Quantitative unique continuation for the heat equation with Coulomb potentials

Zhang, Can

doi:10.3934/mcrf.2018047

Cited by 7 publications

(9 citation statements)

References 30 publications

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“…Such a kind of interpolation inequality have been established for solutions of parabolic equations either in convex bounded domains or in bounded C 2 -smooth domains but with homogeneous Dirichlet boundary conditions; See for instance [5,23,24,25,26,31]. In these papers, the approach for the desired interpolation inequality is mainly based on the parabolic-type Almgren frequency function method, which is essentially adapted from [12,27].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Interpolation inequality at one time point for parabolic equations with time-independent coefficients and applications

Yu¹,

Zhang²

2017

Preprint

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In this paper, we study the Hölder-type interpolation inequality and observability inequality from measurable sets in time for parabolic equations either with L p unbounded potentials or with electric potentials. The parabolic equations under consideration evolve in bounded C 1,1 domains of R N (N ≥ 3) with homogeneous Neumann boundary conditions. The approach for the interpolation inequality is based on a modified reduction method and some stability estimates for the corresponding elliptic operator.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…This is coincident with the assumption (i) in (1.3). However, in view of the electric potential O(|x| −1 ), one could see that the assumption p > N is not optimal (see also [31]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Interpolation inequality at one time point for parabolic equations with time-independent coefficients and applications

Yu¹,

Zhang²

2017

Preprint

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“…Lemma A.4. ( [19] or [27]) Let r > 0, λ > 0, T > 0 and x 0 ∈ Ω n . Denote The following two properties hold:…”

Section: This Along With (A2)-(a4) Implies Thatmentioning

confidence: 99%

A uniform bound on costs of controlling semilinear heat equations on a sequence of increasing domains and its application

Wang

Zhang

2022

ESAIM: COCV

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In this paper, we first prove a uniform upper bound on costs of null controls for semilinear heat equations with globally Lipschitz nonlinearity on a sequence of increasing domains, where the controls are acted on an equidistributed set that spreads out in the whole Euclidean space R N . As an application, we then show the exact null-controllability for this semilinear heat equation in R N . The main novelty here is that the upper bound on costs of null controls for such kind of equations in large but bounded domains can be made uniformly with respect to the sizes of domains under consideration. The latter is crucial when one uses a suitable approximation argument to derive the global null-controllability for the semilinear heat equation in R N . This allows us to overcome the well-known problem of the lack of compactness embedding arising in the study of null-controllability for nonlinear PDEs in generally unbounded domains.

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“…In the Euclidean case, this inequality is called two parabolic-type Hardy inequality, which was obtained by Zhang [18].…”

Section: Hardy Type Inequalities With Exponential Weights On Gmentioning

confidence: 99%

Hardy and Rellich inequalities for anisotropic p-sub-Laplacians

Ruzhansky

Sabitbek

Сураган

2020

Banach J. Math. Anal.

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In this paper we establish the subelliptic Picone type identities. As consequences, we obtain Hardy and Rellich type inequalities for anisotropic p-sub-Laplacians which are operators of the formwhere X i , i = 1, . . . , N , are the generators of the first stratum of a stratified (Lie) group. Moreover, analogues of Hardy type inequalities with multiple singularities and many-particle Hardy type inequalities are obtained on stratified groups.

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Quantitative unique continuation for the heat equation with Coulomb potentials

Cited by 7 publications

References 30 publications

Interpolation inequality at one time point for parabolic equations with time-independent coefficients and applications

Interpolation inequality at one time point for parabolic equations with time-independent coefficients and applications

A uniform bound on costs of controlling semilinear heat equations on a sequence of increasing domains and its application

Hardy and Rellich inequalities for anisotropic p-sub-Laplacians

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