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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.

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

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].

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

Definition 2.1. Given, if

satisfies

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 [

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

where

and satisfies

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

Proof. Let where is the solution of

and satisfies

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

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

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

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. [

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.

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

and

Suppose that following conditions hold:

(D4) If and f satisfy the following inequalities

and

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

and

with

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

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

Using (3.8), we have

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

Since so we have

Multiplying (3.11) by and integrating over

, we have

Using (3.12) and Lemma 3.1, we obtain

From the condition (D5), we see

Using the Jensen’s inequality, we get

Hence, we have

Integrating (3.16) from 0 to, we have

Let then (3.17) becomes

By the condition (D5), we have

Therefore, there exists such that

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

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