Vtyjkyuk

I'm trying to learn about nonlinear evolution equatiions. Can anyone offer a better text in this area? What basic results and knowledge in the theory of nonlinear evolution equatiions must be known for the graduates and researchers? Any reference will be appreciated！

The first thing you should do is establishing well-posedness of the nonlinear evolutionary PDE. After this you can do things like long-time behavior, optimal regularity, numerical analysis, optimal control problems, shape optimization, global attractors etc.

To approach well-posedness there are many methods: Galerkin approximation, semigroup theory, fixed point theorems (Banach, Schauder), monotone theory (Browder-Minty), etc.

First I'd recommend Nonlinear Partial Differential Equations with Applications by Roubicek. Part II of the book handles evolution problems and introduces every auxiliaries you will need later. Moreover, several PDEs and examples are investigated. No semigroup theory here, but the other methods. Nonlinear Evolution Equations by Zheng and Infinite-Dimensional Dynamical Systems by Robinson also fits to this description.

For semigroup theory I'd use One-Parameter Semigroups for Linear Evolution Equations by Engel & Nagel and C0-Semigroup and Applications by Vrabie. Both introduce semigroup theory in general and then also mention nonlinear problems.

For monotone theory I'd use Monotone Operators in Banach Space and Nonlinear Partial Differential Equations by Showalter and Nonlinear Differential Equations of Monotone Types in Banach Spaces by Barbu.

Aside from well-posedness for an analysis for nonlinear evolutionary PDES motivated from physical/biological applications you can look at Abstract Parabolic Evolution Equations and their Applications by Yagi.