Nonglobal existence of solutions for a generalized Ginzburg-Landau equation coupled with a Poisson equation

摘要

In this article, we consider a system of a Ginzburg-Landau equation in u coupled with a Poisson equation in phi, partial derivativeu/partial derivativet = (1 + i alpha)[Deltau + u(2)u + k partial derivative phi/partial derivativex(1)u] + gammau, t is an element of (0,T), x is an element of R-n, -Delta phi = partial derivative/partial derivativex(1)(u(2)), x is an element of R-n, t is an element of (0,T), u(0,x) = u(0)(x), x = (x(1),x(2),...,x(n)) is an element of R-n, where T > 0, alpha, gamma, and k are real parameters, u and phi are, respectively, a complex and rear valued function. For alpha is an element of (-1,1) and k < 1, we prove the existence of initial data u(0) H-1(R-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. (C) 2001 Academic Press. [References: 17]