TY - JOUR

T1 - Landau-Ginzburg type equations in the subcritical case

AU - Hayashi, Nakao

AU - Kaikina, Elena I.

AU - Naumkin, Pavel I.

N1 - Funding Information:
We are grateful to a referee for many useful suggestions and comments. One of the authors (N.H.) is partially supported by Grant-In-Aid for Scienti c Research (B) (no. 12440050), JSPS and two of the authors (E.I.K and P.I.N.) is partially supported by CONACYT.

PY - 2003/2

Y1 - 2003/2

N2 - 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.

AB - 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.

KW - Dissipative nonlinear evolution equation

KW - Landau-Ginzburg equation

KW - Large time asymptotics

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U2 - 10.1142/S0219199703000872

DO - 10.1142/S0219199703000872

M3 - Article

AN - SCOPUS:0037293663

SN - 0219-1997

VL - 5

SP - 127

EP - 145

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

IS - 1

ER -