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

M. F. Prokhorova

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 4, pp. 203-213

Résumé

We investigate equations of the form $D_{t}u = \Delta u + \xi\nabla u$ for an unknown function $u(t,x)$, $t\in\mathbb R$, $x\in X$, where $D_t u = a_0(u,t)+\sum_{k=1}^r a_k(t,u)\partial_t^k u$, $\Delta$ is the Laplace–Beltrami operator on a Riemannian manifold $X$, and $\xi$ is a smooth vector field on $X$. More exactly, we study morphisms from this equation within the category $\mathcal{PDE}$ 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 mappings of $X$ to other smooth manifolds (of the same or smaller dimension). It is shown that a mapping $f\colon X\to Y$ defines a morphism from the equation $D_{t}u = \Delta u + \xi\nabla u$ if and only if, for some vector field $\Xi$ and a metric on $Y$, the equality $(\Delta+\xi\nabla)f^{\ast}v = f^{\ast}(\Delta + \Xi\nabla)v$ holds for any smooth function $v\colon Y\to\mathbb R$. In this case, the quotient equation is $D_{t}v = \Delta v + \Xi\nabla v$ for the unknown function $v(t,y)$, $y\in Y$. It is also shown that, if a mapping $f\colon X\to Y$ is a locally trivial fiber bundle, then $f$ defines a morphism from the equation $D_{t}u = \Delta u$ if and only if fibers of $f$ are parallel and, for any path $\gamma$ on $Y$, the expansion factor of a fiber transferred along the horizontal lift $\gamma$ on $X$ depends on $\gamma$ only.

Détail

Keywords: category of partial differential equations, heat equation, wave equation.
Mots-clés : reaction–diffusion equation