### Abstract

We study the Cauchy problem for the nonlinear Landau-Ginzburg equation { ∂_{t}u - αΔu + β|u|^{σ} = 0, x ∈ R^{n}, t > 0, u(0, x) = u_{0}(x), x ∈ R^{n}, where α, β ∈ C with dissipation condition ℜα > 0. We are interested in the subcritical case σ ∈ (0, 2/n). We assume that θ = | ∫ u_{0}(x)dx| ≠ 0 and ℜδ(α, β) > 0, where δ(α, β) = β|α|^{n - n/2σ}/((2 + σ)|α|^{2} + σα^{2})^{n/2}. Furthermore we suppose that the initial data u_{0} ∈ L^{1} are such that (1 + |x|)^{a}u_{0} ∈ L^{1}, with sufficiently small norm ε = ||(1 + |x|)^{a}u_{0}||_{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, ∞); L^{1}), 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 language | English |
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Pages (from-to) | 127-145 |

Number of pages | 19 |

Journal | Communications in Contemporary Mathematics |

Volume | 5 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2003 Feb |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Communications in Contemporary Mathematics*,

*5*(1), 127-145. https://doi.org/10.1142/S0219199703000872