On the Green’s Matrices of Strongly Parabolic Systems of Second Order

lim t→s+ Γij(t, x, s, ·) = δijδx(·), Γij(·, ·, s, y) = 0 on ∂U = R × ∂Ω, where δij is the Kronecker delta symbol, δ(s,y)(·, ·) and δx(·) are Dirac delta functions. In particular, when U = R, the Green’s matrix (or Green’s function) is called the fundamental matrix (or fundamental solution). We prove that if weak solutions of (P) satisfy an interior Hölder continuity estimate (see Section 2.5 for the precise formulation), then the Green’s matrix exists in U and satisfies several natural growth properties; see Theorem 2.7. Moreover, when U = R, we show that the fundamental matrix satisfies the semi-group property as well as the upper Gaussian bound of Aronson [1]; see Theorem 2.11. The method we use does not involve Harnack type inequalities or the maximum principle, and works for both the scalar and the vectorial situation. Moreover, we