On Inverse Problems for Strongly Degenerate Parabolic Equations under the Integral Observation Condition

2018 ◽  
Vol 58 (12) ◽  
pp. 2002-2017 ◽  
Author(s):  
V. L. Kamynin
Keyword(s):  
Inverse Problems ◽  
Parabolic Equations ◽  
Degenerate Parabolic Equations ◽  
Observation Condition ◽  
Degenerate Parabolic ◽  
Strongly Degenerate ◽  
Integral Observation ◽  
Integral Observation Condition

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).

Inverse problems for linear degenerate parabolic equations by “time-like” Carleman estimate

2015 ◽  
Vol 23 (1) ◽  
pp. 1-21 ◽  
Author(s):  
Atsushi Kawamoto
Keyword(s):  
Inverse Problems ◽  
Parabolic Equations ◽  
Fixed Time ◽  
Carleman Estimate ◽  
Coupled Systems ◽  
Degenerate Parabolic Equations ◽  
Strongly Coupled ◽  
Degenerate Parabolic ◽  
Inverse Source Problems ◽  
Linear Degenerate

AbstractIn this paper, we study inverse problems for multi-dimensional linear degenerate parabolic equations and strongly coupled systems. In particular we discuss the Lipschitz type stability results for the inverse source problems which determine a source term by boundary data on an appropriate sub-boundary and the data on any fixed time. Our arguments are based on the Carleman estimate. Here we prove and use the Carleman estimate with the

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