obtain

(2.10)

From (2.9) and (2.10), we see that is a supsolution to the problem (1.1)-(1.3) defined for all with By using the supand sub-solution argument (c.f. [7]), we know that the solution o the problem (1.1)-(1.3) is global.

Remark 2.1. If the conditions (2.2) and (2.3) hold, the problem (1.1)-(1.3) is called by the problem without strong absorption terms.

3. Blow Up

In the section, we use the convexity method (see [18,19]) to show that the global solution blows up in finite time under some suitable condition. To this end, we define

(3.1)

and

(3.2)

Suppose that following conditions hold:

(D4) If and f satisfy the following inequalities

(3.3)

and

(3.4)

(D5) There exist a constant and a convexity function such that

(3.5)

and

(3.6)

with

(3.7)

Lemma 3.1. If the condition (D4) holds, then , i.e.,

Proof. Multiplying (1.1) by and integrating by parts over, we have

(3.8)

Using (3.8), we have

(3.9)

Using (3.9) and (3.1), we have So, we obtain

Theorem 3.1. Suppose that the conditions (D4) and (D5) hold, then the solution of the problem (1.1)-(1.3) blows up in finite time.

Proof. Using (3.2), we have

(3.10)

Since so we have

(3.11)

Multiplying (3.11) by and integrating over

, we have

(3.12)

Using (3.12) and Lemma 3.1, we obtain

(3.13)

From the condition (D5), we see

(3.14)

Using the Jensen’s inequality, we get

(3.15)

Hence, we have

(3.16)

Integrating (3.16) from 0 to, we have

(3.17)

Let then (3.17) becomes

(3.18)

By the condition (D5), we have

(3.19)

Therefore, there exists such that

(3.20)

From (3.20), we know that the solution of the problem (1.1)-(1.3) must blow up in finite time.

4. Acknowledgements

The present work is supported by National Natural Science Foundation of China under Grant No. 10901047.

REFERENCES