## 1.4. A quick summary of frequently used methods and techniques in second order parabolic equations.

Note:
This summary is mainly based on L. Evans' Partial Differential equations, D Gilbarg, NS Trudinger's Elliptic partial differential equations of second order and Tian Ma's Theory and Methods of Partial Differential Equations (in Chinese).

Parabolic partial differential equations (PDE) of second order has tremendous amounts of applications among our everyday lives, and hence attracts special interests from allover the world, by various research fields. For example:

• Heat Equations modeling heat transfer;
• Diffusion Equations modeling dynamics of free particles;
• Reaction Diffusion Equations modeling chemical reactions;
• Diffusion Predator-Pray Equations modeling biological invasion;
• Diffusion FitzHugh–Nagumo Equations modeling activation-inhibition systems;
• Ginzburg Landau Equations modeling phase transition phenomena;
• Ricci Flow had been proved to be extremely powerful in geometry problems, for example in Poincare's Conjecture and Thurston's Geometrization Conjecture.

A typical second order parabolic PDE takes the following form:

Where $A=\mathrm{\left(a}$ij)0<=i,j<=N is an N-dimensional positive definite matrix. The following form gives a summary of common-used tools and ideas to analyze these PDEs:

Method Usage Comments
Extreme value principles Provide a estimation of solution magnitude using boundary conditions A very classical technique
Regularity estimation Provide a estimation of solution smoothness using boundary conditions A generalization of extreme value principles, utilizing interpolation inequalities in a proper Sobolev space
Green's function method Provide a exact solution of the equation Construct the solution by looking for a fundamental solution, which has lots of nice analytical properties
Fourier transformation Provide a exact solution of the equation To transform differential equations into algebraic ones, and are therefore easier to solve
Fourier series Provide an approximate solution or stability analysis of the equation Fourier series provides a way of approximation, which has great amount of applications in nature science and engineering.
Galerkin methods A powerful tool to prove the uniqueness and existence of the solution Can be used for equations of any type (i.e. elliptic, hyperbolic and dispersive). Also serves as the fundamental idea of finite element methods.
Method of Operator Semigroups Treat differential operators as functional generators, and therefore provide a very clear picture in geometric point of view A universal technique to treat ODEs (ordinary), PDEs (partial), SDEs (stochastic), DDEs (delayed) and stochastic processes.