Parabolic partial differential equation Amangeldi Zhanel & Bolat Togzhan IS 172

Solution Under broad assumptions, parabolic PDEs as given above have solutions for all x,y and t>0. An equation of the form is considered parabolic if L is a (possibly nonlinear) function of u and its first and second derivatives, with some further conditions on L. With such a nonlinear parabolic differential equation, solutions exist for a short timebut may explode in a singularity in a finite amount of time. Hence, the difficulty is in determining solutions for all time, or more generally studying the singularities that arise. This is in general quite difficult, as in the solution of the Poincaré conjecture via Ricci flow.

The heat conduction equation and other diffusion equations are examples. Initial-boundary conditions are used to give

Examples Heat equation Mean curvature flow Ricci flow

Heat equation where α is a positive constant, and Δ or 2 denotes the Laplace operator. In the physical problem of temperature variation, u(x,y,z,t) is the temperature and α is the thermal diffusivity. For the mathematical treatment it is sufficient to consider the case α = 1.

Mean curvature flow so that the surface is nearly planar with its normal nearly parallel to the z axis, this reduces to a diffusion equationdiffusion equation

Ricci flow where is the average (mean) of the scalar curvature (which is obtained from the Ricci tensor by taking the trace) and n is the dimension of the manifold. This normalized equation preserves the volume of the metric.

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