2023/01/19 by Gabriele Grillo, Grillo, Gabriele, Kazuhiro Ishige +5 · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Caveolin-1 and cellular processes #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2301.07942
openalex publication_date 2023/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Stochastic incompleteness of a Riemannian manifold M amounts to the nonconservation of probability for the heat semigroup on M. We show that this property is equivalent to the existence of nonnegative, nontrivial, bounded (sub)solutions to ΔW=ψ(W) for one, hence all, general nonlinearity ψ which is only required to be continuous, nondecreasing, with ψ(0)=0 and ψ>0 in (0,+∞). Similar statements hold for unsigned (sub)solutions. We also prove that stochastic incompleteness is equivalent to the nonuniqueness of bounded solutions to the nonlinear parabolic equation ∂t u =Δϕ(u) with bounded initial data for one, hence all, general nonlinearity ϕ which is only required to be continuous, nondecreasing and nonconstant. Such a generality allows us to deal with equations of both fast-diffusion and porous-medium type, as well as with the one-phase and two-phase classical Stefan problems, which seem to have never been investigated in the manifold setting.