Gradient estimates for some f-heat equations driven by Lichnerowicz’s equation on complete smooth metric measure spaces

Dũng, Nguyễn Tiến; Khánh, Nguyễn Ngọc; Ngô, Quốc Anh

doi:10.1007/s00229-017-0946-3

Cited by 24 publications

(27 citation statements)

References 27 publications

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“…In the static case Liouville type results for the elliptic counterpart of Eq. 1.3 are established that include families of Einstein scalar-field and generalised Lichnerowiz type equations (see [9,11,13,18,30]). Our results here unifies, extends and improves various earlier results in the literature.…”

Section: Preliminariesmentioning

confidence: 99%

“…As primarily here we are concerned with positive solutions u, the nonlinearity = (u) is taken a smooth function, at least of class C 1 , defined on the half-axis u > 0. To showcase our results and to put the problems studied into context, we of course discuss various classes of nonlinearities with specific growth and singularities at infinity or zero, e.g., logarithimic-power type singularities or polynomial power-like growth or a combination of the two, as arising in many contexts and recent applications (see, e.g., [7,13,20,21,37,39,40] and the references therein). However, for the sake of the gradient estimates per se and many of the consequences, we keep the discussion general and do not make any specific structural assumptions on the nonlinearity .…”

Section: Introductionmentioning

confidence: 99%

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Gradient Estimates for a Weighted Γ-nonlinear Parabolic Equation Coupled with a Super Perelman-Ricci Flow and Implications

Taheri

2021

Potential Anal

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This article studies a nonlinear parabolic equation on a complete weighted manifold where the metric and potential evolve under a super Perelman-Ricci flow. It derives elliptic gradient estimates of local and global types for the positive solutions and exploits some of their implications notably to a general Liouville type theorem, parabolic Harnack inequalities and classes of Hamilton type dimension-free gradient estimates. Some examples and special cases are discussed for illustration.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Gradient Estimates for a Weighted Γ-nonlinear Parabolic Equation Coupled with a Super Perelman-Ricci Flow and Implications

Taheri

2021

Potential Anal

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“…In 2006, Souplet and Zhang [38] proved a local elliptic form by adding a logarithmic correction term. Recently, many authors extended the Li-Yau and Hamilton-Souplet-Zhang gradient estimates to the other heat-type equations; see for example [3,11,12,13,18,42,43,44,50,51] and references therein.…”

Section: Then There Does Not Exist a Minimizer Of The Weighted Yamabementioning

confidence: 99%

Gradient estimates for a nonlinear parabolic equation and Liouville theorems

2018

manuscripta math.

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We establish local elliptic and parabolic gradient estimates for positive smooth solutions to a nonlinear parabolic equation on a smooth metric measure space. As applications, we determine various conditions on the equation's coefficients and the growth of solutions that guarantee the nonexistence of nontrivial positive smooth solutions to many special cases of the nonlinear equation. In particular, we apply gradient estimates to discuss some Yamabe-type problems of complete Riemannian manifolds and smooth metric measure spaces.

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“…Let us recall that when s = 1, (1.1) and (1.2) are known as the Lichnerowicz equations (see, for example, [3,4,9]). Recently, this type of equation has been extensively studied (see, for example, [1,10,12] for elliptic problems and [5,14,15] for parabolic problems). In [10,12], it was proved that if p > 1 then equation (1.2) with s = 1 admits only the trivial positive solution u = 1.…”

Section: Introductionmentioning

confidence: 99%

Uniform Lower Bound and Liouville Type Theorem for Fractional Lichnerowicz Equations

Duong

Nguyen²,

Nguyen³

2021

Bull. Aust. Math. Soc.

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We study the fractional parabolic Lichnerowicz equation $$ \begin{align*} v_t+(-\Delta)^s v=v^{-p-2}-v^p \quad\mbox{in } \mathbb R^N\times\mathbb R \end{align*} $$ where $p>0$ and $ 0<s<1 $ . We establish a Liouville-type theorem for positive solutions in the case $p>1$ and give a uniform lower bound of positive solutions when $0<p\leq 1$ . In particular, when v is independent of the time variable, we obtain a similar result for the fractional elliptic Lichnerowicz equation $$ \begin{align*} (-\Delta)^s u=u^{-p-2}-u^p \quad\mbox{in }\mathbb R^N \end{align*} $$ with $p>0$ and $0<s<1$ . This extends the result of Brézis [‘Comments on two notes by L. Ma and X. Xu’, C. R. Math. Acad. Sci. Paris349(5–6) (2011), 269–271] to the fractional Laplacian.

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Gradient estimates for some f-heat equations driven by Lichnerowicz’s equation on complete smooth metric measure spaces

Cited by 24 publications

References 27 publications

Gradient Estimates for a Weighted Γ-nonlinear Parabolic Equation Coupled with a Super Perelman-Ricci Flow and Implications

Gradient Estimates for a Weighted Γ-nonlinear Parabolic Equation Coupled with a Super Perelman-Ricci Flow and Implications

Gradient estimates for a nonlinear parabolic equation and Liouville theorems

Uniform Lower Bound and Liouville Type Theorem for Fractional Lichnerowicz Equations

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