TY - JOUR

T1 - Mapping properties of heat kernels, maximal regularity and semi-linear parabolic equations on noncompact manifolds

AU - Mazzucato, Anna L.

AU - Nistor, Victor

N1 - Funding Information:
A. Mazzucato was partially supported by NSF grant DMS-0405803. V. Nistov was partially supported by NSF grant DMS-0200808.

PY - 2006/12

Y1 - 2006/12

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=34748883072&partnerID=8YFLogxK

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U2 - 10.1142/S0219891606000938

DO - 10.1142/S0219891606000938

M3 - Article

AN - SCOPUS:34748883072

VL - 3

SP - 599

EP - 629

JO - Journal of Hyperbolic Differential Equations

JF - Journal of Hyperbolic Differential Equations

SN - 0219-8916

IS - 4

ER -