The main result for the global solution is the following theorem.
Theorem 2.1 Let u be a solution of problem (1.1). Assume that the following conditions (i)-(iv) are satisfied:
-
(i)
for any ,
(2.1)
-
(ii)
for any ,
(2.2)
-
(iii)
(2.3)
-
(iv)
(2.4)
Then the solution u to problem (1.1) must be a global solution and
(2.5)
where
(2.6)
and is the inverse function of H.
Proof Consider the auxiliary function
(2.7)
Now we have
(2.8)
(2.9)
and
(2.10)
It follows from (2.9) and (2.10) that
(2.11)
By (1.1), we have
(2.12)
Substitute (2.12) into (2.11), to get
(2.13)
With (2.8), we have
(2.14)
Next, we substitute (2.14) into (2.13) to obtain
(2.15)
In view of (2.7), we have
(2.16)
Substituting (2.16) into (2.15), we get
(2.17)
The assumptions (2.1) and (2.2) guarantee that the right-hand side of (2.17) is nonnegative, i.e.,
(2.18)
By applying the maximum principle [24], it follows from (2.18) that P can attain its nonnegative maximum only for or . For , by (2.4), we have
We claim that P cannot take a positive maximum at any point . In fact, suppose that P takes a positive maximum at a point , then
(2.19)
With (1.1) and (2.16), we have
(2.20)
Next, by using the fact that , for any , it follows from (2.20) that
which contradicts with inequality (2.19). Thus we know that the maximum of P in is zero, i.e.,
and
(2.21)
For each fixed , integration of (2.21) from 0 to t yields
(2.22)
which implies that u must be a global solution. Actually, if that u blows up at finite time T, then
Passing to the limit as in (2.22) yields
and
which contradicts with assumption (2.3). This shows that u is global. Moreover, it follows from (2.22) that
Since H is an increasing function, we have
The proof is complete. □