On a sunny day (Thu, 29 Aug 2013 10:18:17 -0700) it happened "Guv Bob"
wrote in
:

When 12VDC is applied to the input, I need a 12VDC output for 5 =
seconds.
+12
|
-----------------------------------------------------|
| | | |
| --- | ===
| / \ 1N4001 Load --- 1000uF / 16V
| 1u poly --- D2 | | C2
=== C1 |-------------- ///
| |--- d
-------------| | IRLZ34
| 4M7 | |--- s T1
[ ] R --- |
| / \ D1 ///
/// ---
| 1N4148
///

When power is off C1 is discharged.
When power comes on the gate of MOSFET T1 is posive via C1,
and C1 slowly charges via R1,
R.C time, select a suitable capacitor and resistor.
The gate voltage slowly drops, and then T1 stops conducting.
When power goes off, then C1 is discharged again via D1.

D2 and C2 are needed to prevent transients on the drain (load line) of T1 piercing the gate electrode isolation layer.

IRLZ34 is a logic level (2 to 3 V gate on IIRC) MOSFET.

Alternative solution:
12 V light bulb - wakes up cat - cat drinks from cup - cup get lighter - balance turns - ball rolls, hits bell - you wake up, and swtich of the power.