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$\begin{equation*} (a+b[u]_{s,A}^{2θ-2})(-Δ)_A^su+V(x)u=f(x,|u|)u\,\, \text{in $\mathbb{R}^N$},\end{equation*}$ |

$s∈ (0,1)$ |

$N>2s$ |

$a∈ \mathbb{R}^+_0$ |

$b∈ \mathbb{R}^+_0$ |

$θ∈[1,N/(N-2s))$ |

$A:\mathbb{R}^N\to\mathbb{R}^N$ |

$V:\mathbb{R}^N\to \mathbb{R}^+$ |

$(-Δ )_A^s$ |

$p$ |

$\begin{align*}M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy\right)(-\Delta)_p^su = \lambda|u|^{p_s^*-2}u+f(x,u)~~\mbox{in }\,\,\mathbb{R}^N,\end{align*}$ |

$M(t) = a+bt^{\theta-1}$ |

$t\geq 0$ |

$a\geq 0, b>0,\theta>1$ |

$(-\Delta)_p^s$ |

$p$ |

$0<s<1$ |

$1<p<N/s$ |

$p_s^* = Np/(N-ps)$ |

$\lambda>0$ |

$f:\mathbb{R}^N\times\mathbb{R}\rightarrow\mathbb{R}$ |

$f$ |

$\lambda_0>0$ |

$\lambda<\lambda_0$ |

$k\in\mathbb{N}$ |

$\Lambda_k>0$ |

$k$ |

$\lambda<\Lambda_k$ |

$(-\Delta)_p^s$ |

$f$ |

In this paper, we study an anomalous diffusion model of Kirchhoff type driven by a nonlocal integro-differential operator. As a particular case, we are concerned with the following initial-boundary value problem involving the fractional $p$-Laplacian $\left\{ \begin{array}{*{35}{l}} {{\partial }_{t}}u+M([u]_{s, p}^{p}\text{)}(-\Delta)_{p}^{s}u=f(x, t) & \text{in }\Omega \times {{\mathbb{R}}^{+}}, {{\partial }_{t}}u=\partial u/\partial t, \\ u(x, 0)={{u}_{0}}(x) & \text{in }\Omega, \\ u=0\ & \text{in }{{\mathbb{R}}^{N}}\backslash \Omega, \\\end{array}\text{ }\ \ \right.$ where $[u]_{s, p}$ is the Gagliardo $p$-seminorm of $u$, $Ω\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary $\partialΩ$, $1 < p < N/s$, with $0 < s < 1$, the main Kirchhoff function $M:\mathbb{R}^{ + }_{0} \to \mathbb{R}^{ + }$ is a continuous and nondecreasing function, $(-Δ)_p^s$ is the fractional $p$-Laplacian, $u_0$ is in $L^2(Ω)$ and $f∈ L^2_{\rm loc}(\mathbb{R}^{ + }_0;L^2(Ω))$. Under some appropriate conditions, the well-posedness of solutions for the problem above is studied by employing the sub-differential approach. Finally, the large-time behavior and extinction of solutions are also investigated.

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