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I'm a physicist, and while working on a problem I ended up with the following PDE describing my physical property $u(x,t)$, $$ \frac{\partial u}{\partial t} + \frac{\partial}{\partial x} \left( u^3 + a u^3 \frac{\partial^3 u}{\partial x^3} \right) = \frac{b}{u} $$ where $a$ and $b$ are constants. The property $u$ should physically always be positive.

Now, in order to look up helpful material on analysis and numerical solutions, I would need to classify/name this kind of PDE. Does this PDE belong to some known class/type/family of equations? If it's helpful, perhaps disregard the right hand side term, and we can call it an "Equation of type X with a source term".

I would also be interesting in useful ways of rewriting this PDE, of course.

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    It's been awhile since I've looked at PDEs, but this seems to be some sort of non-linear, driven, advection-hyperdiffusion PDE. Advection refers to overall bulk movement ($u^3$ term) and the hyperdiffusion comes from the fourth derivative. – 2017-02-13
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    The tag ([tag:differential-equations]) is intended for questions about ordinary differential equations, there is a separate tag for [tag:pde]s; see the [tag-wiki](http://math.stackexchange.com/tags/differential-equations/info) and the tag-excerpt. (The tag-excerpt is also shown when you are adding a tag to a question.) – 2017-02-13

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