In the present chapter we establish some fundamental properties for a general family of parabolic equations.

Let Y be a subset of L∞(R × D,RN2+2N+1) × L∞(R × ∂D,R) satisfying (A1-1)-(A1-3) (see in Section 1.3). We may write a = (aij , ai, bi, c0, d0) for a = ((aij)Ni,j=1, (ai)

N i=1, (bi)

N i=1, c0, d0) ∈ Y if no confusion occurs. For a

given a = (aij , ai, bi, c0, d0), we may assume that aij(t, x), ai(t, x), bi(t, x), and c0(t, x) are defined and bounded for all (t, x) ∈ R × D, and d0(t, x) is defined and bounded for all (t, x) ∈ R× ∂D.