Extremals for sharp GNS inequalities on compact manifolds

Abreu, Emerson; Ceccon, Jurandir; Montenegro, Marcos

doi:10.1007/s10231-014-0426-2

Cited by 3 publications

(2 citation statements)

References 30 publications

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“…We shall mention some examples below without trying to give an exhaustive list. Sharp constants in GNI in R n were studied in [2,3,4,13,15,16,17,18,28,29,30,41] and in GNI on Riemannian manifolds in [1,10,11]. The sharp constant in an anisotropic GNI with fractional derivatives was found in [21].…”

Section: Introductionmentioning

confidence: 99%

A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation

Фила

Winkler

2019

Advances in Mathematics

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We establish a Gagliardo-Nirenberg-type inequality in R n for functions which decay fast as |x| → ∞. We use this inequality to derive upper bounds for the decay rates of solutions of a degenerate parabolic equation. Moreover, we show that these upper bounds, hence also the Gagliardo-Nirenberg-type inequality, are sharp in an appropriate sense.Theorem 1.2 If p ≥ 1 and u 0 ∈ q 0 >0 L q 0 (R n ), then for any δ > 0 one can find C(δ) > 0 such that for the minimal solution u of (1.6) we have u(·, t) L ∞ (R n ) ≤ C(δ)t − 1 p +δ for all t > 0.(1.8)

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Section: Introductionmentioning

confidence: 99%

A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation

Фила

Winkler

2019

Advances in Mathematics

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“…[5], or, to pick some examples from Geneviève Raugel's work: [26, (2.21)], [27, (6.14), (8.15)], [15, proof of (2.32)]). Various extensions of the classical GNI are available, for instance: It has been studied in Besov spaces ( [21,22]), on Riemannian manifolds ( [4,6]), with a BMO term ( [8,28,20,29,23]), in Orlicz spaces ( [17,18,16]), for noninteger derivatives ( [24]), with weighted ( [10]) and anisotropic ( [11]) terms, and extremal functions and optimal constants have been determined ( [30,9,3,1,19]). For a relation with mass transport theory see [7,2].…”

Section: Introductionmentioning

confidence: 99%

A Gagliardo–Nirenberg Type Inequality for Rapidly Decaying Functions

Фила

Lankeit

2020

J Dyn Diff Equat

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We improve the Gagliardo-Nirenberg inequalityFila and M. Winkler: A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation, Adv. Math., 357 (2019), https://doi.org/10.1016/j.aim.2019.106823] for rapidly decaying functions (ϕ ∈ W 1,r (R n ) \ {0} with finite K = R n L(ϕ)) by specifying the dependence of C on K and by allowing arbitrary r ≥ 1. Mathematics Subject Classification (MSC 2010): 35A23; 26D10

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On nonlinear damped wave equations for positive operators. I. Discrete spectrum

Ruzhansky¹,

Tokmagambetov²

2019

Differential Integral Equations

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In this paper we study a Cauchy problem for the nonlinear damped wave equations for a general positive operator with discrete spectrum. We derive the exponential in time decay of solutions to the linear problem with decay rate depending on the interplay between the bottom of the operator's spectrum and the mass term. Consequently, we prove global in time well-posedness results for semilinear and for more general nonlinear equations with small data. Examples are given for nonlinear damped wave equations for the harmonic oscillator, for the twisted Laplacian (Landau Hamiltonian), and for the Laplacians on compact manifolds.

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Extremals for sharp GNS inequalities on compact manifolds

Cited by 3 publications

References 30 publications

A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation

A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation

A Gagliardo–Nirenberg Type Inequality for Rapidly Decaying Functions

On nonlinear damped wave equations for positive operators. I. Discrete spectrum

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