In many cases of interest a nonlinear evolution partial differential equation can be written as an infinite-dimensional dynamical system $$ du/dt+A(t)u=0 $$ on a suitable functional space $X$, where $A(t)$ is a nonlinear operator, cf. e.g. the classical paper of Kato Nonlinear semigroups and evolution equations.

I would greatly appreciate references to recent surveys or books (especially those aimed at nonexperts as much as possible) of the results on nonlinear evolution PDEs obtained in this framework including the case when the original evolution PDEs are of order greater than two.

The only relatively recent reference I found so far is the survey Partial Differential Equations in the 20th Century by Brezis and Browder but there this topic is only cursorily mentioned. Perhaps I use wrong keywords for googling or something :(

Many thanks in advance for your help.