The Ginzburg-Landau Equation With Parameter ? With Nonhomogeneous Dirichlet Boundary Control Conditions

In this article, we study the weakly damped defocusing semilinear Ginzburg-Landau equations with a parameter ? perturbation on bounded domains at the H1-energy level with inhomogeneous Dirichlet control acting on a portion of the boundary. First, we introduce the dynamic extension method for homogenizing the inhomogeneous boundary input and construct approximate solutions using mono-tone operator theory to prove the existence of weak solutions. Then, using multipllier techniques we prove the exponential decay of solutions under the assumption that the boundary control also decays in a similar fashion. Finally, we prove that the solution is uniqueness.