A Neumann boundary value problem for a generalized Ginzburg-Landau equation

We study the following generalized 1D Ginzburg-Landau equation on @W=(0,~)x(0,~):u"t=(1+i@m)u"x"x+(a"1+ib"1)|u|^2u"x+(a"2+ib"2)u^2u@?"x-(1+i@n)|u|^4u with initial and Neumann boundary conditions u(x,0)=h(x),u"x(0,t)=P(t). Under suitable conditions, we prove that there is a unique weak solution that exists for all time.

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