A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian

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 $\{ {array}{*{35}{l}} {{∂ }ₜ}u+M([u]s, pᵖ)(-Δ)ₚˢu=f(x, t) & in Ω × {{R}⁺}, {{∂ }ₜ}u=∂ u/∂ t, \\ u(x, 0)={{u}₀}(x) & in Ω, \\ u=0\ & in {{R}N} Ω, \\{array} \ \ .$ where $[u]s, p$ is the Gagliardo $p$-seminorm of $u$, $Ω⊂ R^N$ is a bounded domain with Lipschitz boundary $∂Ω$, $1 < p < N/s$, with $0 < s < 1$, the main Kirchhoff function $M:R⁺₀ → R⁺$ is a continuous and nondecreasing function, $(-Δ)_p^s$ is the fractional $p$-Laplacian, $u_0$ is in $L^2(Ω)$ and $f∈ L^2loc(R⁺_0;L^2(Ω))$. 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.