Well-posedness of nonlocal Ginzburg–Landau type equations

The Cauchy problem for linear and nonlinear Ginzburg–Landau type equations is studied. The equation includes variable coefficients with convolution terms. Assuming enough smoothness of the initial data and certain growth conditions on coefficients, the existence, uniqueness of local and global solution, \(L^{p}\) $L^p$ -regularity, and blow-up properties of solutions are established.