Qualitative properties of fractional convolution elliptic and parabolic operators in Besov spaces

The maximal \(B_{p,q}^{s}\) $B_{p,q}^s$ -regularity properties of a fractional convolution elliptic equation is studied. Particularly, it is proven that the operator generated by this nonlocal elliptic equation is sectorial in \( B_{p,q}^{s}\) $B_{p,q}^s$ and also is a generator of an analytic semigroup. Moreover, well-posedeness of nonlocal fractional parabolic equation in Besov spaces is obtained. Then by using the \(B_{p,q}^{s}\) $B_{p,q}^s$ -regularity properties of linear problem, the existence, uniqueness of maximal regular solution of corresponding fractional nonlinear equation is established.