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In this paper, the ultimate bound set and globally exponentially attractive set of a generalized Lorenz system are studied according to Lyapunov stability theory and optimization theory. The method of constructing Lyapunov-like functions applied to the former Lorenz-type systems (see, e.g. Lorenz system, Rossler system, Chua system) isn't applicable to this generalized Lorenz system. We overcome this difficulty by adding a cross term to the Lyapunov-like functions that used for the Lorenz system to study this generalized Lorenz system. The authors in [D. Li, J. Lu, X. Wu, G. Chen, Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system, Journal of Mathematical Analysis and Applications 323 (2006) 844-853] obtained the ultimate bound set of this generalized Lorenz system but only for some cases with $0 ≤ α < \frac{1}{{29}}.$ The ultimate bound set and globally exponential attractive set of this generalized Lorenz system are still unknown for $\alpha \notin \left[ {0, \frac{1}{{29}}} \right).$ Comparing with the best results in the current literature [D. Li, J. Lu, X. Wu, G. Chen, Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system, Journal of Mathematical Analysis and Applications 323 (2006) 844-853], our new results fill up the gap of the estimate for the case of $\frac{1}{{29}} ≤ α < \frac{{14}}{{173}}.$ Furthermore, the estimation derived here contains the results given in [D. Li, J. Lu, X. Wu, G. Chen, Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system, J. Math. Anal. Appl. 323 (2006) 844-853] as special case for the case of $0 ≤ α < \frac{1}{{29}}.$

It was showed that the generalized Camassa-Holm equation possible development of singularities in finite time, and beyond the occurrence of wave breaking which exists either global conservative or dissipative solutions. In present paper, we will further investigate the uniqueness of global conservative solutions to it based on the characteristics. From a given conservative solution $u = u(t,x)$, an equation is introduced to single out a unique characteristic curve through each initial point. By analyzing the evolution of the quantities $u$ and $v = 2 \arctan u_x$ along each characteristic, it is obtained that the Cauchy problem with general initial data $u_0∈ H^1(\mathbb{R})$ has a unique global conservative solution.

$\begin{eqnarray*}\label{1}\left\{\begin{array}{llll}u_t = Δ u-χ_{1}\nabla·(u\nabla v)+μ_{1}u(1-u-a_{1}w), &x∈ Ω, ~~~t>0, \\0 = Δ v-v+w, &x∈Ω, ~~~t>0, \\w_t = Δ w-χ_{2}\nabla·(w\nabla z)+μ_{2}w(1-a_{2}u-w), &x∈ Ω, ~~~ t>0, \\0 = Δ z-z+u, &x∈Ω, ~~~t>0, \\\end{array}\right.\end{eqnarray*}$ |

$Ω\subset R^n$ |

$n≥2$ |

$χ_{i}$ |

$μ_{i}$ |

$a_{i}$ |

$(i = 1, 2)$ |

$χ_{i}$ |

$μ_{i}$ |

$a_{i}$ |

$(i = 1, 2)$ |

$(u_{0}, w_{0})$ |

$\frac{χ_{i}}{μ_{i}}$ |

$a_{1}, a_{2}∈ (0, 1)$ |

$μ_{1}$ |

$μ_{2}$ |

$(u, v, w, z)$ |

$\left(\frac{1-a_{1}}{1-a_{1}a_{2}}, \frac{1-a_{2}}{1-a_{1}a_{2}}, \frac{1-a_{2}}{1-a_{1}a_{2}}, \frac{1-a_{1}}{1-a_{1}a_{2}}\right)$ |

$L^{∞}(Ω)$ |

$t\to ∞$ |

$a_{1}≥1>a_{2}>0$ |

$μ_{2}$ |

$\left(0, 1, 1, 0\right)$ |

$\left\{\begin{array}{llll}u_t=\Delta u-\chi_1\nabla\cdot( u\nabla w)+\mu_1u(1-u-a_1v),\quad &x\in \Omega,\quad t>0,\\v_t=\Delta v-\chi_2\nabla\cdot( v\nabla w)+\mu_2v(1-a_2u-v),\quad &x\in\Omega,\quad t>0,\\w_t=\Delta w- w+u+v,\quad &x\in\Omega,\quad t>0,\\\end{array}\right.$ |

$\begin{eqnarray*}\left\{\begin{array}{llll}n_t+u·\nabla n = \nabla·(D(n)\nabla n)-\nabla·(n \mathcal{S}(x, n, c)·\nabla c)\\ +ξ n-μ n^{2}, &x∈ Ω, &t>0, \\c_{t}+u·\nabla c = Δ c-c+n, &x∈Ω, &t>0, \\u_{t}+\nabla P = Δ u+n\nablaφ, &x∈Ω, &t>0, \\\nabla· u = 0, &x∈Ω, &t>0\end{array}\right.\end{eqnarray*}$ |

$ \begin{equation*}{\label{1.3}}\begin{split}|\mathcal{S}(x, n, c)|\leq C_{\mathcal{S}}(1+n)^{-α}\end{split}\end{equation*}$ |

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