Buying shares in stages

DCA (in my opinion) is not an investing strategy. It describes what happens naturally when people invest money periodically (like in a 401(k)). They reduce the risk of large losses if the stock goes down, but that comes at potential opportunity cost if the stock goes up.

If you are planning to buy-and-hold and you think the stock is going to go up (which obviously you do or you wouldn't buy it), then I would just invest the lump sum and let whatever happens happen. If the market goes down, it will (or at least always has) come back. With DCA you're reducing risk at the beginning but in the long run your cost basis should be about the same either way. Plus, since the market tends to go up more than it goes down, odds are you're actually better off lump-sum investing.

For sake of argument, though, let's answer your actual question:

once I buy say 50% of the shares and the stock goes up say 30% in 1 week, I don't want to dollar cost average (DCA) 30% higher. ... Should I just live with the 50% stake, or should I wait till the stocks goes lower?

It depends on what you think the stock will do going forward. If you think the stock will still go up, then you should buy it. If you think it will go back down,then you could wait until it does and then buy lower.

Let's look at Tesla. Say you bought half of the Tesla stock you planned to buy on Jan 1st. In one month it went up 75% from 443 to 780. Would you buy more then? If you did, you'd have seen it go from 780 back to 360 at its low point in mid-March. Ouch

But suppose you bought that 50% on April 1st at the height of the COVID crisis. In one month it went up 45% from 482 to 701. Obviously it will go back down again, right? It hasn't. If you did not buy more, you'd have missed out on the 169% increase it's experienced since then.

By the way, you'd have the same decision to make if the stock went down. Would you be willing to buy more after the stock dropped by 30%? Or are you "catching a falling knife"?

So the only way to decide what to do after a gain is decide if you think it will go up more from that.