= Solution
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.
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