Neumann inhomogeneous boundary value problem for the n + 1 complex Ginzburg–Landau equation

We study the following Neumann inhomogeneous boundary value problem for the complex Ginzburg–Landau equation on Ω⊂Rn(n⩽3):ut=(a+iα)Δu-(b+iβ)|u|2u(a,b,t>0)Ω⊂Rn(n⩽3):ut=(a+iα)Δu-(b+iβ)|u|2u(a,b,t>0) under initial condition u(x, 0) = h(x) for x ∈ Ω   and Neumann boundary condition ∂u∂n=K(x,t) on ∂Ω where h, K are given functions. Under suitable conditions, we prove the existence of a global solution in H1.