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.