MAPPING PROPERTIES OF HEAT KERNELS, MAXIMAL REGULARITY, AND SEMI-LINEAR PARABOLIC EQUATIONS ON NONCOMPACT MANIFOLDS

Abstract

Let [Formula: see text] be a second order, uniformly elliptic, positive semi-definite differential operator on a complete Riemannian manifold of bounded geometry M, acting between sections of a vector bundle with bounded geometry E over M. We assume that the coefficients of L are uniformly bounded. Using finite speed of propagation for L, we investigate properties of operators of the form [Formula: see text]. In particular, we establish results on the distribution kernels and mapping properties of e-tLand (μ + L)s. We show that L generates a holomorphic semigroup that has the usual mapping properties between the Ws,p-Sobolev spaces on M and E. We also prove that L satisfies maximal Lp–Lq-regularity for 1 < p, q < ∞. We apply these results to study parabolic systems of semi-linear equations of the form ∂tu + Lu = F(t, x, u, ∇ u).