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, semipositive-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 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).