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### Open Access Journals

DCDS-S

In this paper, we investigate the bifurcations of nonlinear waves described by the Gardner equation $u_{t}+a u u_{x}+b u^{2} u_{x}+\gamma u_{xxx}=0$. We obtain some new results as follows: For arbitrary given parameters $b$ and $\gamma$, we choose the parameter $a$ as bifurcation parameter. Through the phase analysis and explicit expressions of some nonlinear waves, we reveal two kinds of important bifurcation phenomena. The first phenomenon
is that the solitary waves with fractional expressions can be bifurcated from three types of nonlinear waves which are solitary waves with hyperbolic expression and two types of periodic waves with elliptic expression and trigonometric expression respectively. The second phenomenon is that the kink waves can be bifurcated from the solitary waves and the singular waves.

DCDS

In this paper, motivated by [13], we use the Littlewood-Paley theory to investigate the Cauchy problem of the Boltzmann equation. When the initial data is a small perturbation of an equilibrium state, under the Grad's angular cutoff assumption, we obtain the unique global strong solution to the Boltzmann equation for the hard potential case in the Chemin-Lerner type spaces $C([0,\infty);\widetilde{L}^{2}_{\xi}(B_{2,r}^{s}))$ with $1\leq r\leq2$ and $s>3/2$ or $s=3/2$ and $r=1$.
Besides, we also prove the Lipschitz continuity of the solution map. Our results extend some previous works on the Boltzmann equation in Sobolev spaces.

CPAA

In this paper, we apply the Mountain Pass Lemma of
Ambrosetti-Rabinowitz [2] to study the existence of new
periodic solutions with a prescribed energy for a class of second
order Hamiltonian conservative systems.

KRM

Boundedness and large time behavior of an attraction-repulsion chemotaxis model with logistic source

In this paper, we study an attraction-repulsion Keller-Segel chemotaxis model with logistic source

$\begin{cases} u_{t}=Δ u-χ\nabla·(u\nabla v)+ξ\nabla·(u\nabla w)+f(u), &x∈Ω,\ t>0,\\ v_{t}=Δ v+α u-β v, &x∈Ω,\ t>0,\\ w_{t}=Δ w+γ u-δ w, &x∈Ω,\ t>0,\\\end{cases}\;\;\;\;(*)$ |

in a smooth bounded domain

, with homogeneous Neumann boundary conditions and nonnegative initial data

satisfying suitable regularity, where

and

is a smooth growth source satisfying

and

$Ω \subset \mathbb{R}^n(n≥ 1)$ |

$(u_0,v_0,w_0)$ |

$χ≥ 0,ξ≥ 0,α, β, γ, δ>0$ |

$f$ |

$f(0)≥ 0$ |

$f(u)≤ a-bu^θ, \ \ u≥ 0,\ \ \mathrm{with~some} \ \ a≥ 0,b>0,θ≥1.$ |

When

(i.e. repulsion cancels attraction), the boundedness of classical solution of system (*) is established if the dampening parameter

and the space dimension

satisfy

$χα=ξγ$ |

$θ$ |

$n$ |

$\begin{cases} θ > \max\{1,3-\frac6n\}, &\text{when }\ \ 1≤ n≤ 5,\\ θ≥ 2, &\text{when }\ \ 6≤ n≤ 9,\\ θ>1+\frac{2(n-4)}{n+2}, &\text{when} \ \ \ n≥10.\\\end{cases}$ |

Furthermore, when

and repulsion cancels attraction, by constructing appropriate Lyapunov functional, we show that if

, the solution

exponentially stabilizes to the constant stationary solution

in the case of

. Our results implies that when repulsion cancels attraction the logistic source play an important role on the solution behavior of the attraction-repulsion chemotaxis system.

$f(u)=μ u(1-u)$ |

$μ>\frac{χ^2α^2(β-δ)^2}{8δβ^2}$ |

$(u,v,w)$ |

$(1,\frac{α}{β},\frac{γ}{δ})$ |

$1≤ n≤ 9$ |

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