Describing terms of a PDE I have seen words like 'dispersive', 'advective' and 'diffusive' when describing certain terms in PDEs - however I am unsure on the exact intuition behind this terminology and how one is to know how a term behaves prior to finding the solution?
For example how do we know if a 3rd order term is advective or dispersive?
Thank you
 A: Ok, here are my 2 cents.

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*DISSIPATIVE (or dispersive, or diffusive, they are all synonyms)

Well, the classic dissipative equation is the heat one: $\partial_t u = \Delta u$. Forget about boundary conditions etc. Test the equation with $u$ itself and you get $\frac{d}{dt} ||u||_{L^2 }^2 = -||\nabla u||_{L^2} ^2 $. So what is happening? If $u$ represents the heat, it's saying that the total heat (i.e. in $L^2$ norm) is decreasing. I.e. it is DISSIPATING energy. It is losing energy. The only case where this does not happen is when you have thermal equilibrium, i.e. $u=constant \implies \nabla u \equiv 0$.

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*TRANSPORT (or advective)

The name says it: you have a mass somewhere, you transport it in time. The transport equation is $\partial_t u= v \cdot \partial_x u$ and you can see a solution of it is $u(x+ vt)$. You have a mass in $x=0$ and you have a velocity $v$? Very well, then after a time $t$ your mass will be in $x=vt$.

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*REACTION

This one is the 0-th order term so to speak, and depends on $u$ only. Generally it's written like $\partial_t u =R(u)$ for some $R$. It's trickier because it really depends on what $R$ is. For example if you want a reaction with positive feedback, take $R(u)=u$. If you start with $u>0$, then you have $u(t)= u_0 e^t$ as a solution which grows indefinitely. In general it's trickier, for example in the study of bacteria populations, you have to take into account the food availability, or if it's a system of equations (see Lotka-Volterra model) there might be some interplays. It's really difficult to give you an exhaustive answer because it really depends.
To conclude,if you combine all together you get the so called advectiondiffusion-reaction equation, which reads
$$ \partial_t u  = \nu \Delta u + v \cdot \nabla u + R(u)$$
where $\nu$ is sometimes called "viscosity" (in hydrodynamic settings) or "thermal diffusivity" (in heat equation) or whatever second order effect it makes you lose energy.
By the way,you can construct a dissipative third order equation easily. Take for instance:
$$ \partial_t \partial_x u = \partial_x ^3 u$$
Then you have the dispersion of the term $\partial_x u$. The question is: is this useful somewhere/somehow? I don't know to be honest.
