Degenerate Kirchhoff-type diffusion problems involving the fractional p-Laplacian. (October 2017) - Pan, Ning; Zhang, Binlin; Cao, Jun
Abstract: In this paper we study the existence of a global solution for a diffusion problem of Kirchhoff type driven by a nonlocal integro-differential operator. As a particular case, we consider the following parabolic equation involving the fractional p -Laplacian: ∂ t u + [ u ] s, p ( λ − 1 ) p ( − Δ ) p s u = | u | q − 2 u, in Ω × R +, ∂ t u = ∂ u / ∂ t, u ( x, 0 ) = u 0 ( x ), in Ω, u ( x, t ) = 0, in ( R N ∖ Ω ) × R 0 +, where [ u ] s, p is the Gagliardo p –seminorm of u, Ω ⊂ R N is a bounded domain with Lipschitz boundary ∂ Ω, p < q < N p / ( N − s p ) with 1 < p < N / s and s ∈ ( 0, 1 ), 1 ≤ λ < N / ( N − s p ), ( − Δ ) p s is the fractional p -Laplacian. Under some appropriate assumptions, we obtain the existence of a global solution for the problem above by the Galerkin method and potential well theory. It is worth pointing out that the main result covers the degenerate case, that is the coefficient of ( − Δ ) p s can vanish at zero.,