| Abstract: |
| In this talk, we will present some of the recent developments in the study of qualitative properties of
solutions to various fractional elliptic and parabolic equations
$$
{\cal L} u = f(t, u(x,t)),
$$
where ${\cal L}$ is a fractional elliptic or parabolic operator assuming one of the following forms
$$ (\Delta)^s, \;\; \partial_t + (-\Delta)^s, \;\; \partial_t^\alpha + (-\Delta)^s, \;\; (\partial_t -\Delta)^s.$$
We will illustrate the extent of non-locality of these operators and explain the differences among them. Then we will present some
of our recent results on qualitative properties of solutions including monotonicity, symmetry, uniqueness, nonexistence, and a priori estimates. |
|