## 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: