Existence and decay estimates of solutions to complex Ginzburg–Landau type equations

This paper deals with the initial-boundary value problem (denoted by (CGL)) for the complex Ginzburg–Landau type equation ∂u∂t−(λ+iα)Δu+(κ+iβ)|u|q−1u−γu=0 with initial data u0∈Lp(Ω)u0∈Lp(Ω) in the case 10λ>0, α,β,γ,κ∈Rα,β,γ,κ∈R. There are a lot of studies on local and global existence of solutions to (CGL) including the physically relevant case q=3q=3 and κ>0κ>0. This paper gives existence results with precise properties of solutions and rigorous proof from a mathematical point of view. The physically relevant case can be considered as a special case of the results. Moreover, in the case κ<0κ<0, local and global existence of solutions with the decay estimate ‖u(t)‖Lp(Ω)≤c1e−c2t‖u(t)‖Lp(Ω)≤c1e−c2t (c1,c2c1,c2 are positive constants) is obtained under some conditions. The key to the local existence is to construct a semigroup {et[(λ+iα)Δ]}{et[(λ+iα)Δ]} and its LpLp-LqLq estimate. On the other hand, the key to the global existence is to derive estimates for solutions by using a kind of interpolation inequality with Re〈|v|p−2v,−(λ+iα)Δv〉.