Nonlocal separable elliptic and parabolic equations and applications

The regularity properties of nonlocal anisotropic elliptic equations with parameters are investigated in abstract weighted $L_{p}$ spaces. The equations include the variable coefficients and abstract operator function $ A=A\left( x\right) $ in a Banach space $E$ in leading part. We find the sufficient growth assumptions on $A$ and appropriate symbol polynomial functions that guarantee the uniformly separability of the linear problem. It is proved that the corresponding anisotropic elliptic operator is sectorial and is also the negative generator of an analytic semigroup. By using these results, the existence and uniqueness of maximal regular solution of the nonlinear nonlocal anisotropic elliptic equation is obtained in weighted $ L_{p}$ spaces. In application, the maximal regularity properties of the Cauchy problem for degenerate abstract anisotropic parabolic equation in mixed $L_{\mathbf{p}}$ norms, the boundary value problem for anisotropic elliptic convolution equation, the Wentzel–Robin type boundary value problem for degenerate integro-differential equation and infinite systems of degenerate elliptic integro-differential equations are obtained.