A simple approach to the Cauchy problem for complex Ginzburg–Landau equations by compactness methods

This paper is concerned with the Cauchy problem (CGL) in L2(RN) for complex Ginzburg–Landau equations with Laplacian Δ and nonlinear term |u|q−2u multiplied by the complex coefficients λ+iα and κ+iβ, respectively (q⩾2, λ>0, κ>0, α,β∈R). The global existence of strong solutions to (CGL) is established without any upper restriction on q⩾2 but with some restriction on α/λ and β/κ. The result corresponds to Ginibre and Velo (1996) [3, Proposition 5.1], which is technically proved by combining convolution (regularizing) methods with compactness (localizing) methods, while our proof here is fairly simplified. The key to our proof is the Cauchy problem (CGL)R which is (CGL) with Δ replaced with Δ−VR, where VR(x):=(|x|−R)2 (|x|>R), VR(x):=0 (|x|⩽R). The solvability of (CGL)R is a direct consequence of Okazawa and Yokota (2002) [16, Theorem 4.1], . Taking the limit of global strong solutions to (CGL)R as R→∞ yields a global strong solution to (CGL). The result gives also an unbounded version of Okazawa and Yokota (2002) [16, Theorem 1.1 with p=2] for the initial–boundary value problem on bounded domains.