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

Anna L. Mazzucato, Victor Nistor

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

Abstract

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 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).

Original languageEnglish (US)
Pages (from-to)599-629
Number of pages31
JournalJournal of Hyperbolic Differential Equations
Volume3
Issue number4
DOIs
StatePublished - Dec 2006

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematics(all)

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