It is well known that the equation in (1.1) is degenerate if and singular if , and therefore there is no classical solution in general. To state the definition of the weak solution, we first define the class of nonnegative testing functions
where .
Definition 2.1 A function is called a sub-solution (super-solution) of Problem (1.1) in if the following conditions hold:
-
(i)
in Ω,
-
(ii)
on ,
-
(iii)
for almost every and every ,
(2.1)
A function is called a local solution of (1.1) if it is both a sub-solution and a super-solution for some and is called a solution of (1.1) if it is a local solution of (1.1) in for any .
Local existence of weak solutions of (1.1) can be obtained by utilizing the methods of standard regularization (see [9]) and the continuity of the solutions can be derived by the arguments similar to that in [18]. Moreover, Problem (1.1) admits global solutions when the initial data are small (see [1]). Since the regularization procedure is crucial in what follows, we shall sketch the outline. Consider the regularized problem
(2.2)
where may be chosen sufficiently small in such a way that there exists a solution of (2.2) on for every , and is bounded independently of k. Furthermore, for , and a super-solution (sub-solution) comparison theory holds for (2.2) (see [1, 19]).
Since is monotone in k, we may define , and it is easy to see that is a solution of (1.1). Furthermore, if u is a solution of (1.1), then we have
where we use the fact on ∂ Ω to derive this inequality. With and defined so that
and
we have
Thus, we can choose the appropriate test function ξ as in [1, 19] to obtain . If u is a sub-solution of (1.1), the above argument shows that . Thus is the maximal solution of (1.1), and this solution satisfies a sub-solution comparison principle.
Before proving our main results, we give a comparison principle for the solution of Problem (1.1), which is similar to Proposition 2.3 in [9] and can be proved by modifying the above arguments (see also [1, 10, 19]).
Proposition 2.1 Let u and v be a nonnegative bounded sub-solution and a nonnegative super-solution of (1.1), respectively. If either and u is bounded from the above or and v has a positive lower bound, then in if in Ω.
Proof of Theorem 1.2 Case (i): with . For any bounded smooth domain such that , let be the unique solution of the following elliptic problem:
(2.3)
By the comparison principle for linear elliptic problem we know in Ω. Set , and . It is well known from the strong maximum principle that .
By continuity, we can choose a suitable domain with such that . Define , where satisfies
(2.4)
Since , it follows from the theory in ODEs that is nonincreasing and for all
Then it can be verified that is a super-solution of (1.1). In fact, because and , we know that satisfies the following inequalities (in the weak sense):
(2.5)
In addition, on , for any , and by the choice of A. Moreover, there exists a positive constant such that in . Therefore, by applying Proposition 2.1 to (1.1) we see that for , which implies . The arbitrariness of and ensure that . Furthermore, let , then satisfies (1.1) with the initial condition . By the aforementioned proof, we see that with any . From the relation of the extinction time of to A, it follows that for any , i.e. for any .
Case (ii): . Let ϕ, be the same as Case (i) and denote . Set with , then it is easy to verify that is a super-solution of (1.1) when is sufficiently small such that in Ω. Applying Proposition 2.1 to Problem (1.1) in for any we obtain in , which implies that . Therefore, satisfies
By the choice of k and it is easily verified that . Thus, by the results of Case (i), we can conclude that the solution vanishes in finite time when the initial data are suitably small. The proof of this theorem is complete. □
Proof of Theorem 1.3 We first prove the case with . Set where satisfies the following ordinary differential equation:
(2.6)
Since and , we know by integrating the ODE that vanishes at some finite time . Moreover, as in the proof of Theorem 1.2, it can be verified that is a super-solution of (1.1). Thus, by applying Proposition 2.1 to and for any we know that also vanishes at .
