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In the convention $A{v}_{xx}+2B{v}_{xy}+C{v}_{yy}=\cdots$, the coefficients of the principal symbol areThus the second-order partial differential equation isThere is no region of elliptic partial differential equation type. The terms hyperbolic, parabolic and elliptic refer to the pointwise classification of a second-order partial differential equation. At $(0,0)$ every second-order coefficient vanishes, so the principal symbol has rank zero and the equation is degenerate rather than genuinely parabolic there. Moreover the right side is $2$, so no twice continuously differentiable classical solution can satisfy the equation at the origin.

For a second-order scalar partial differential equation, classification refers to its principal second-derivative form. An elliptic partial differential equation has a definite principal form, exemplified by Poisson's equation; it typically determines a spatial field from boundary data. A parabolic partial differential equation has a distinguished first-order evolution direction and diffusive second spatial derivatives, exemplified by $u_t=D{u}_{xx}$. Its initial data relax or spread with time. Thus the pseudo-time equation is parabolic, while its steady-state spatial equation is elliptic.

For $A{u}_{xx}+2B{u}_{xy}+C{u}_{yy}+\cdots =0$, the sign of $\Delta =B^2-AC$ distinguishes a hyperbolic partial differential equation when $\Delta >0$, an elliptic partial differential equation when $\Delta <0$, and a parabolic partial differential equation when $\Delta =0$ and the principal symbol has rank one. If $A=B=C=0$, the second-order principal symbol has rank zero and the differential order drops; this fully degenerate case must be checked separately for compatibility with the lower-order terms and forcing.