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### Open Access Journals

DCDS

In this paper we mainly study the Cauchy problem for a generalized Camassa-Holm equation. First, by using the Littlewood-Paley decomposition and transport equations theory, we establish the local well-posedness for the Cauchy problem of the equation in Besov spaces. Then we give a blow-up criterion for the Cauchy problem of the equation. we present a blow-up result and the exact blow-up rate of strong solutions to the equation by making use of the conservation law and the obtained blow-up criterion. Finally, we verify that the system possesses peakon solutions.

DCDS

In this paper, we consider a chemotaxis-competition system of parabolic-elliptic-parabolic-elliptic type

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

with homogeneous Neumann boundary conditions in an arbitrary smooth bounded domain

,

, where

,

and

are positive constants. It is shown that for any positive parameters

,

,

and any suitably regular initial data

, this system possesses a global bounded classical solution provided that

are small. Moreover, when

and the parameters

and

are sufficiently large, it is proved that the global solution

of this system exponentially approaches to the steady state

in the norm of

as

; If

and

is sufficiently large, the solution of the system converges to the constant stationary solution

as time tends to infinity, and the convergence rates can be calculated accurately.

$Ω\subset R^n$ |

$n≥2$ |

$χ_{i}$ |

$μ_{i}$ |

$a_{i}$ |

$(i = 1, 2)$ |

$χ_{i}$ |

$μ_{i}$ |

$a_{i}$ |

$(i = 1, 2)$ |

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

$\frac{χ_{i}}{μ_{i}}$ |

$a_{1}, a_{2}∈ (0, 1)$ |

$μ_{1}$ |

$μ_{2}$ |

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

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

$L^{∞}(Ω)$ |

$t\to ∞$ |

$a_{1}≥1>a_{2}>0$ |

$μ_{2}$ |

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

keywords:
Chemotaxis
,
Lotka-Volterra-type competition
,
boundedness
,
Logistic source
,
asymptotic stability

## Year of publication

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