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

Abstract

We study the following Neumann inhomogeneous boundary value problem for the complex Ginzburg–Landau equation on Ω ⊂ R n ( n ⩽ 3 ) : u t = ( a + i α ) Δ u - ( b + i β ) | u | 2 u ( 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 H 1.