Degenerate Kirchhoff-type diffusion problems involving the fractional p-Laplacian. (October 2017)

• Record Type: Journal Article
• Authors: Pan, Ning; Zhang, Binlin; Cao, Jun
• Is Part Of: Nonlinear analysis. Volume 37(2017)
• Page Start: 56
• Page End: 70
• Publication Date: 2017-10
• Subjects: Fractional p-Laplacian -- Parabolic equations -- Potential well -- Kirchhoff-type -- Galerkin solution; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2017.02.004
• Languages: English
• ISSNs: 1468-1218

Abstract

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.