In this section, our argument makes use of the following Sobolev-type inequality
(3.1)
valid in for a nonnegative function v that vanishes on ∂ Ω. In this section, our results restricted to for proof of (3.1), see [8].
We consider the auxiliary function defined as
(3.2)
with
(3.3)
we assume that for all ,
(3.4)
for some constant β. In (3.2)-(3.3), n is subjected to restrictions
For convenience, we set
(3.6)
and compute
(3.7)
with
(3.8)
due to (3.4), we obtain
(3.9)
By using Hölder’s inequality,
(3.10)
and Sobolev-type inequality (3.1), we obtain
(3.11)
where Γ is defined in (3.1). Joining (3.11) and (3.9), we obtain
(3.12)
with arbitrary . Choosing , the first eigenvalue of problem (2.1), we have
(3.13)
by the Rayleigh principle. By using (3.13) in the last factor of (3.12), we obtain
(3.14)
with
(3.15)
Suppose that β is small enough to satisfy the condition
and that initial data is small enough to satisfy the condition
Then either remains negative for all time, or there exists a first time such that
(3.18)
Then we obtain the differential inequality
(3.19)
Integrating this differential inequality, we obtain
(3.20)
This result is summarized in the next theorem.
Theorem 2 Let Ω be a bounded domain in , and assume that the data of problem (1.1) satisfy conditions (3.4), (3.16), (3.17). Then the auxiliary function defined in (3.2) satisfies (3.20), and exists for all time .