Caption: A cartoon of false vacuum decay during an inflation era. The x-axis is size of the inflaton and the y-axis is the energy of inflaton for each size point.
Features:
But in eternal inflation that region becomes a pocket universe in the whole universe (which is eternal and infinite) and consists of the false-vacuum universe and infinitely many pocket universes.
The pocket universes are usually thought to have somewhat different physics (e.g., different values for fundamental physical constants, different elementary particles, and maybe other things).
If physics does vary pocket universes ordinary physics (atomic physics, nuclear physics, and maybe quantum chromodynamics: the physics of quarks) may be emergent physics---an accident of the way our pocket universe was initiated by its initial random quantum fluctuation.
However, our most fundamental physical laws (e.g., quantum mechanics, quantum field theory, general relativity as macroscopic scale emergent theory from quantum gravity, theory of everything (TOE), thermodynamics statistical mechanics, TOE-Plus) are assumed to be constant (but NOT their fundamental physical constants, etc.) or we have NO guidance at all for the universe well beyond the observable universe either in space or time. We would NOT even be able to speculate about the "beyond"---except in philosophy.
At least in some fraction of pocket universes where the physics is rich enough all kinds of the complex structures can develop like life as we know it.
But we have a name for it anyway: the theory of everything (TOE).
This is NOT a good name in yours truly's view since it is NOT a theory of everything---but no one ever listens to yours truly.
Some theorists think that string theory (which is a candidate for TOE) could accommodate wildly varying ordinary physics between the pocket universes, but it may NOT (see, e.g., Livio, M. 2013, How Can We Tell If a Multiverse Exists?).
However, the density of mass-energy in the false vacuum may be so high that quantum gravity governs false-vacuum universe. Since we do NOT have an established quantum gravity theory, referring to quantum gravity may mean we know nothing much about the false-vacuum universe.
On the other hand, maybe they just expand forever or eventually become static embedded in the also expanding forever false-vacuum universe.
If the false vacuum also expand forever, it may keep most pocket universes causally disconnected (i.e, they never affect each other in any way) forever.
However, wormholes actually exist, they may make causal connections between the pocket universes---but idea is highly speculative.
Yours truly thinks that the speculation is no, it does NOT evolve globally.
Does its entropy increase? If does NOT evolve globally, no. Unless as some have theorized (e.g., Sean Carroll (1966--)) ever increasing entropy is a fundamental fact of the whole universe.
Thus, inflation explains two important features of the observable universe. This has always been a strong argument for the inflation paradigm, but NOT for any particular version of inflation.