I know that the traditional classification of PDEs into parabolic, elliptic, and hyperbolic is applicable for the second order equations. However, I often see remarks about parabolicity of higher order PDEs in various articles. In particular, the equations of the thin film family are said to have the "degenerating parabolicity". For example, this paper introduces the following degenerate parabolic nonlinear fourth order equation (in $1\mathrm D$): $$ u_t + \nabla\cdot \left(\left\lvert u\right\rvert^p\,\nabla \Delta u\right) = 0. $$

How do we formally define the parabolic, elliptic, and hyperbolic classification of high order nonlinear PDES?


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