Landau-Ginzburg type equations in the subcritical case

Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin

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17 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)127-145
Number of pages19
JournalCommunications in Contemporary Mathematics
Issue number1
Publication statusPublished - 2003 Feb


  • Dissipative nonlinear evolution equation
  • Landau-Ginzburg equation
  • Large time asymptotics


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