Existence and Nonexistence of Global Solutions of a Fully Nonlinear Parabolic Equation ()
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.