Existence and stabilization results for a singular parabolic equation involving the fractional Laplacian

2017/09/06 by Jacques Giacomoni, Giacomoni, J., Tuhina Mukherjee +3
Computer Science · Mathematics · #35A15 #35R09 #35R11 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1709.01906

openalex publication_date 2017/09/06 · openalex created_date 2017/09/15 · openalex updated_date 2026/07/28

Abstract

In this article, we study the following parabolic equation involving the fractional Laplacian with singular nonlinearity (P_t^s) \ \beginsplit u_t + (-Δ)^s u amp;= u^-q + f(x,u), u gt;0 in (0,T) × Ω, u amp;= 0 in (0,T) × (\mb Rⁿ ∖Ω), u(0,x)amp;=u₀(x) in \mb Rⁿ, \endsplit . where Ω is a bounded domain in \mbRⁿ with smooth boundary ∂ Ω, n> 2s, s ∈ (0,1), q>0, q(2s-1)0. We suppose that the map (x,y)∈ Ω× \mb R⁺ ↦ f(x,y) is a bounded below Carathéodary function, locally Lipschitz with respect to second variable and uniformly for x ∈ Ω it satisfies \beginequation \limsup_{y → +∞} (f(x,y))/(y)

Citations

• The Brezis-Nirenberg result for the fractional Laplacian
• Regularity of the obstacle problem for a fractional power of the laplace operator

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