scholarly journals Simultaneous reconstruction of time-dependent coefficients in the parabolic equation from over-specification conditions

2021 ◽  
Vol 12 ◽  
pp. 100197
Author(s):  
M.J. Huntul
Keyword(s):  
Parabolic Equation ◽  
Time Dependent ◽  
Simultaneous Reconstruction

scholarly journals Doubly Degenerate Parabolic Equation with Time-Dependent Gradient Source and Initial Data Measures

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Lihua Deng ◽  
Xianguang Shang
Keyword(s):  
Parabolic Equation ◽  
Initial Data ◽  
Degenerate Parabolic Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Local Existence ◽  
Time Dependent ◽  
Priori Estimates ◽  
The Cauchy Problem ◽  
Degenerate Parabolic

This paper is devoted to the Cauchy problem for a class of doubly degenerate parabolic equation with time-dependent gradient source, where the initial data are Radon measures. Using the delicate a priori estimates, we first establish two local existence results. Furthermore, we show that the existence of solutions is optimal in the class considered here.

Time-dependent singularities in a semilinear parabolic equation with absorption

2016 ◽  
Vol 18 (05) ◽  
pp. 1550077 ◽  
Author(s):  
Jin Takahashi ◽  
Eiji Yanagida
Keyword(s):  
Parabolic Equation ◽  
Singular Point ◽  
Fundamental Solution ◽  
Laplace Equation ◽  
Space Variable ◽  
The Other ◽  
Time Dependent ◽  
Singular Solutions ◽  
Semilinear Parabolic Equation ◽  
Semilinear Parabolic

This paper concerns solutions with time-dependent singularities for a semilinear parabolic equation with a superlinear absorption term. Here, by time-dependent singularity, we mean a singularity with respect to the space variable whose position depends on time. It is shown that if the power of the nonlinearity is in some range, then any singularity is removable. On the other hand, in other range, two types of time-dependent singular solutions exist: One resembles the fundamental solution of the Laplace equation near the singular point, and the other has a stronger singularity.

scholarly journals Eulerian time-stepping schemes for the non-stationary Stokes equations on time-dependent domains

2022 ◽  
Author(s):  
Erik Burman ◽  
Stefan Frei ◽  
Andre Massing
Keyword(s):  
Parabolic Equation ◽  
Stokes Equations ◽  
A Priori ◽  
Time Dependent ◽  
Numerical Examples ◽  
Time Stepping ◽  
Discretisation Error ◽  
Moving Domains ◽  
Theoretical Results ◽  
Norm Error

AbstractThis article is concerned with the discretisation of the Stokes equations on time-dependent domains in an Eulerian coordinate framework. Our work can be seen as an extension of a recent paper by Lehrenfeld and Olshanskii (ESAIM: M2AN 53(2):585–614, 2019), where BDF-type time-stepping schemes are studied for a parabolic equation on moving domains. For space discretisation, a geometrically unfitted finite element discretisation is applied in combination with Nitsche’s method to impose boundary conditions. Physically undefined values of the solution at previous time-steps are extended implicitly by means of so-called ghost penalty stabilisations. We derive a complete a priori error analysis of the discretisation error in space and time, including optimal $$L^2(L^2)$$ L 2 ( L 2 ) -norm error bounds for the velocities. Finally, the theoretical results are substantiated with numerical examples.

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