When the flight date like in one or two days, and there are unsold seats, is it a possibility that airlines reduce the ticket price in the flight day or so?
Generally not, but this can be very different between airlines and routes. Typically airlines set a price for a seat on a given flight; if the seats sell fast, they may even increase prices; if the opposite is true, prices indeed can go down a bit as the flight date approaches. (Caveat: prices can also differ based on one's IP address, browser cookies and other things, but I don't want to go into that discussion.) This mostly applies to off-season leisure flights, e.g. from Germany to a Mediterranean island. If no one is willing to fly at the posted price, lowering the prices is better than flying an empty airplane.
However, flights on major routes, such as between Western Europe and North America (and many others), are never off-season and are almost always fully booked. Airlines don't have to worry about a lack of demand and don't have to attract extra customers by low prices. Quite the contrary, prices go steeply up as the flight date approaches, because there are always some people who want to travel on short notice for business or family reasons. With no other realistic options for intercontinental travel, there will always be people willing to pay obscene amounts of money for a seat.
On the other hand, cheap last-minute deals are quite common in charter flights, in which the seats can be bought only from a travel agency with a package holiday. If someone cancels the holiday (never getting the full 100% money back), the travel agency will be happy to sell it again at a reduced price, because they want to avoid empty seats and hotel rooms (which they already paid the airlines and hotels for!) and in total might even get more revenue than the originally listed holiday price.
Yes, lower-cost last minute tickets are made available sometimes. Pricing of airline tickets is very complex and is done to maximize the price each customer pays. The general pattern is that tickets start getting sold about 11 months before the flight for some price towards the cheaper end of the spectrum. Price increases partly due to customers buying cheaper tickets first but mostly due to reduce availability as the flight fills up. The airline also counts on those who tickets close to the departure date to be more desperate.
With modern flight scheduling, the vast majority of flights fly completely full or with very few available seats left. For this reason, last minute flights are almost always more expensive. However, if a flight turns out to be less popular or there are lots of cancellations due to some events (but the entire flight is not cancelled) than ticket prices go down. If you are looking for cheap last minute tickets, you would have to be flexible with your dates and your destination. Basically keep looking and you may find it. Price Alerts websites are great for that when they allow daily alerts.
Price can also get reduced at the times too. A few months ago I started tracking the cost of a flight to Japan for specific dates (between work and school we had only a few days leeway). The tracked price started around $2000 CAD back in May, in went up some weeks later and decided to book when the tracker reported $2400. I was using a weekly tracker, so this was already out of data and we booked for $2800 just a day after getting the alert. About three weeks later, we got a notice from the airline about the schedule changing by a couple of mins. I decided to go through the booking process to see what the new price was and - to my surprise - it had gone down to $1900 CAD! The difference was that the first time, there were fewer than 10 seats available on the plane, the second time, the plane was about half-empty! They had changed planes for a larger model and so the extra seat availability made the price go down. I kept checking the price out of curiosity every week and went up $7007 CAD a week before the trip! Now, a few days before departure, the price this morning is $3800 CAD. So, if they couldn't fill it at $7007, then they try at a lower price.
I agree with the other answers that it's uncommon for prices to decrease within the 48 hours before the flight. I'd add that some flight search engine shows the price history, which may help you assess the frequency of last-minute discount for your itinerary. E.g., Google Flight:
Based on the price history, one can try to establish some trends, e.g. from https://nomadicfoodist.com/when-do-prices-drop-for-flights/:
Airline pricing isn't driven by econ 101 S&D. There are a couple reasons for that. The biggest reason is that there are only a few airline providers and the other is that the marginal cost of flying one extra person is pretty close to $0 which is quite far from the average cost.
From a game theory perspective, and in general, one of the worst things an airline could do for itself is decrease the price of a ticket as it got closer to actual takeoff. The reason is that it would give people an incentive to wait to book.
Another thing to consider is that there are (simplifying things a bit) 2 groups of last minute ticket buyers. There are people who are value conscious and are only buying the last minute ticket because it just so happened to get very cheap. The other group is the people who found out last minute that something happened and they just have to fly. The latter group is the opposite of value conscious, they're going to buy a seat at (nearly) any price.
The first group is small because flying somewhere is not just about the cost of the plane ticket. You've got to consider the coordination of getting time off of work, you've got to have a hotel, maybe a rental car, etc. Hypothetically, if an airline offered you a seat for a flight that leaves this afternoon that goes to your favorite city, could you just drop everything else in your life to jump on that deal? Probably not, or at least that'd be the case for most people. Airlines know this and so offering last minute cheap flights isn't going to make a big difference in how many people will buy otherwise empty seats.
There's not really much to say about the people that find out last minute they just have to fly somewhere. Maybe it's a sick relative or a business event but those people can't curtail their last minute travel needs just because the price is high.
All of these reasons combine to give airlines a very strong incentive to keep prices high or the same as travel dates approach.
Mostly not, companies try to get the most from desperate people who need to catch that flight at last second and they can pay a lot for a ticket (like people who are going to a funeral), so better to have 2 persons happy to pay a lot rather than having 8 people who fly just because it's cheaper.
But most of the time, they keep prices high for marketing reasons and don't mind filling the flight, also because last minute passengers make hard flights programming for airlines and, if they begin reducing ticket prices for last minute bookings, this phenomenon will increase making things worse...
Anyway, for some routes, probably the harder to fill, it's actually possible to find cheap tickets at the last second. (For instance, if you book now, in August, a ticket from Europe to Chennai for the next 2/3 days, you can find very good prices...)
You may be lucky, maybe not. 25 to 30 years ago this was a thing, especially in the UK and offers were constantly streamed by teletext which was text transmitted between analog TV frames.
Bucket shops would buy buckets of tickets to sell, and any surplus was sold cheaply close to the departure date. The same business model as scalpers, and this still happens with Asian hotel rooms (The "C" people) that are bought in blocks. Somehow this became frowned upon as if tainted by similar "smokers weekends" to Amsterdam.
Not so common nowadays. But people have tribal memory...