Mapping properties of heat kernels, maximal regularity and semi-linear parabolic equations on noncompact manifolds

Let L:C^∞ (M; E) → C^∞ (M; E) 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 f(√L). In particular, we establish results on the distribution kernels and mapping properties of e^-tL and (μ + L)^s. We show that L generates a holomorphic semigroup that has the usual mapping properties between the W^s,p-Sobolev spaces on M and E. We also prove that L satisfies maximal L^p- L^q-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).