Examples: chemotaxis, porous medium equation, aggregation with long term interaction
Global existence and blow up mechanism
Navier-Stokes equations: weak and strong solutions. Galerkin method and L^{2} contraction approach
Euler equations: finite time blow up. The BKM criterion
Detour: Sobolev spaces, compensated compactness and weak solutions

"On the motion of a viscous liquid filling space" J. Leray, 1934 [ pdf file ]

"Euler eqs: local existence and singularity formation" M. Lopes, H. Nussenzveig and Y. Zheng, [ pdf file ]

Maximum principle, Comparison Principle, Large time behavior
Global existence, Asymptotic decay and blow up mechanism
Applications: Image processing, Navier-Stokes equations

The nonlinear wave equation and related problems

Linear and semi-linear wave equations; Schrödinger and Klein-Gordon equations

Lawrence C. Evans,
Partial Differential Equations (Graduate Studies in Mathematics, V. 19), AMS Robert McOwen, Partial Differential Equations, Methods and Applications,
Chapters 10-12 Heinz-Otto Kreiss and Jens Lorenz,
Initial-Boundary Value Problems and the Navier-Stokes Equations Randy LeVeque,
Numerical Methods for Hyperbolic Conservation Laws Peter Lax, Hyperbolic
Conservation Laws and the Mathematical Theory of Shock Waves, CBMS, 1972 Joel Smoller,
Shock Waves and Reaction-Diffusion Equations, Springer, 1994 Constantine Dafermos,
Hyperbolic Conservation Laws in Continuum Mechanics, Springer, 2005 Richtmyer & Morton,
Finite Difference Methods for Initial Value Problems, 1967 DeVore, & Lorentz,
Constructive approximation, Springer-Verlag, Berlin, 1993