# Factorization of the reaction-diffusion equation, the wave equation, and other equations

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## Abstract

We investigate equations of the form *D* _{ t } *u* = Δ*u* + *ξ∇* _{ u } for an unknown function *u*(*t, x*), *t* ∈ ℝ, *x* ∈ *X*, where *D* _{ t } *u* = *a* _{0}(*u, t*) + Σ _{ k=1} ^{ r } *a* _{ k }(*t, u*)*∂* _{ t } ^{ k } *u*, Δ is the Laplace-Beltrami operator on a Riemannian manifold *X*, and *ξ* is a smooth vector field on *X*. More exactly, we study morphisms from this equation within the category *PDE* of partial differential equations, which was introduced by the author earlier. We restrict ourselves to morphisms of a special form—the so-called *geometric morphisms*, which are given by maps of *X* to other smooth manifolds (of the same or smaller dimension). It is shown that a map *f*: *X* → *Y* defines a morphism from the equation *D* _{ t } *u* = Δ*u* + *ξ∇* _{ u } if and only if, for some vector field Ξ and a metric on *Y*, the equality (Δ + *ξ*∇)*f*v* = *f**(Δ + Ξ∇)*v* holds for any smooth function *v*: *Y* → ℝ. In this case, the quotient equation is *D* _{ t } *v* = Δ*v* + Ξ∇*v* for an unknown function *v*(*t, y*), *y* ∈ *Y*. It is also shown that, if a map *f*: *X* → *Y* is a locally trivial bundle, then *f* defines a morphism from the equation *D* _{ t } *u* = Δ*u* if and only if fibers of *f* are parallel and, for any path *γ* on *Y*, the expansion factor of a fiber translated along the horizontal lift *γ* to *X* depends on *γ* only.

## Keywords

category of partial differential equations reaction-diffusion equation heat equation wave equation.## Preview

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