Nonlocal abstract Ginzburg–Landau-type equations and applications

We study a nonlocal abstract Ginzburg–Landau type equation. The equation includes variable coefficients with convolution terms and an abstract linear operator function $A$ in a Fourier-type Banach space $E$. For sufficiently smooth initial data, assuming growth conditions for the operator $A$ and the coefficient $a$, the existence and uniqueness of the solution and the $L^p$ -regularity properties are established. We obtain the existence and uniqueness of the solution, and the regularity of different classes of nonlocal Ginzburg–Landau-type equations by choosing the space $E$ and operator $A$ that occur in a wide variety of physical systems.