scholarly journals An observability estimate for parabolic equations from a measurable set in time and its applications

2013 ◽  
Vol 15 (2) ◽  
pp. 681-703 ◽  
Author(s):  
Kim Dang Phung ◽  
Gengsheng Wang
Keyword(s):  
Parabolic Equations ◽  
Observability Estimate ◽  
Measurable Set

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).

scholarly journals Regularity of solutions for quasi-linear parabolic equations

1976 ◽  
Vol 60 ◽  
pp. 151-172
Author(s):  
Yoshiaki Ikeda
Keyword(s):  
Dirichlet Problem ◽  
Linear Equation ◽  
Bounded Domain ◽  
Parabolic Equations ◽  
Space Time ◽  
Second Order ◽  
Regularity Of Solutions ◽  
Measurable Set ◽  
Euclidian Space ◽  
Linear Parabolic Equations

Let Ω be a bounded domain in n-dimensional Euclidian space En (n ≧ 2), and consider the space-time cylinder Q = Ω × (0, T] for some fixed T > 0. In this paper we deal with the Cauchy and Dirichlet problem for a second order quasi-linear equation(1.1) ut — div A(x, t, u, ux) + B(x, t, u, ux) = 0 for (x, t) ∈ Q,(1.2) u(x, 0) = (ϕ)(x) in Ω and u(x, t) = tψ(x, t) for (x, t) ∈ Γ = ∂Ω × (0, T] ,where ∂Ω is a boundary of Ω which satisfies the following condition (A) : Condition (A). There exist constants ρ0 and »0 both in (0,1) such that, for any sphere K(ρ) with center on ∂Ω and radius ρ ≦ ρ0, the inequality meas [K(ρ) ∩ Ω] ≦ (1 — λ0) × meas E(ρ) holds, where meas E means the measure of a measurable set E.

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>

Hypothesis on the Solvability of Parabolic Equations with nonlocal Condition

2002 ◽  
Vol 7 (1) ◽  
pp. 93-104 ◽  
Author(s):  
Mifodijus Sapagovas
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Nonlocal Condition ◽  
Nonlocal Conditions ◽  
Numerical Algorithms ◽  
Uniqueness Of The Solution

Numerous and different nonlocal conditions for the solvability of parabolic equations were researched in many articles and reports. The article presented analyzes such conditions imposed, and observes that the existence and uniqueness of the solution of parabolic equation is related mainly to ”smallness” of functions, involved in nonlocal conditions. As a consequence the hypothesis has been made, stating the assumptions on functions in nonlocal conditions are related to numerical algorithms of solving parabolic equations, and not to the parabolic equation itself.

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