Non-linear parabolic PDE notation and conditions I am told that for a non-linear PDE
$$u_t = f(x,t,u,u_x,u_{xx})$$
is parabolic if, writing $f = f(x,t,z,p,q)$, $$\frac{\partial f}{\partial q} > 0.$$
Is this the usual definition of parabolic in the field?
Also, suppose I have
$$u_t = \frac{1}{u_x}$$ or $$u_t = u_x$$
then do I need $u_x$ to be bounded for parabolicity? Bounded means what exactly? I saw it somewhere but it was not explained to me.
Thanks.
 A: There exist various definitions, or rather, different shades of parabolicity. The one (that $\partial f/\partial q$ does not change sign) bartgol is referring to is typically given when classifying second order equations. In some books (e.g., Qing Han's new book, if I remember correctly) this notion is called degeneracy, in order to avoid confusion with the "time-irreversible" notion of parabolicity (which seems to be used in your source). The latter is the idea that parabolic equations should generalize the heat equation, not the backward heat equation. This general class roughly includes all equations that are even order in space when written in a first order form in time, and are well-posed. For linear equations, the most general form of parabolicity that I know of is called Shilov parabolicity. Slightly more restrictive class is that of Petrowsky parabolicity, which arises when you want to have a classification in terms of the principal part in space. For nonlinear equations you linearize around a solution to transfer those notions.
A: You don't need 
$$\frac{\partial f}{\partial q} > 0.$$
Indeed, the two equations
$$u_t = u_{xx}\qquad u_t=-u_{xx}$$
(which correspond to a forward and backward heat equation respectively) are both parabolic. The important thing is that the sign of $f_q$ does not change. Its sign will then determine only if the equation is forward or backward in time (that is if you need to give an initial or a final condition).
I don't understand the second question. "Do I need $u_x$ to be bounded..." in order for what? What property do you want your solution/equation to enjoy?
