Is Linearization a method to solve nonlinear PDEs? I know almost nothing in PDEs theory. In my studding on a geometric problem I led to the Ricci flow on Riemannian metrics. At procedure of solving the flow we see the concept of linearization. I wounder to know:

what is the linearization method on solveing PDEs, and the motivation behind it?

 A: Linearization does not provide you with a solution to a nonlinear differential equation, let alone a solution valid anywhere except at the points you linearize at. 
What you are in fact doing is approximating the nonlinear equation with a linear on at a specific point. The further you get from that point the worse the approximation gets. The main reason for doing this is not to obtain a solution, but to look at the behavior of the solution at those points. Usually you will solve the differential equation for the points at which all the derivatives are equal to $0$, meaning that those points are fixed points and there is no motion at those points. Then you would look at the eigenvalues of the matrix of first order terms in the Taylor series. These eigenvalue will give you a sense of the stability at these points.  
In addition to the links in the comments I would check out this one for stability theory
But any linearization is not going to be extendable over the domain. There are numerical methods to solve nonlinear PDEs, they won't give you a closed form formula though.
