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.