## 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
^{r}_{
k=1}
*a*
_{
k
}(*t, u*)*∂*
^{k}_{
t
}
*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.

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Original Russian Text © M.F. Prokhorova, 2013, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2013, Vol. 19, No. 4.

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Prokhorova, M.F. Factorization of the reaction-diffusion equation, the wave equation, and other equations.
*Proc. Steklov Inst. Math.* **287, **156–166 (2014). https://doi.org/10.1134/S0081543814090156

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### Keywords

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