Nonlinear parabolic systems

We examine existence, uniqueness, and regularity of solutions to nonlinear parabolic systems. We begin with an approach to strongly parabolic quasilinear equations using techniques very similar to those applied to hyperbolic systems in Chapter 5, moving on to symmetrizable quasilinear parabolic systems in §7.2. It turns out that another approach, making stronger use of techniques of Chapter 3, yields sharper results. We explore this in §7.3, treating there completely nonlinear as well as quasilinear systems. For a class of scalar equations in divergence form, we make contact with the DeGiorgi-Nash-Moser theory and show how some global existence results follow.