THE COMPLEX GINZBURG-LANDAU EQUATION

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We have considered initial value problem for abstract evolution equations of the formdu/dt + (λ+ iα)∂φ(u) + (κ + iβ)∂ψ(u) - γu = 0, t>0 ; u(0) =u_0,where ∂φ, ∂ψ are subdifferentials of convex functions φ, ψ on a Hilbert space X.Let Ω ⊂ R^N be a bounded domain. Setting X : = L^2(Ω), φ (u) : = (1/p)‖∇u‖^p_<L^p>, u ∈ W^<1, p>_0(Ω) ; ψ(v) : =(1/q)‖v‖^q_<L^q>, v ∈ L^q(Ω), we have ∂ψ(u) = |u|^<q-2>u, u ∈ D(∂ψ) : = L^<2(q-1)>(Ω), ∂φ(u) = -Δ_pu : =-div(|∇u|^<p-2>∇u), u ∈ D(∂φ) : = {u ∈ W^<1, p>_0(Ω) ; Δ_pu∈ L^2(Ω)} and hence(CGL)_p ∂u/∂t - (λ+iα)・Δ_pu + (κ+iβ)|u|^<q-2>u - γu = 0, t>0 ; u(O) : u_0.Here we have assumed that p 【greater than or equal】 2, q 【greater than or equal】 2. We have revealed that the global in time solvability of (CGL)_p is classified as follows according to the pair (α/λ, β/κ) ∈ R^2 and the initial value. In any case we assume that |α|/λ 【less than or equal】 1/c_p : = 2√<p-1>/(p-2).1. Existence of weak solutions for any (α/λ, β/κ), u_0 ∈ L^2(Ω)(no uniqueness, in general)2. Existence of strong solutions for (α/λ, β/κ) belonging to "(CGL) region" : = {(x, y) ∈ R^2 ; xy 【greater than or equal】 0 or |xy| - 1 < (|x| + |y|)/c_q} and u_0 ∈ W^<1, p>_0(Ω) ∩ L^q(Ω) (no uniqueness, in general)3. Unique existence of strong solutions for (α/λ, β/κ) with |β|/κ 【less than or equal】 1/c_q and u_0 ∈ L^2(Ω) (smoothing effect of solution operators).