scholarly journals A Strange Term in the Homogenization of Parabolic Equations with Two Spatial and Two Temporal Scales

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
L. Flodén ◽  
A. Holmbom ◽  
M. Olsson Lindberg
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Time Derivative ◽  
Local Problem ◽  
Temporal Scales ◽  
Space And Time ◽  
Temporal Oscillation ◽  
Spatial Oscillations ◽  
Spatial Oscillation

We study the homogenization of a parabolic equation with oscillations in both space and time in the coefficienta(x/ε,t/ε2)in the elliptic part and spatial oscillations in the coefficientρ(x/ε)that is multiplied with the time derivative∂tuε. We obtain a strange term in the local problem. This phenomenon appears as a consequence of the combination of the spatial oscillation inρ(x/ε)and the temporal oscillation ina(x/ε,t/ε2)and disappears if either of these oscillations is removed.

scholarly journals A Note on Parabolic Homogenization with a Mismatch between the Spatial Scales

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Liselott Flodén ◽  
Anders Holmbom ◽  
Marianne Olsson Lindberg ◽  
Jens Persson
Keyword(s):  
Spatial Scale ◽  
Parabolic Problem ◽  
Spatial Scales ◽  
Time Derivative ◽  
Local Problem ◽  
Linear Parabolic Problem ◽  
Spatial Oscillations ◽  
Spatial Oscillation

We consider the homogenization of the linear parabolic problem which exhibits a mismatch between the spatial scales in the sense that the coefficient of the elliptic part has one frequency of fast spatial oscillations, whereas the coefficient of the time derivative contains a faster spatial scale. It is shown that the faster spatial microscale does not give rise to any corrector term and that there is only one local problem needed to characterize the homogenized problem. Hence, the problem is not of a reiterated type even though two rapid scales of spatial oscillation appear.

scholarly journals On non-local problems for parabolic equations

1984 ◽  
Vol 93 ◽  
pp. 109-131 ◽  
Author(s):  
J. Chabrowski
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Linear Parabolic Equation ◽  
Local Problem ◽  
Local Problems ◽  
Non Local

The main purposes of this paper are to investigate the existence and the uniqueness of a non-local problem for a linear parabolic equationin a cylinder D = Ω × (0, T].

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.

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.

scholarly journals Smoothing properties in multistep backward difference method and time derivative approximation for linear parabolic equations

2005 ◽  
Vol 2005 (4) ◽  
pp. 523-536
Author(s):  
Yubin Yan
Keyword(s):  
Difference Method ◽  
Parabolic Equations ◽  
Parabolic Problem ◽  
Approximation Scheme ◽  
Time Derivative ◽  
Nonsmooth Data ◽  
Smoothing Property ◽  
Linear Parabolic Equations ◽  
Linear Parabolic Problem ◽  
Data Error

A smoothing property in multistep backward difference method for a linear parabolic problem in Hilbert space has been proved, where the operator is selfadjoint, positive definite with compact inverse. By using the solutions computed by a multistep backward difference method for the parabolic problem, we introduce an approximation scheme for time derivative. The nonsmooth data error estimate for the approximation of time derivative has been obtained.

Intrinsic scaling method for doubly nonlinear parabolic equations and its application

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Masashi Misawa ◽  
Kenta Nakamura
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Large Data ◽  
Nonlinear Parabolic Equations ◽  
Sobolev Type ◽  
Scaling Method ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear ◽  
Doubly Nonlinear Equation ◽  
Doubly Nonlinear Parabolic Equation

Abstract In this article, we consider a fast diffusive type doubly nonlinear parabolic equation, called 𝑝-Sobolev type flows, and devise a new intrinsic scaling method to transform the prototype doubly nonlinear equation to the 𝑝-Sobolev type flows. As an application, we show the global existence and regularity for the 𝑝-Sobolev type flows with large data.

scholarly journals Numerical Solution of a Kind of Fractional Parabolic Equations via Two Difference Schemes

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Abdon Atangana ◽  
Dumitru Baleanu
Keyword(s):  
Parabolic Equation ◽  
Fractional Calculus ◽  
Numerical Solution ◽  
Parabolic Equations ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Implicit Schemes ◽  
The Stability

A kind of parabolic equation was extended to the concept of fractional calculus. The resulting equation is, however, difficult to handle analytically. Therefore, we presented the numerical solution via the explicit and the implicit schemes. We presented together the stability and convergence of this time-fractional parabolic equation with two difference schemes. The explicit and the implicit schemes in this case are stable under some conditions.

Complete blow-up for quasilinear degenerate parabolic equations

2000 ◽  
Vol 130 (4) ◽  
pp. 877-908 ◽  
Author(s):  
R. Suzuki
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Parabolic Equations ◽  
Degenerate Parabolic Equation ◽  
Dirichlet Boundary ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Degenerate Parabolic Equations ◽  
Degenerate Parabolic ◽  
Image Position

Non-negative post-blow-up solutions of the quasilinear degenerate parabolic equation in RN (or a bounded domain with Dirichlet boundary condition) are studied. Various sufficient conditions for complete blow-up of solutions are given.

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