Solutions with snaking singularities for the fast diffusion equation

Фила, Марек; King, John R.; Takahashi, Jin; Yanagida, Eiji

doi:10.1090/tran/8479

Cited by 3 publications

References 29 publications

(24 reference statements)

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Solutions with moving singularities for a one-dimensional nonlinear diffusion equation

Fila,

Takahashi,

Yanagida

2024

Math. Ann.

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The aim of this paper is to study singular solutions for a one-dimensional nonlinear diffusion equation. Due to slow diffusion near singular points, there exists a solution with a singularity at a prescribed position depending on time. To study properties of such singular solutions, we define a minimal singular solution as a limit of a sequence of approximate solutions with large Dirichlet data. Applying the comparison principle and the intersection number argument, we discuss the existence and uniqueness of a singular solution for an initial-value problem, the profile near singular points and large-time behavior of solutions. We also give some results concerning the appearance of a burning core, convergence to traveling waves and the existence of an entire solution.

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Solutions with moving singularities for a one-dimensional nonlinear diffusion equation

Fila,

Takahashi,

Yanagida

2024

Math. Ann.

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The very singular solution for the Anisotropic Fast Diffusion Equation and its consequences

Vázquez

2024

Nonlinear Analysis

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Anisotropic and isotropic persistent singularities of solutions of the fast diffusion equation

Фила¹,

Macková²,

Takahashi³

et al. 2021

Preprint

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The aim of this paper is to study a class of positive solutions of the fast diffusion equation with specific persistent singular behavior. First, we construct new types of solutions with anisotropic singularities. Depending on parameters, these solutions either solve the original equation in the distributional sense, or they are not locally integrable in space-time. We show that the latter also holds for solutions with snaking singularities, whose existence has been proved recently by M. Fila, J.R. King, J. Takahashi, and E. Yanagida. Moreover, we establish that in the distributional sense, isotropic solutions whose existence was proved by M. Fila, J. Takahashi, and E. Yanagida in 2019, actually solve the corresponding problem with a moving Dirac source term. Last, we discuss the existence of solutions with anisotropic singularities in a critical case.

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Solutions with snaking singularities for the fast diffusion equation

Abstract: We construct solutions of the fast diffusion equation, which exist for all t ∈ R and are singular on the set Γ(t) := {ξ(s); −∞ < s ≤ ct}, c > 0, where ξ ∈ C 3 (R; R n ), n ≥ 2. We also give a precise description of the behavior of the solutions near Γ(t).

Cited by 3 publications

References 29 publications

Solutions with moving singularities for a one-dimensional nonlinear diffusion equation

Solutions with moving singularities for a one-dimensional nonlinear diffusion equation

The very singular solution for the Anisotropic Fast Diffusion Equation and its consequences

Anisotropic and isotropic persistent singularities of solutions of the fast diffusion equation

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