Solutions with moving singularities for equations of porous medium type

Фила, Марек; Takahashi, Jin; Yanagida, Eiji

doi:10.1016/j.na.2018.08.016

Cited by 10 publications

(16 citation statements)

References 15 publications

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“…In this paper, we consider the case where both the position and strength of the singularity depend on time. See [7,14,20,22] for related works on such dynamic singularities.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the evolution equation with a dynamic Hardy-type potential

et al. 2021

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Motivated by the celebrated paper of Baras and Goldstein (1984), we study the heat equation with a dynamic Hardy-type singular potential. In particular, we are interested in the case where the singular point moves in time. Under appropriate conditions on the potential and initial value, we show the existence, non-existence and uniqueness of solutions, and obtain a sharp lower and upper bound near the singular point. Proofs are given by using solutions of the radial heat equation, some precise estimates for an equivalent integral equation and the comparison principle.

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“…In this paper, we consider the case where both the position and strength of the singularity depend on time. See [7,14,20,22] for related works on such dynamic singularities.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the evolution equation with a dynamic Hardy-type potential

et al. 2021

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“…The snaking solutions are the simplest representatives of a much more general class, that is in turn illustrative of very wide-ranging issues of non-uniqueness in the equation of fast diffusion (cf. [6] and references therein), whereby the head of the snake can be specified to take any path leaving in its wake a line singularity.…”

Section: Discussionmentioning

confidence: 99%

“…It was shown in [6] that for m > (n − 2)/(n − 1) and n ≥ 3, under some assumptions on given functions θ, k, u 0 , there are solutions of (1.3)-(1.5) which behave as k(t)|x−θ(t)| −(n−2)/m near x = θ(t). This corresponds to the singularity of the steady stateũ.…”

Section: Setmentioning

confidence: 99%

Solutions with snaking singularities for the fast diffusion equation

Фила

King

Takahashi

et al. 2021

Trans. Amer. Math. Soc.

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“…As χ ε → 1 a.e. in supp ϕ and by (8) and boundedness of u each of the functions ∇u · ∇ϕ, uu 3 r ϕ, ϕ t u belongs to L 1 (supp ϕ),…”

Section: Relation Between Classical and Weak Solutionsmentioning

confidence: 99%

“…solutions with standing singularities were considered in [4,5,6,10,16,23] for various ranges of m > 0, m = 1, and some results on moving singularities for the same equation can be found in [8].…”

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confidence: 99%