In this section, our argument makes use of the following Sobolev-type inequality
valid in for a nonnegative function v that vanishes on ∂ Ω. In this section, our results restricted to for proof of (3.1), see .
We consider the auxiliary function defined as
we assume that for all ,
for some constant β. In (3.2)-(3.3), n is subjected to restrictions
For convenience, we set
due to (3.4), we obtain
By using Hölder’s inequality,
and Sobolev-type inequality (3.1), we obtain
where Γ is defined in (3.1). Joining (3.11) and (3.9), we obtain
with arbitrary . Choosing , the first eigenvalue of problem (2.1), we have
by the Rayleigh principle. By using (3.13) in the last factor of (3.12), we obtain
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
Then we obtain the differential inequality
Integrating this differential inequality, we obtain
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 .