Ukraine

Showing weakness usually doesn't end well for authoritarian leaders and at the same time they aren't the ones currently suffering the consequences. In some regards it might even have expanded the powers they have internally (at least in the short term).

So in some twisted way there is some logic from their point of view. The fault really lies in starting this whole thing under the assumption it would be an easy win, while not having a way to back out again, while saving face.

This war was lost over a year ago. None of their aims were achievable after that point.

To understand why it continues you have to understand two things:

Putin will cease to be leader if he ends this.

Putin believes in luck. The longer this continues the more likely of some random event going in his favour or of the West getting bored and withdrawing their support.

He's intentionally dragging out the conflict. The tactics they use are designed to do nothing more than slow down how much territory is lost over time.

Every so often they attack to achieve a little 'victory'. This is then twisted into a massive victory on every tv across Russia. This plus huge pay offerings, gives them endless volunteers. Particularly from rural regions where education is lacking.

Putin does not care how many of his own people he kills. They are nothing to him. The tactic putin is using is based on throwing 10, 20, 100 or whatever number to kill one Ukrainian. That is one less Ukrainian. That is what matters to him.