Cold start will run short time then die battery not strong enough to realy spin moter

Many things to consider but, let's try the simple things. You didn't say, so I am going to assume the mower is at least newish, not 30 years old. Let's say you have less than 300 hours on the engine. Let's further assume it worked last summer, or last week, but now it's acting up. Let's also verify the battery is in good shape and fully charged.

A mechanically functional (meaning all the internal parts are working, i.e. pistons, valves, etc) engine requires 3 things, air, fuel, spark. So the quickest way to start your diagnostics is to remove the cover over the air filter and then remove the air filter. I do not recommend spraying Quick Start.

1. remove air filter
2. put throttle on fast run (do not choke)
3. pour an ounce or two of fresh gas into the carburetor
4. crank engine (15 to 20 seconds, rest to cool the starter, repeat 3 times if it does not fire at first).

Removing the air filter and placing the throttle at full speed, no choke, eliminates the "air" question. Dumping in the fuel eliminated all fuel related questions. What's left? Spark. If the engine fires and runs a bit, then dies, you have eliminated the spark question. The engine started so Spark and Air are good; you have to track down a fuel problem.

If the engine does not fire, you have eliminated the Air & Fuel questions; you must tract down a Spark problem.

Tracking down fuel problems.

1. Make sure there is plenty of fresh fuel in tank.
2. Remove fuel line from filter and remove the filter, gas should pour out of the hose coming from the tank. If it does not, remove gas tank cap. If fuel flows now, you need to unclog the cap vent or replace the cap. If the fuel still does not flow then blow air from the filter end of the hose back up through fuel line to blow out blockage inside the gas tank. Once you have fuel flowing out the line to the filter, PUT ON A FRESH FILTER. Try cranking the engine. If it still does not start move on to the carburetor.
3. Note: many systems have a fuel shut off solenoid. This is a small black cylinder on the bottom of the carburetor. It has two wires running to it. It should make an audible click when you turn the key on.
4. NOTE: if battery voltage drops below 9 volts when cranking or while the engine is running, the fuel solenoid will not function. That means the engine will not start or will die soon after starting. So check your battery voltage before, during, and after cranking. If the battery voltage is good before but drops below 9 volts while cranking this is generally a sign the battery has one or more bad cells. Replace the battery. If the battery is in fact good and the voltage is still dropping you may have a bad starter motor and it is drawing to much amperage.
5. Ck to see that the carburetor bowl is full of fuel, if not...you may have to disassemble and clean the carb at this point.

Tracking down Spark problems.

1. verify battery fully charged.
2. verify the fuel cutoff solenoid clicks when you turn the key switch
3. get fresh spark plugs and verify gap
4. install plugs and crank.Perhaps verify spark with a spark testor

If these simple steps do not solve the spark problem, you have a Spark problem that moves into the arena of complex. You will need to check coil gap and coil performance. If it still doesn't start you will need to diagnose the electrical system for wiring and safety switch problems, key switch problems, fuses, etc.

Once you get it running be sure to replace the Air Filter with a new one.

These simple steps solve most starting and running questions and problems.