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



2021 ◽  
Vol 14 (1) ◽  
pp. 204-233
Author(s):  
Quincy Stevene Nkombo ◽  
Fengquan Li

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.



2020 ◽  
Vol 26 ◽  
pp. 2
Author(s):  
P. Cannarsa ◽  
P. Martinez ◽  
J. Vancostenoble

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).







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