Semilinear Strongly Degenerate Parabolic Equations with a New Class of Nonlinearities

2016 ◽  
Vol 45 (3) ◽  
pp. 507-517
Author(s):  
Dao Trong Quyet ◽  
Le Thi Thuy ◽  
Nguyen Xuan Tu
Keyword(s):  
Parabolic Equations ◽  
Degenerate Parabolic Equations ◽  
New Class ◽  
Degenerate Parabolic ◽  
Strongly Degenerate

Multiresolution schemes for strongly degenerate parabolic equations in one space dimension

2007 ◽  
Vol 23 (3) ◽  
pp. 706-730 ◽  
Author(s):  
Raimund Bürger ◽  
Alice Kozakevicius ◽  
Mauricio Sepúlveda
Keyword(s):  
Parabolic Equations ◽  
Space Dimension ◽  
Degenerate Parabolic Equations ◽  
Degenerate Parabolic ◽  
Strongly Degenerate ◽  
Multiresolution Schemes ◽  
One Space Dimension

scholarly journals Radon Measure-valued Solutions for Nonlinear Strongly Degenerate Parabolic Equations with Measure Data

2021 ◽  
Vol 14 (1) ◽  
pp. 204-233
Author(s):  
Quincy Stevene Nkombo ◽  
Fengquan Li
Keyword(s):  
Parabolic Equations ◽  
Radon Measure ◽  
Degenerate Parabolic Equations ◽  
Measure Data ◽  
Initial Value ◽  
Parabolic Capacity ◽  
Nonlinear Degenerate Parabolic Equations ◽  
Degenerate Parabolic ◽  
Strongly Degenerate ◽  
Nonlinear Degenerate

In this paper, we prove the existence of Radon measure-valued solutions for nonlinear degenerate parabolic equations with nonnegative bounded Radon measure data. Moreover, we show the uniqueness of the measure-valued solutions when the Radon measure as a forcing term is diffuse with respect to the parabolic capacity and the Radon measure as a initial value is diffuse with respect to the Newtonian capacity. We also deduce that the concentrated part of the solution with respect to the Newtonian capacity depends on time.

scholarly journals The cost of controlling strongly degenerate parabolic equations

2020 ◽  
Vol 26 ◽  
pp. 2
Author(s):  
P. Cannarsa ◽  
P. Martinez ◽  
J. Vancostenoble
Keyword(s):  
Parabolic Equations ◽  
Complex Analysis ◽  
Degenerate Parabolic Equations ◽  
Control Cost ◽  
Locally Distributed Control ◽  
Degenerate Parabolic ◽  
The Moment ◽  
The Cost ◽  
Strongly Degenerate ◽  
Degeneracy Parameter

We consider the typical one-dimensional strongly degenerate parabolic operator Pu = ut − (xαux)x with 0 < x < ℓ and α ∈ (0, 2), controlled either by a boundary control acting at x = ℓ, or by a locally distributed control. Our main goal is to study the dependence of the so-called controllability cost needed to drive an initial condition to rest with respect to the degeneracy parameter α. We prove that the control cost blows up with an explicit exponential rate, as eC/((2−α)2T), when α → 2− and/or T → 0+. Our analysis builds on earlier results and methods (based on functional analysis and complex analysis techniques) developed by several authors such as Fattorini-Russel, Seidman, Güichal, Tenenbaum-Tucsnak and Lissy for the classical heat equation. In particular, we use the moment method and related constructions of suitable biorthogonal families, as well as new fine properties of the Bessel functions Jν of large order ν (obtained by ordinary differential equations techniques).

scholarly journals Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

2007 ◽  
Vol 60 (3-4) ◽  
pp. 365-385 ◽  
Author(s):  
Raimund Bürger ◽  
Ricardo Ruiz ◽  
Kai Schneider ◽  
Mauricio A. Sepúlveda
Keyword(s):  
Parabolic Equations ◽  
Degenerate Parabolic Equations ◽  
Fully Adaptive ◽  
Discontinuous Flux ◽  
Degenerate Parabolic ◽  
Strongly Degenerate ◽  
Multiresolution Schemes

Boundary value problems for strongly degenerate parabolic equations

1997 ◽  
Vol 22 (1-2) ◽  
pp. 17-38 ◽  
Author(s):  
M. Lavrentiev Jr. ◽  
P. Broadbridge ◽  
V. Belov
Keyword(s):  
Boundary Value Problems ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Degenerate Parabolic Equations ◽  
Degenerate Parabolic ◽  
Strongly Degenerate
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