Comparing with the problem under a general homogeneous Dirichlet boundary condition, the existence of weight function in the boundary condition has a great influence on the global and non-global existence of solutions.
Theorem 1 Suppose that , then the solution of problem (1.5)-(1.7) blows up in finite time for arbitrary and sufficiently large initial data.
Proof In order to prove the blow-up result, we need to establish a self-similar blow-up solution. Let
where , , and
It is obvious that blows up at as t approaches T. Set
An explicit calculation yields
Then there exists satisfying
and
Therefore we have
and
In view of the above, this gives
We will discuss the problem for two cases.
Case 1. , . We need to show that for sufficiently small T,
That is,
Let δ be sufficiently large, satisfying . By the condition , , and , we just have to make the following equality hold:
It is obvious that it holds for .
For , choose , such that
Case 2. , choose and we can get
and
(3.1)
-
(i)
; we have , then
Since , choose and to satisfy . However,
thus we find
Then
in the sense of and .
In order to get the result, we have to show that
Note that and , we choose
in which case we can get the result.
-
(ii)
; we have and . Since and , we know that , and
(3.2)
Let , , it is obvious that and . Substituting equation (3.2) into equation (3.1) gives
However, since and , it follows that
Finally, we need to show that
Because and , we have to show
In other words, we just need the following inequality:
to hold. So choose T to be small enough, and we can get
For , ,
choose , then is the lower solution of problem (1.5)-(1.7). This implies that the solution blows up in finite time for large enough initial data. □
Theorem 2 Suppose that for . If , then the solution of problem (1.5)-(1.7) blows up in finite time for all strictly positive initial dates with T sufficiently large.
Proof Consider the following problem:
(3.3)
As , we know , and . Therefore, the solution of equation (3.3) is an upper solution of the following problem:
Since and , the solution of this problem blows up in finite time if .
It is obvious that the solution of problem (3.3) is a lower solution of problem (1.5)-(1.7) when and . By Proposition 1, is a blow-up solution. □
Suppose that the solution of problem (1.5)-(1.7) with blows up in finite time, and let . We suppose that the initial data satisfies the following assumptions:
(H1) .
(H2) There exists a constant such that .
Theorem 3 Suppose that for . If , satisfies condition (H1)-(H2), and is the blow-up solution of problem (1.5)-(1.7) in finite time T with , then the blow-up rate is
where and .
Remark 2 Choose , , , and , one can easily verify that satisfies (C1)-(C2), conditions in Theorem 3 are thus valid.
Lemma 1 If satisfies condition (H1)-(H2), , then there exists a positive constant such that .
Proof It is obvious that is Lipschitz continuous and differentiable almost everywhere. By equation (1.5) with and , it yields
and thus
Integrating the result above over , we can obtain the conclusion. □
Proof of Theorem 3 Let , where is sufficiently small. Since , we have
so we can choose to be small enough and thus obtain
On the other hand, as , we get
Let , By Jensen’s inequality, this gives
and
It follows from the assumptions of (H1)-(H2) that . Therefore, it is easy to deduce that for . That is, and integrating this over yields . Combining the results with Lemma 1, we obtain the desired result. □