Galerkin FEM for Fractional Order Parabolic Equations with Initial Data in H − s , 0 ≤ s ≤ 1

2013 ◽  
pp. 24-37
Author(s):  
Bangti Jin ◽  
Raytcho Lazarov ◽  
Joseph Pasciak ◽  
Zhi Zhou
Keyword(s):  
Initial Data ◽  
Fractional Order ◽  
Parabolic Equations

scholarly journals Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations

2020 ◽  
Vol 10 (1) ◽  
pp. 353-370 ◽  
Author(s):  
Hans-Christoph Grunau ◽  
Nobuhito Miyake ◽  
Shinya Okabe
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Fourth Order ◽  
Parabolic Equations ◽  
Sufficient Conditions ◽  
Nonlinear Parabolic Equations ◽  
Heat Equations ◽  
Cauchy Problems ◽  
The Cauchy Problem ◽  
Positivity Of Solutions

Abstract This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity. The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space. The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation.

Convergence to travelling fronts in semilinear parabolic equations

1978 ◽  
Vol 80 (3-4) ◽  
pp. 213-234 ◽  
Author(s):  
Franz Rothe
Keyword(s):  
Asymptotic Behavior ◽  
Initial Data ◽  
Parabolic Equations ◽  
Step Function ◽  
Wave Type ◽  
All Solutions ◽  
Asymptotic Speed ◽  
Rate Of Decay ◽  
Travelling Fronts ◽  
Semilinear Parabolic

SynopsisWe study the convergence to the stationary state for the parabolic equation u, = uxx + F(u). There exist wave-type solutions u(x, t) = φ(x − ct) for a continuum of velocities c. In the asymptotic behavior of this equation was investigated for a step function as initial data. In this paper we obtain the asymptotic behavior for a large class of monotone initial data.All solutions with initial data in this class evolve to wave-type solutions, where the rate of decay of the initial data determines the asymptotic speed.

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).

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