Basic math. If your birthrate stabilizes below replacement level then you go extinct because each generation is smaller than the one before. The only way you don't go extinct is if your birthrate over the long haul is at or above replacement level.
Do I need to seriously believe that there is possible to extinct for humanity because of some people are not willing to make children? Are there any chances for this to happen before heat death of the Universe if to extrapolate Japan's birth trend to all human population?
No, because not all parts of humanity are equally averse to having children. So as long as it is physically possible for people to have kids and survive until adulthood, natural selection will eventually reassert itself. Things could get dicey for a while until then, though?
If global fertility rate becomes that like Japan, we'll have a halving of the population every generation. This leads to a global population of 78 million after 10 generations. Also a miserable population because each person needs to support two elderly people while before it was two people supporting one elderly. That's a 4x increase in work.
But what if we adjust the figure to take into account those percent Japanese who want children but cannot afford them? When the population begins to "extinct" because of those who can but do not want to reproduce, those who want to but cannot now will be able to do it.
Also the overbearing burden to support the elderly with a smaller younger population will apply to every generation (barring temporary baby booms) since each generation will be smaller than the previous one.
I think the end result will be that elderly won't be supported anymore. But let to work until they die. Cultures can change, assisted suicide or simply not keeping people alive past certain point might come reality.
I said they obviously will stabilize. They always do. Everything always does. At a certain point the population is just so low that having children becomes highly advantageous again. There's very very few biological system where the entire population just dies out from lack of reproduction (despite all other resources to do so being available).
On the other hand: there is absolutely 0 evidence to suggest that the population wouldn't stabilize. There are sub populations of humans that already have stabilized populations, this is true throughout the animal kingdom. If it's just preferences to not have kids, then those people will see their genetic line stopped and the people with preferences to have kids will see their genes dominate. If it's environmental, then our path will look a lot like that of other animals experiencing environmental shifts: certainly tough, but if you can adapt fast enough you'll be ok.
Right now, the main reason people don't have kids who want them is finances. The likely outcome here looks like the economy will winnow as a consequence until having kids becomes a positive financial decision (and that might mean going back as far as subsistence living in a worst case scenario). Regardless, the population will stabilize short of an asteroid or other catastrophe.
There's not a single precedent ever for what you suggest. The population stabilizing is the default for all of nature.
Your argument about sub populations can be just as easily applied to humans as a whole. Perhaps humans will be replaced by another species.
If you look at nature more coarsely, you don’t see population stabilization. Instead you see speciation and extinction due to random and constant changes. Perhaps there will be an environmental event that we would otherwise survive that will do us in because of our low population.
For example, the limit of 0.99^n is 0.