Nonglobal Existence of Solutions for a Generalized Ginzburg Landau Equation Coupled with a Poisson Equation

In this article, we consider a system of a Ginzburg Landau equation in u coupled with a Poisson equation in , u Ž1 i . u u 2u k u u, t Ž0, T., x n , t x1 Ž u 2., x n , t Ž0, T., x1 uŽ0, x. u0Žx., x Žx1, x2 , . . . , xn. n , where T 0, , , and k are real parameters, u and are, respectively, a complex and real valued function. For Ž 1, 1.and k 1, we prove the existence of initial data u0 H1Ž n., n 1 or 2, for which the solution u is nonglobal. Our method uses energy arguments. We establish differential inequalities having only nonglobal solutions