The structure of the category of parabolic equations

2005/12/05 by Marina Prokhorova, Prokhorova, Marina · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #18B99 #35K05 #35K10 #35K55 #35K57 #58J70 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Category Theory (math.CT) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Waves and Solitons #math.AP #math.CT #msc:18B99 #msc:35K05 #msc:35K10 #msc:35K55 #msc:35K57 #msc:58J70

paper · pdf · doi:10.48550/arxiv.math/0512094

33 pages. Text improved and extended

openalex publication_date 2005/12/05 · arxiv created 2009/05/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define here the category of partial differential equations. Special cases of morphisms from an object (equation) are symmetries of the equation and reductions of the equation by a symmetry groups, but there are many other morphisms. We are mostly interested in a subcategory that arises from second order parabolic equations on arbitrary manifolds. We introduce a certain structure in this category enabling us to find the simplest representative of every quotient object of the given object, and develop a special-purpose language for description and study of structures of this kind. An example that deals with nonlinear reaction-diffusion equation is discussed in more detail.

Cited by

• Conservation Laws and Potential Symmetries of Linear Parabolic Equations
• Reduction operators of linear second-order parabolic equations

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