Satraps each ruled a piece of land within the empire, but what happened when they didn't agree about their own inner borders? Did they always just go to a higher Satrap to decide, or to the emparor if needed? Or did it happen that fights broke out between Satraps in order to redefine their own borders?
Since you mentioned satraps, I'm guessing you're asking about the Achaemenid Persian Empire and not the much earlier Akkadian Empire, but pre-emptive apologies if I misunderstood.
This is actually a hard question to get a straightforward answer for. The sources do seem to imply that Satraps, or local governors of lower ranks (more on that below), did fight each other over territory, but the period where we have the most information about inter-satrap politics is also a period of revolts and somewhat unique circumstances in Persian history.
So first, the terminology, just so that everyone can be on the same page while reading this. We don't have a great sense of what exactly "Satrap" means. Literally, the Old Persia word xshasathrapa means "protector of the kingdom," but that doesn't tell us much about their role. We actually don't have many Persian uses of the word in the surviving record, but most of the ones we do have associate them with a full "province" listed in the lands ruled by the Great Kings in their inscriptions.
The Greeks used the title more fluidly, but also never had a firm grasp on the internal structure of the Empire. Basically any governor could be called a Satrap in Greek, but when they wanted to make it clear that it was a lower ranking governor, the Greeks identified Persian local rulers as "Hyparkhs." We don't have any obvious Persian equivalent, but I'll use Hyparkh here as a convenient way to distinguish between them and the "full" Satraps. There's also a smattering of local rulers, ruling in local styles, ranging from Greek tyrants to elected councils or magistrates to petty kings who paid tribute to Persia.
The Greeks started getting more involved in regional Persian political drama in the late 5th Century BCE, and, as a result, the last century of Achaemenid regional politics is the best documented. The problem with that is this was a period of particular instability in the western Empire, which skews the information available.
By far, the easiest way to get a general view of local politics, at least at the northwestern fringe of the Empire in this period, is to look at Xenophon's Anabasis. In his story of the infamous March of the 10,000 after the Battle of Cunaxa, Xenophon recalls the many minor regional systems he and the other Greek mercenaries encountered in Armenia and the rest of eastern Anatolia. Every local town, village, kinglet, and tribe was armed and prepared to fight. Many lived or traveled between fortified settlements to defend from hostile neighbors, and the Greek cities like Trapezos hired the 10,000 (or rather their remnants) to fight their non-Greek neighbors. On occasion, the 10,000 threw in with one local faction to secure allies to fight their way through a hostile factions territory. Far removed from the King, or even the Satraps, local groups carried on their own conflicts and grudges with violence, and so long as they paid taxes they were undisturbed by the powers that be.
During the Peloponnesian War, the Tissaphernes and Pharnabazus II (Satraps of Lydia and Hellespontine Phrygia respectively) were tasked with aiding Sparta and retaking the Greek cities along their coastlines that had been captured by the Athenians in preceding decades. Thucydides, Xenophon, and Plutarch all report constant Spartan frustrations with both Satraps because they were just as invested in outmaneuvering one another and claiming Greek cities along their border to boost their provinces' tax revenue. There aren't any accounts of Tissaphernes and Pharnabazus coming directly to blows with each other, but as cities passed back and forth between Perso-Spartan and Athenian control, both Tissaphernes and Pharnabazus used the Spartan fleets under their command to claim the same cities. They also sent conflicting reports back to the court of Darius II trying to bolster his support for their claims.
When a Spartan embassy brought their own report to Darius, explaining their problems with the pair of Satraps, Darius sent his 16 year old son, Cyrus the Younger, to serve as Karanos, a sort-of Super Satrap or absolute military governor with power over multiple provinces to settle things down. This worked to end the Peloponnesian War. Cyrus' willingness to just dump funds into Sparta and ability to override the Satraps' pettier concerns helped end the war in Sparta's favor. Part of Cyrus' effectiveness was simply that he didn't really need to care about his standing in the provinces. He was a prince with vast wealth beyond his position as governor and probably thought that he was going to be made king soon. That didn't happen and it lead to one of the first clear examples of two Persian governors openly duking it out over their borders.
To become Karanos, Cyrus also had to be a Satrap, and Darius II gave him Lydia, demoting Tissaphernes to Hyparkh of Caria, roughly the southern half of the Lydian province. Tissaphernes hated this, and hated Cyrus for it. When Darius died and named his eldest son, Artaxerxes II heir rather than Cyrus, Tissaphernes revealed that Cyrus was trying to assassinate Artaxerxes. Instead of the desired effect of getting Cyrus executed, Artaxerxes just sent Cyrus back to Lydia with a promise that there would not be a third chance. In an apparent attempt to limit Cyrus' personal power, he gave Caria, and thus Tissaphernes, direct control of the Ionian Greek cities.
The problem was that the Greeks liked Cyrus and had lingering dislike of Tissaphernes from his screwing around in the recent war. They revolted, not from Persia, but from Tissaphernes specifically, calling on Cyrus to claim their cities for himself. So Cyrus and Tissaphernes went to war. As Karanos and Satrap of Lydia, there really wasn't anyone short of Artaxerxes himself to appeal to, and both sides sent messengers to the royal court. Artaxerxes wanted to avoid escalating war in the northwest and focus on other concerns, namely a growing revolt in Egypt, so he sided with Cyrus. However, the problem never completely went away. Tissaphernes stubbornly held onto the most important Ionian city of Miletus, right at the northern edge of Caria, and Cyrus continued a half-hearted siege for months or years (its not directly stated when exactly this conflict started or finished).
The problem here, regarding the specific requirement in the OP, is that Cyrus was planning to rebel, but wasn't open about it yet. Apparently nobody was all that concerned with a Satrap and his nominally subservient Hyparkh fighting over territory, but Cyrus almost certainly instigated the revolt to build up his support in anticipation of a revolt, which finally occurred in 401 BCE, leading to Cyrus' death and Tissaphernes returning as Satrap of Lydia.
The following 40 years are an absolute mess of local politics in the region that really demonstrates the blurry line between inter-governor conflict and open rebellion. We get a weird story devoid of context from Hellespontine Phrygia. When the Hyparkh of the Aeolian Greek cities died around 399 BCE, his wife, Mania, successfully convinced Satrap Pharnabazus to let her take over her husband's position. In his Hellenica, Xenophon reports that Mania commanded troops in the field from a chariot or a carriage as they conquered various Greek cities in southern Aeolis/northern Ionia. He doesn't explain why she was doing this. It doesn't make sense for it to be connected to the Spartan invasion because Xenophon explicitly says that Sparta struggled to take territory in the first years of their invasion. Either these cities were rebelling on their own, or Pharnabazus was openly seizing territory from Tissaphernes, continuing their established rivalry over those Greek border towns. Xenophon just doesn't say.
Cyprus provides one of the most instructive examples. The Persians always took a relatively hands off approach to the island, allowing a dozen petty kings to continue ruling their city-states so long as they paid tribute. It was apparently entirely acceptable for these kings to go to war with one another over Cypriot territory or overthrow rivals in their own cities so long as they recognized the overarching Persian authority. That's exactly how Evagoras became king of Cypriot Salamis in 411, and proceeded to attack his neighbors, demanding that they pay tribute to him in addition to Persia. Nobody seems to have cared until 401, when Evagoras took Cyrus' revolt as opportunity to stop paying tribute altogether. A warning in 395 got him back on program for a bit, but he almost immediately started assisting Sparta in their new war with Persia and by 391 Evagoras revolted.