Non-existence and instantaneous extinction of solutions for singular\n nonlinear fractional diffusion equations

2015/05/12 by Matteo Bonforte, Bonforte, Matteo, Antonio Segatti +2
Computer Science · Mathematics · #26A33 #35J60 #35J75 #35K55 #35K67 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1505.03167

openalex publication_date 2015/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show non-existence of solutions of the Cauchy problem in \ℝ^N\nfor the nonlinear parabolic equation involving fractional diffusion \∂_tν + (-\Δ)^s \φ(u)= 0, with 0<s<1 and very singular nonlinearities\n\φ . More precisely, we prove that when \φ(u)=-1/uⁿ with n>0, or\n\φ(u) = \log u, and we take nonnegative L¹ initial data, there is no\n(nonnegative) solution of the problem in any dimension N\≥ 2. We find the\nrange of non-existence when N=1 in terms of s and n. The range of\nexponents that we find for non-existence both for parabolic and elliptic\nequations are optimal. Non-existence is then proved for more general\nnonlinearities \φ, and it is also extended to the related elliptic problem\nof nonlinear nonlocal type: u + (-\Δ)^s \φ(u) = f with the same type of\nnonlinearity \φ.\n

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