In this section, we give some extinction properties of nonnegative nontrivial weak solutions of problem (1)-(3) stated in the following theorems. The corresponding decay estimates to the solutions will be presented in the proofs of the theorems for brief expressions instead of in the statements.
Theorem 1 Suppose that and . Then the nonnegative nontrivial weak solution of problem (1)-(3) vanishes in finite time for any nonnegative initial data provided that either or λ is sufficiently small.
Proof We first consider the case that . Multiplying both sides of (1) by u and integrating the result over Ω, we have
By Hölder’s inequality, we get the inequality
In particular, if , we get the inequality
(5)
from the two expressions above. By using the Sobolev embedding inequality, one can show that there exists an embedding constant such that
i.e.,
In particular, if , then the inequality above turns out to be
(6)
Since , we get , and hence, inequality (5) becomes
(7)
By Lemma 2, we get the inequality
(8)
where . Since , and , it can be easily seen that .
It then follows from (8) and Young’s inequality that
(9)
where and will be determined later. If we choose
then and . From (9) we have
(10)
By inequalities (7) and (10), we get the inequality
Here, we can choose and λ or small enough so that
Setting , we have .
By Lemma 1, we then obtain
where , which give the decay estimates in finite time for .
Secondly, we consider the case that . If , multiplying both sides of (1) by () and integrating the result over Ω, we get
By Hölder’s inequality, we have the inequality
In particular, if , we then get the inequality
(11)
by the two expressions above. By the Sobolev embedding inequality, one can see that there exists an embedding constant such that
(12)
Using Hölder’s inequality again, we have the inequality
(13)
From inequalities (11), (12), and (13), we then obtain the inequality
(14)
By Lemma 2, we can also have
(15)
where . Since and , one can easily see that . Then it follows from (15) and Young’s inequality that
(16)
where and will be determined later. If we choose
then and . We then have the inequality
(17)
by (16). From inequalities (14) and (17), we can also obtain the inequality
Here, we can choose and λ or small enough so that . Setting , we have from the inequality above. By Lemma 1, we obtain that
where , which give the decay estimates in finite time for such that .
If , one can show that there exists an embedding constant such that
by multiplying both sides of (1) by () and integrating the result over Ω, and the Sobolev embedding inequality. By using the inequality above and a similar argument as above, the following decay estimates can be obtained:
provided that
where . □
Theorem 2 If , then the nonnegative nontrivial weak solution of problem (1)-(3) vanishes in finite time provided that , or λ is sufficiently small, and , where if , then , and if , then .
Proof Assume that . If , multiplying both sides of (1) by u and integrating the result over Ω, we have the equation
(18)
By (10) and (18), and using Hölder’s inequality, we get the inequality
Choosing small enough so that , we obtain the inequality
Hence, we have the inequality
provided that
and
where , from which and a similar argument as the one used in the proof of Theorem 1, the following decay estimates can be obtained:
where .
If and , multiplying both sides of (1) by () and integrating the result over Ω, we get the equation
(19)
By (17) and (19), and using Hölder’s inequality, we obtain the inequality
Choosing small enough so that , we have
Therefore, we obtain the inequality
provided that
and
where , which yields the following decay estimates:
where .
Since , we have , and hence, if , then .
Assume that . If is the first eigenvalue of the boundary problem
and , , is an eigenfunction corresponding to the eigenvalue , then for sufficiently small , it can be easily shown that is an upper solution of problem (1)-(3) provided that , . We then have for by the comparison principle. Therefore, from equation (19), we can obtain the inequality
from which the following decay estimates can be obtained:
provided that
where . □
Remark 2 Since the Sobolev embedding inequality cannot be used in the proof of Theorem 2, it is not necessary to consider the cases that and , when . In addition, if , the conditions in Theorem 2 imply that .
Theorem 3 Suppose that and . Then the nonnegative nontrivial weak solution of problem (1)-(3) vanishes in finite time for any nonnegative initial data provided that β is sufficiently large.
Proof We first consider the case that . Multiplying both sides of (1) by u and integrating the result over Ω, and using Hölder’s inequality, we get
By Lemma 2, we have the inequality
(20)
where . Since , we have . It then follows from (20) and Young’s inequality that
(21)
where will be determined later. From (18) and (21), one can see that
We then obtain the inequality
by (6) and the inequality above, where . We can choose small enough so that . Once is fixed, we may choose β large enough so that
Hence, we have the inequality
from which the following decay estimates can be obtained by a similar argument as the one used in the proof of Theorem 1:
where .
Secondly, we consider the case that . If , multiplying both sides of (1) by () and integrating the result over Ω, and then using Hölder’s inequality, we get
By Lemma 2, it can be shown that
(22)
where . Since , we have . It then follows from (22) and Young’s inequality that
(23)
where will be determined later. From (19) and (23), one can see that
By (12), (13), and the inequality above, we can obtain the inequality
where . We can choose small enough so that . Once is fixed, we can choose β large enough so that
Hence, we can obtain the inequality
from which the following decay estimates can be obtained:
where .
Similarly, one can obtain the following decay estimates for :
where . □
Remark 3 One can see from Theorems 1-3 that the extinction of nonnegative nontrivial weak solutions to problem (1)-(3) occurs when .
Remark 4 Theorems 1-3 all require , λ, or to be sufficiently small or β to be sufficiently large.