Landau-Ginzburg type equations in the subcritical case

We study the Cauchy problem for the nonlinear Landau-Ginzburg equation { ∂_tu - αΔu + β|u|^σ = 0, x ∈ Rⁿ, t > 0, u(0, x) = u₀(x), x ∈ Rⁿ, where α, β ∈ C with dissipation condition ℜα > 0. We are interested in the subcritical case σ ∈ (0, 2/n). We assume that θ = | ∫ u₀(x)dx| ≠ 0 and ℜδ(α, β) > 0, where δ(α, β) = β|α|^{n - n/2σ}/((2 + σ)|α|² + σα²)^n/2. Furthermore we suppose that the initial data u₀ ∈ L¹ are such that (1 + |x|)^au₀ ∈ L¹, with sufficiently small norm ε = ||(1 + |x|)^au₀||₁, where a ∈ (0, 1). Also we assume that σ is sufficiently close to 2/n. Then there exists a unique solution of the Cauchy problem (*) such that u(t, x) ∈ C((0, ∞); L^∞) ∩ C([0, ∞); L¹), satisfying the following time decay estimates for large t > 0 ||u(t)||_∞ ≤ Cε〈t〉^1/σ. Note that in comparison with the corresponding linear case the decay rate of the solutions of (*) is more rapid.