Category of nonlinear parabolic equations and its structure
Abstract:
We define the notion of "map admitted by the system" for PDE and PDE
systems. This notion generalizes the main notion of Lie group analysis:
"group admitted by the system". Considering equations as objects, and admitted
maps as morphisms, we obtain the category of PDEs. We consider more explicitly
the category of second order parabolic equations, which are posed on arbitrary
manifolds, and we investigate its structure. It is proposed a special language,
intended for description and investigation of such kind of structures.
We illustrate use of obtained category structure on the example of nonlinear
reaction-diffusion equation.