Raspberry Pi crashes

Take a look at the RasPi Troubleshooting Wiki (Display).  A likely cause is that RasPi's 5V has drooped and the HDMI signals are no longer strong enough for your monitor.  You can check 5V by measuring between TP1 and TP2 with a multimeter.  It should be at least 4.8V for reliable operation.  On my RasPi, I can run down to 4.65V but below that I first see a bunch of green pixels on the screen and if the voltage drops even lower HDMI drops out entirely.

A common way for a RasPi to misbehave after a few minutes is polyfuse F3.  A polyfuse provides a small resistance, and if you exceed its rated current it heats up and the resistance increases, which causes the RasPi 5V to drop low enough for it to fail.  When you turn off power, the polyfuse cools and slowly returns to something near its original state.  It may be that your board has an iffy polyfuse and its resistance is now too high.

See the RasPi Troubleshooting Wiki (Power Problems) to see whether that's your problem.  If it is, you can replace F3 with a polyfuse or a single-blow fuse, which has lower resistance.  You can also just bridge across it with a wire, but that loses F3's protection.  Any board modification voids the warranty, so be careful.

Another possibility is a bad solder joint or PC board trace which fails when the board heats up.  These are not usually repairable, so it's better to return the board if it's still under warranty.

When RasPi isn't doing anything, you should only see the red PWR light.  The green ACTivity light (labeled OK on RasPi rev 1.0) flashes when RasPi accesses the SD card but is otherwise off.

Take a look at the RasPi Troubleshooting Wiki (Display).  A likely cause is that RasPi's 5V has drooped and the HDMI signals are no longer strong enough for your monitor.  You can check 5V by measuring between TP1 and TP2 with a multimeter.  It should be at least 4.8V for reliable operation.  On my RasPi, I can run down to 4.65V but below that I first see a bunch of green pixels on the screen and if the voltage drops even lower HDMI drops out entirely.

A common way for a RasPi to misbehave after a few minutes is polyfuse F3.  A polyfuse provides a small resistance, and if you exceed its rated current it heats up and the resistance increases, which causes the RasPi 5V to drop low enough for it to fail.  When you turn off power, the polyfuse cools and slowly returns to something near its original state.  It may be that your board has an iffy polyfuse and its resistance is now too high.

See the RasPi Troubleshooting Wiki (Power Problems) to see whether that's your problem.  If it is, you can replace F3 with a polyfuse or a single-blow fuse, which has lower resistance.  You can also just bridge across it with a wire, but that loses F3's protection.  Any board modification voids the warranty, so be careful.

Another possibility is a bad solder joint or PC board trace which fails when the board heats up.  These are not usually repairable, so it's better to return the board if it's still under warranty.

When RasPi isn't doing anything, you should only see the red PWR light.  The green ACTivity light (labeled OK on RasPi rev 1.0) flashes when RasPi accesses the SD card but is otherwise off.