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

ANNA L. MAZZUCATO

Mathematics Department, Pennsylvania State University, University Park, PA 16802, USA

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and

VICTOR NISTOR

Mathematics Department, Pennsylvania State University, University Park, PA 16802, USA

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https://doi.org/10.1142/S0219891606000938Cited by:12 (Source: Crossref)

Abstract

Let 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 . 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).

Keywords:

AMSC: 35K55, 35L10, 58J35, 58J45

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