The time-periodic solution to a 2D generalized Ginzburg-Landau equation
This paper is devoted to study the 2D generalized Ginzburg-Landau equation \[ u_t=\rho u+(1+i\nu) \Delta u-(1+i\mu) |u|^{2\sigma} \cdot u+\alpha \lambda_1\cdot \nabla\bigl( |u|^2\cdot u\bigr)+ \beta( \lambda_2 \cdot\nabla u)|u|^2+f,\tag{1} \] where \(\Omega=[0,L] \times [0,L]\), \(\rho,\sigma,L\) are positive numbers, \(\alpha,\beta, \nu,\mu\) are real numbers, and \(\lambda_1,\lambda_2\) are real vectors. Assuming that the function \(f\) is \(\omega\)-periodic in time and \(u(x+T,t)= u(x,t)\), the authors prove both existence and uniqueness of the time periodic solution of (1). To this end, they use both the Galerkin method and Leray-Schauder fixed point arguments.
scientific article; zbMATH DE number 1763702