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 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 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.

21.—Runge's Theorem for Parabolic Equations in Two Space Variables

1975 ◽  
Vol 73 ◽  
pp. 307-315 ◽  
Author(s):  
David Colton
Keyword(s):  
Parabolic Equation ◽  
Strong Solution ◽  
Parabolic Equations ◽  
Entire Solution ◽  
Second Order ◽  
Maximum Norm ◽  
Cylindrical Domain ◽  
Order Parabolic Equation ◽  
Compact Subsets

SynopsisLet u be a real valued strong solution defined in a cylindrical domain of a linear second-order parabolic equation in two space variables with entire coefficients. Then it is shown that on compact subsets of its domain of definition u can be approximated arbitrarily closely in the maximum norm by an entire solution of the parabolic equation.

On the solutions of the first boundary value problem for the linear parabolic equations

1988 ◽  
Vol 108 (3-4) ◽  
pp. 339-355 ◽  
Author(s):  
Eugenio Sinestrari ◽  
Wolf von Wahl
Keyword(s):  
Parabolic Equation ◽  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Second Order ◽  
Inhomogeneous Term ◽  
Hölder Continuous ◽  
Space And Time ◽  
Order Parabolic Equation ◽  
Linear Parabolic Equations

SynopsisThe first boundary value problem for a linear second order parabolic equation is studied under the assumption that the inhomogeneous term is continuous in space and time and Hölder-continuous only with respect to the space variables.

scholarly journals Non-autonomous perturbations of a non-classical non-autonomous parabolic equation with subcritical nonlinearity

2017 ◽  
Vol 2 (1) ◽  
pp. 31-60 ◽  
Author(s):  
Matheus C. Bortolan ◽  
Felipe Rivero
Keyword(s):  
Asymptotic Behavior ◽  
Parabolic Equation ◽  
Bounded Domain ◽  
Small Perturbation ◽  
Parabolic Equations ◽  
Smooth Bounded Domain ◽  
Subcritical Nonlinearity

AbstractIn this work we study the continuity of four different notions of asymptotic behavior for a family of non-autonomous non-classical parabolic equations given by$$\begin{array}{} \displaystyle \left\{ \begin{array}{*{20}{l}} {{u_t} - \gamma \left( t \right)\Delta {u_t} - \Delta u = {g_\varepsilon }\left( {t,u} \right),{\;\text{in}\;}\Omega } \hfill \\ {u = 0,{\;\text{on}\;}\partial \Omega {\rm{.}}} \hfill \\ \end{array}\right. \end{array}$$in a smooth bounded domain Ω ⊂ ℝn, n ⩾ 3, where the terms gε are a small perturbation, in some sense, of a function f that depends only on u.

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.

Sign in / Sign up

Export Citation Format

Share Document