scholarly journals Observability estimate for the parabolic equations with inverse square potential

2021 ◽  
Vol 6 (12) ◽  
pp. 13525-13532
Author(s):  
Guojie Zheng ◽  
◽  
Baolin Ma ◽  
◽  
Keyword(s):  
Bounded Domain ◽  
Open Subset ◽  
Parabolic Equations ◽  
Positive Measure ◽  
Measure Theory ◽  
Observability Estimate

<abstract><p>This paper investigates an observability estimate for the parabolic equations with inverse square potential in a $ C^2 $ bounded domain $ \Omega\subset\mathbb{R}^d $, which contains $ 0 $. The observation region is a product set of a subset $ E\subset(0, T] $ with positive measure and a non-empty open subset $ \omega\subset\Omega $ with $ 0\notin\omega $. We build up this estimate by a delicate result in measure theory in <sup>[<xref ref-type="bibr" rid="b7">7</xref>]</sup> and the Lebeau-Robbiano strategy.</p></abstract>

scholarly journals Observability Estimate for the Fractional Order Parabolic Equations on Measurable Sets

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Guojie Zheng ◽  
M. Montaz Ali
Keyword(s):  
Parabolic Equation ◽  
Bounded Domain ◽  
Fractional Order ◽  
Parabolic Equations ◽  
Positive Measure ◽  
Measure Theory ◽  
Order Parabolic Equation ◽  
Observability Estimate

We establish an observability estimate for the fractional order parabolic equations evolved in a bounded domainΩofℝn. The observation region isF×ω, whereωandFare measurable subsets ofΩand (0,T), respectively, with positive measure. This inequality is equivalent to the null controllable property for a linear controlled fractional order parabolic equation. The building of this estimate is based on the Lebeau-Robbiano strategy and a delicate result in measure theory provided in Phung and Wang (2013).

Interior h1 Estimates for Parabolic Equations with LMO Coefficients

2010 ◽  
Vol 62 (1) ◽  
pp. 202-217
Author(s):  
Lin Tang
Keyword(s):  
Bounded Domain ◽  
Parabolic Equations ◽  
A Priori

AbstractIn this paper we establish a priori h1-estimates in a bounded domain for parabolic equations with vanishing LMO coefficients.

Global blow-up of a nonlinear heat equation

1986 ◽  
Vol 104 (1-2) ◽  
pp. 161-167 ◽  
Author(s):  
A. A. Lacey
Keyword(s):  
Heat Equation ◽  
Parabolic Equations ◽  
Finite Time ◽  
Positive Measure ◽  
Blow Up ◽  
Nonlinear Heat Equation ◽  
Semilinear Parabolic Equations ◽  
Semilinear Parabolic

SynopsisSolutions to semilinear parabolic equations of the form ut = Δu + f(u), x in Ω, which blow up at some finite time t* are investigated for “slowly growing” functions f. For nonlinearities such as f(s) = (2 +s)(ln(2 +s))1+b with 0 < b < l,u becomes infinite throughout Ω as t→t* −. It is alsofound that for marginally more quickly growing functions, e.g. f(s) = (2 + s)(ln(2 +s))2, u is unbounded on some subset of Ω which has positive measure, and is unbounded throughout Ω if Ω is a small enough region.

Feynman formula for a class of second-order parabolic equations in a bounded domain

2008 ◽  
Vol 78 (1) ◽  
pp. 590-595 ◽  
Author(s):  
Ya. A. Butko ◽  
M. Grothaus ◽  
O. G. Smolyanov
Keyword(s):  
Bounded Domain ◽  
Parabolic Equations ◽  
Second Order ◽  
Feynman Formula

On the uniqueness of inverse problems for the reduced wave equation with unknown embedded obstacles

2021 ◽  
Author(s):  
Jiaqing Yang ◽  
Meng Ding ◽  
Keji Liu
Keyword(s):  
Wave Equation ◽  
Inverse Problems ◽  
Bounded Domain ◽  
Open Subset ◽  
Wave Equations ◽  
A Priori ◽  
Singularity Analysis ◽  
Piecewise Constant ◽  
Small Domain ◽  
Dirichlet To Neumann Map

Abstract In this paper, we consider inverse problems associated with the reduced wave equation on a bounded domain Ω belongs to R^N (N >= 2) for the case where unknown obstacles are embedded in the domain Ω. We show that, if both the leading and 0-order coefficients in the equation are a priori known to be piecewise constant functions, then both the coefficients and embedded obstacles can be simultaneously recovered in terms of the local Dirichlet-to-Neumann map defined on an arbitrary small open subset of the boundary \partial Ω. The method depends on a well-defined coupled PDE-system constructed for the reduced wave equations in a sufficiently small domain and the singularity analysis of solutions near the interface for the model.

