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What year is your Corolla? how many miles do you have on the Corolla? When the blower motor switch is turned on and you press the AC switch does a little light in the AC switch illuminate or does it flash once you turn it on?
Thanks for clarifying that I misread the info above and it was very late. Do you have anything else that is not working in the car? Any recent work done? Do you have any test equipment like a volt meter or test light?
With out any test lights or volt meter it is difficult to say for certain or to tell you what to check. Start with the fuses under the hood and in the fuse box inside the car. You can remove the glove box like you are going to change the in cabin air filter then pull the in cabin air filter out and reach in and see if the blower motor cage will spin by hand. then turn the switch on and see if it will start if you assist it. Do you hear the relay click when the fan switch is turned on?
Relays rarely go bad on this for the blower fan. there is more likely a power supply problem. I would not throw a relay at it just yet. The surge in the engine is the compressor engaging this is normal. If you can even get a simple 12 volt test light that can help to do some checks. Let me know and try the in cabin filter/ blower inspection.