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 + i α ) Δ u + ( κ + i β ) | u | q − 1 u − γ u = 0 with initial data u 0 ∈ L p ( Ω ) in the case 1 q 1 + 2 p / N , where Ω is bounded or unbounded in R N , λ > 0 , α , 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 = 3 and κ > 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 0 , local and global existence of solutions with the decay estimate ‖ u ( t ) ‖ L p ( Ω ) ≤ c 1 e − c 2 t ( c 1 , c 2 are positive constants) is obtained under some conditions. The key to the local existence is to construct a semigroup { e t [ ( λ + i α ) Δ ] } and its L p - L q 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 − 2 v , − ( λ + i α ) Δ v 〉 .