scholarly journals Continuous dependence and uniqueness for lateral Cauchy problems for linear integro-differential parabolic equations

2017 ◽  
Vol 25 (5) ◽  
pp. 617-631
Author(s):  
Alfredo Lorenzi ◽  
Luca Lorenzi ◽  
Masahiro Yamamoto
Keyword(s):  
Open Subset ◽  
Time Domain ◽  
Parabolic Equations ◽  
Continuous Dependence ◽  
Initial Conditions ◽  
Carleman Estimates ◽  
Lateral Boundary ◽  
Cauchy Problems ◽  
Additional Information ◽  
Continuous Dependence Results

AbstractVia Carleman estimates we prove uniqueness and continuous dependence results for lateral Cauchy problems for linear integro-differential parabolic equations without initial conditions. The additional information supplied prescribes the conormal derivative of the temperature on a relatively open subset of the lateral boundary of the space-time domain.

scholarly journals The Mazur property and completeness in the space of Bochner-integrable functions L1(μ, X)

1992 ◽  
Vol 34 (2) ◽  
pp. 201-206
Author(s):  
G. Schlüchtermann
Keyword(s):  
Banach Space ◽  
Measure Space ◽  
Positive Measure ◽  
Dual Space ◽  
Measure Theory ◽  
Locally Convex Space ◽  
Continuous Functional ◽  
Injective Tensor Product ◽  
Integrable Functions ◽  
Locally Convex

A locally convex space (E, ) has the Mazur Property if and only if every linear -sequential continuous functional is -continuous (see [11]).In the Banach space setting, a Banach space X is a Mazur space if and only if the dual space X* endowed with the w*-topology has the Mazur property. The Mazur property was introduced by S. Mazur, and, for Banach spaces, it is investigated in detail in [4], where relations with other properties and applications to measure theory are listed. T. Kappeler obtained (see [8]) certain results for the injective tensor product and showed that L1(μ, X), the space of Bochner-integrable functions over a finite and positive measure space (S, σ, μ), is a Mazur space provided X is also, and ℓ1 does not embed in X.

scholarly journals Smoothness of solutions of parabolic equations in regions with edges

1981 ◽  
Vol 84 ◽  
pp. 159-168 ◽  
Author(s):  
A. Azzam ◽  
E. Kreyszig
Keyword(s):  
Parabolic Equation ◽  
Boundary Value Problem ◽  
Bounded Domain ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Smoothness Of Solutions ◽  
Initial Boundary Value ◽  
Simply Connected

We consider the mixed initial-boundary value problem for the parabolic equationin a region Ω × (0, T], where x = (x1, x2) and Ω ⊂ R2 is a simply-connected bounded domain having corners.

scholarly journals NULL CONTROLLABILITY OF DEGENERATE NONAUTONOMOUS PARABOLIC EQUATIONS

2019 ◽  
pp. 311
Author(s):  
Abbes Benaissa ◽  
Abdelatif Kainane Mezadek ◽  
Lahcen Maniar
Keyword(s):  
Parabolic Equation ◽  
Open Subset ◽  
Parabolic Equations ◽  
Diffusion Coefficients ◽  
Adjoint System ◽  
Null Controllability ◽  
Carleman Estimates ◽  
One Dimensional ◽  
Neumann Type ◽  
The One

In this paper we are interested in the study of the null controllability for the one dimensional degenerate non autonomous parabolic equation$$u_{t}-M(t)(a(x)u_{x})_{x}=h\chi_{\omega},\qquad  (x,t)\in Q=(0,1)\times(0,T),$$ where $\omega=(x_{1},x_{2})$ is asmall nonempty open subset in $(0,1)$, $h\in L^{2}(\omega\times(0,T))$, the diffusion coefficients $a(\cdot)$ isdegenerate at $x=0$ and $M(\cdot)$ is non degenerate on $[0,T]$. Also the boundary conditions are considered tobe Dirichlet or Neumann type related to the degeneracy rate of $a(\cdot)$. Under some conditions on the functions$a(\cdot)$ and $M(\cdot)$, we prove some global Carleman estimates which will yield the  observability inequalityof the associated adjoint system and equivalently the null controllability of our parabolic equation.

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