TIME ANALYTICITY FOR NONLOCAL PARABOLIC EQUATIONS.

In this paper, we investigate pointwise time analyticity of solutions to nonlocal parabolic equations in the settings of ℝ^d and a complete Riemannian manifold M. On the one hand, in ℝ^d, we prove that any solution u=u(t,x) to ..., where ... is a nonlocal operator of order α, is time analytic in (0,1] if u satisfies the growth condition ... for any ... and ... . We also obtain pointwise estimates for ..., where ... is the fractional heat kernel. Furthermore, under the same growth condition, we show that the mild solution is the unique solution. On the other hand, in a manifold M, we also prove the time analyticity of the mild solution under the same growth condition and the time analyticity of the fractional heat kernel when M satisfies the Poincaré inequality and the volume doubling condition. Moreover, we also study the time and space derivatives of the fractional heat kernel in Rd using the method of Fourier transform and contour integrals. We find that when ..., the fractional heat kernel is time analytic at t=0 when x≠0, which differs from the standard heat kernel. As corollaries, we obtain a sharp solvability condition for the backward nonlocal parabolic equations and time analyticity of some nonlinear nonlocal parabolic equations with power nonlinearity of order p . These results are related to those in [9] and [22], which deal with local equations. [ABSTRACT FROM AUTHOR]