In the case , let satisfy the following ODE:
(2.7)
where . Similar to the first case, it is well known that vanishes in finite time since and is a super-solution of (1.1) provided that is small enough such that . Applying Proposition 2.1 to and guarantees the finite time extinction of . This completes the proof of Theorem 1.3. □
Proof of Theorem 1.4 (i) Consider first the case . Let be the first eigenvalue of the following eigenvalue problem:
(2.8)
and () be the corresponding eigenfunction. We may normalize such that . Denote and let satisfy the ODE problem
(2.9)
It is easy to check that is nondecreasing and bounded from above by . Set . We shall show that is a sub-solution of (1.1) when is sufficiently small. In fact, simple computations show that
and
For to be a sub-solution of (1.1), it suffices to show that
which follows from
(2.10)
where . It is easy to see that (2.10) is valid for sufficiently small since .
Next, we turn our attention to construct a super-solution of (1.1). Set , where . Then it is not hard to see that is a super-solution and . Therefore, by an iteration process, one can obtain a solution of Problem (1.1), which satisfies . Indeed, define and iteratively to be a solution of the problem
subject to the boundary and initial conditions as that in (1.1). By applying the comparison technique used in the proof of Lemma 2.1 in [1, 12] we know that the function , for every and , is a solution of (1.1). Because does not vanish, neither does .
-
(ii)
The case can be treated similarly to Case (i).
-
(iii)
Finally we consider the case with . Let satisfy the following ODE:
(2.11)
Then is nondecreasing and satisfies . (The upper bound of can be obtained by contradiction arguments and the monotonicity of follows immediately as the upper bound is derived.) As in the proof of Case (i), we can construct a non-extinction sub-solution with sufficiently small.
To construct a super-solution, consider the following eigenvalue problem:
where is a bounded domain with smooth boundary . Let and () be its first eigenvalue and the corresponding eigenfunction, respectively. We may normalize such that . Denote and . Set , then we shall show that is a super-solution of (1.1) provided that is suitably large. Indeed, if , we know that on , in Ω and satisfies the following inequalities (in the weak sense):
Moreover, by the choice of k. Therefore, by applying the monotonicity iteration process we can obtain a non-extinction solution of (1.1) satisfying . The proof of Theorem 1.4 is complete. □
Proof of Theorem 1.5 The proof of this theorem is similar to that of Theorem 1.4, so we only sketch the outline here. Set where is defined in (1.2) and satisfies the following ODE problem:
(2.12)
Since and , it is well known that is nondecreasing and bounded above by . Then is a sub-solution of (1.1) if is sufficiently small. On the other hand, the super-solution can be chosen to be a large positive constant L satisfying . It can be observed that is a pair of sub-solution and super-solution of (1.1) satisfying . Therefore, by monotonicity iteration, we know that (1.1) admits at least one solution such that . Since in , cannot vanish at any finite time. The proof of Theorem 1.5 is complete. □
Proof of Theorem 1.6 (i) Let be any solution of (1.1). It can be verified that, for the case , a sufficiently large constant L is a super-solution of (1.1). Therefore, we know that in . For convenience, in the following proof, we assume that the weak solution is appropriately smooth; otherwise, we can consider the corresponding regularized problem, and the same result can also be obtained through an approximate process (see [15]). Multiplying equation (1.1) by and integrating by parts over Ω yield the identity
(2.13)
Recall the embedding theorem
Combining this result with (2.13) and using Hölder’s inequality on the right hand side of (2.13) one obtains
(2.14)
Noticing that and , we see from (2.14) that
which implies
This shows that tends to 0 exponentially as .
-
(ii)
Let , where satisfies
(2.15)
Since , is nonincreasing and for . Noticing and , one can see that is a super-solution of (1.1) provided that in Ω. By using the arguments similar to that of the proof of Case (i) of Theorem 1.2 we can show that any solution of Problem (1.1) vanishes in finite time.
-
(iii)
Finally we consider the case . First we construct a non-extinction sub-solution of (1.1). Set , where , α are two positive constants to be determined. Noticing that , it is easily verified that when , is a sub-solution of (1.1) if and if is so small such that . When , for to be sub-solution of (1.1) it is reasonable to choose first so small such that and then . Next, since and is bounded, we can choose a sufficiently large constant to be a sup-solution of (1.1). Therefore, by monotonicity iteration, we can obtain a solution of (1.1) satisfying . Since does not vanish at any finite time, neither does . The proof of Theorem 1.6 is complete. □