Introduction

In this memoir we aim at presenting, in a unified way, various and intercon-

nected basic aspects of the regularity theory of solutions to general, non-linear

second order parabolic systems with polynomial p-growth

p ≥ 2 .

Specifically, we are dealing with systems of the type

(0.1) ut − div a(x, t, u, Du) = 0

and related non-homogeneous ones

ut − div a(x, t, u, Du) + H = 0 , H ∈ D (ΩT ) ,

under natural p-growth and ellipticity assumptions on the vector field

a: ΩT ×

RN

×

RNn

→

RNn

that is

(0.2) |a(x, t, u, w)| ≤ L(1 +

|w|2)

p−1

2

holds for every (x, t) ∈ ΩT , u ∈

RN

and w ∈

RNn.

The system in question is

considered in the cylindrical domain

ΩT = Ω × (−T, 0) ,

where Ω denotes a bounded domain in

Rn,

n ≥ 2, and T 0; in the following Du

will always denote the gradient with respect tot he space variables x, i.e. Du(x, t) ≡

Dxu(x, t). The precise assumptions will be every time specified when presenting

the various theorems obtained in the paper; such assumptions will be in most of

the cases natural and optimal with respect to the implied conclusions. The notion

of (weak) solution adopted here, and in the rest of the paper, prescribes of course

that a map

u ∈

C0(−T,

0;

L2(Ω, RN

)) ∩

Lp(−T,

0; W

1,p(Ω, RN

))

is a (weak) solution to (0.1), under the assumption (0.2) for p ≥ 2, iff

ΩT

u · ϕt − a(x, t, u, Du),Dϕ dx dt = H, ϕ

holds for everyϕ ∈ C0

∞(ΩT

,

RN

).

We shall provide three basic types of interconnected regularity results for so-

lutions to (0.1), and in fact one of our aims is also to give a unifying approach to

different regularity issues, emphasizing the interactions between the various techni-

cal aspects of the problems treated. We shall give optimal partial regularity results

for solutions, the first results on the Hausdorff dimension size of the singular sets

ix