Although most natural phenomena we know of are observed to have behaviors described by at most second order ordinary/partial differential equations, but there are behaviors that require third order differential equations. One example is in motion and it is called jerk. Jerk is the third order derivative of the position/displacement function of a moving object, where the first and second order derivatives represent velocity and acceleration.
You can start from KdV equation: U_t+UU_x+U_xxx=0. It is nonlinear PDE, derived in hydrodynamics, but it has analytical solution. If we linearize it, it will contain 3-rd derivative, which comes from dispersive effects.
In general, I would recommend this book of Whitham, which has many DE for different problems in physics and applied mathematics. See https://archive.org/details/LinearAndNonlinearWaves
There are some applications in which third order ODEs are formed. They are in the momentum tranport of viscoelastic fluids in porous channels, plates etc.
Sure, there are lots of them in astrodynamics: the Clohessy-Wiltsjire equations (relative motion about a circular orbit), the Tschauner-Hempel equations (relative motion about an ellipse), the two-body equation after the substitution y=1/r. There are others,
Colleagues have already pointed a lot of processes that can be modelled through 3rd order differential equations, ordinary and partial. Let me add one PDE example, emerging in porous media flows. So-called non-equilibrium models involving dynamic capillarity and/or play-type hysteresis, lead to equations where the time derivative of the unknown quantity (saturation) is combined with second order derivatives in space.
The resulting model is a nonlinear version of the one mentioned above by Roman Poznanski, and of the Benjamin–Bona–Mahony equation.
Unfortunately, the example proposed by L.Herrera for the temperature is physically inconsistent, in that it has three characteristic roots: one real negative root and two complex conjugate roots with POSITIVE real part. Hence, the corresponding solutions blow up.
@Giorgi, The equation (3) in my answer follows from the standard continuity equation. This is a rough approximation because as we know, the continuity equation depends on the transport equation, and it might happen that it is incompatible, with the proposed transport equation. The point is that the kernel (2) leads to a third order differential equation for the temperature. Probably some how different from (3), but still a third order equation.
Specific applications to material systems contravening the fading memory paradigm, may be found in: L. Herrera. Causal heat conduction contravening the fading memory paradigm. Entropy 21, 950, (2019).