Ancient solutions to nonlocal parabolic equations

In this paper, we develop a systematical approach in applying the method of moving planes to study qualitative properties of solutions for nonlocal parabolic equations. We first establish a series of key ingredients in the proofs, such as maximum principles for antisymmetric functions, narrow region principles, maximum principles near infinity, and the Hopf lemma, then we demonstrate how these new tools can be employed to derive the radial symmetry and monotonicity of ancient positive solutions to fractional parabolic equations in the whole space. Our results are in particular applicable to entire solutions, by which we mean solutions defined for all t∈R. We believe that the new ideas and methods introduced here can be adapted to study many other nonlocal parabolic problems with more general operators and nonlinear terms.