My 3 year old SRX400 just started doing the same thing. Being an electrical engineer, I took it apart.
As anticipated, the power supply circuit was busted. The 2 large capacitors had bulges in them - "teapo" brand often referred to as "cheapo" had expired of natural causes. You wouldn't believe how many of those I pulled out of electronics over the years. They are very poorly made and normally last 3 years or so, even under ideal conditions.
I replaced the caps, but that didn't fix the problem either. So when the capacitors died, something else got pulled along. I did some additional testing and... cutting the long story short, I had to connect pin #3 of AP1510 step-down converter to the ground through a 220K resistor. The current controller is the small 8-pin surface mounted microchip sitting right by the power plug. Total cost of repair - about 80 cents in parts.
At least the manufacturer didn't go cheap on the capacitor ratings, like others. Samsung is the worst offender of all. Not only do they use "teapo" caps in their circuits, but also rate them closely to the circuit voltage, so sure, they blow up in no time!
The quick way to test whether the same solution would work for you, if you're not an electrician, is to take the router apart (open box, disconnect antennas, remove front and back metal shields from the main board). Locate the capacitors (2 big barrel-like things close to where the power plug is). If they're bulging, then there's a problem, they'll have to be replaced either way (they cost about 30 cents each retail).
On the back of the main board there is a removable wireless module (that draws the most current out of the capacitor plagued area). Remove the board and turn the router back on. If the light stays on longer, then replacing caps may be enough.
If after replacing the caps the light still goes off, then get a metal needle. Locate the AP1510 chip - the tiny black thing by the power plug. Hold a large metal part anywhere on the board with one hand (shielding around network plugs, for instance), and touch pin number 3 of the microchip with the needle. If the router resets (if the lights start blinking), keep the needle on pin 3 till the time when it would normally turn itself off. If it stays on, then your problem is exactly like mine. Human body offers about 200-300K resistance, so it's ideal for testing. Don't worry, the voltages on the board are way too low to harm anyone.
You'll need to solder pin 3 to the ground through a 220K ohm resistor (they're about 20 cent retail).
In reality there are some micro-components that could be replaced, but I do not have the equipment for micro-soldering, so I go big and cheap instead. My solution is guaranteed to prolong the life of SRX400 until it's physically obsolete.
If you have any more questions, post a follow-up.