scholarly journals Singular parabolic equations with interior degeneracy and non smooth coefficients: The Neumann case

2018 ◽  
Vol 0 (0) ◽  
pp. 1-17
Author(s):  
Genni Fragnelli ◽  
◽  
Dimitri Mugnai ◽  
Keyword(s):  
Parabolic Equations ◽  
Smooth Coefficients ◽  
Singular Parabolic Equations ◽  
Interior Degeneracy

scholarly journals Carleman estimates for singular parabolic equations with interior degeneracy and non-smooth coefficients

2017 ◽  
Vol 6 (1) ◽  
pp. 61-84 ◽  
Author(s):  
Genni Fragnelli ◽  
Dimitri Mugnai
Keyword(s):  
Parabolic Equations ◽  
Spatial Domain ◽  
Carleman Estimates ◽  
Dirichlet Problems ◽  
Smooth Coefficients ◽  
Degenerate Parabolic ◽  
Singular Parabolic Equations ◽  
Interior Degeneracy

AbstractWe establish Carleman estimates for singular/degenerate parabolic Dirichlet problems with degeneracy and singularity occurring in the interior of the spatial domain. Our results are completely new, since this situation is not covered by previous contributions for degeneracy and singularity on the boundary. In addition, we consider non-smooth coefficients, thus preventing the use of standard calculations in this framework.

scholarly journals An Inverse Source Problem for Singular Parabolic Equations with Interior Degeneracy

2018 ◽  
Vol 2018 ◽  
pp. 1-16
Author(s):  
Khalid Atifi ◽  
Idriss Boutaayamou ◽  
Hamed Ould Sidi ◽  
Jawad Salhi
Keyword(s):  
Numerical Solution ◽  
Parabolic Equations ◽  
Measurement Data ◽  
Gradient Methods ◽  
Stability Estimate ◽  
Inverse Source Problem ◽  
Output Least Squares ◽  
Singular Parabolic Equations ◽  
Interior Degeneracy ◽  
Least Squares Approach

The main purpose of this work is to study an inverse source problem for degenerate/singular parabolic equations with degeneracy and singularity occurring in the interior of the spatial domain. Using Carleman estimates, we prove a Lipschitz stability estimate for the source term provided that additional measurement data are given on a suitable interior subdomain. For the numerical solution, the reconstruction is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. The Fréchet differentiability of the Tikhonov functional and the Lipschitz continuity of the Fréchet gradient are proved. These properties allow us to apply gradient methods for numerical solution of the considered inverse source problem.

scholarly journals A Boundary Estimate for Singular Sub-Critical Parabolic Equations

2020 ◽  
Author(s):  
Ugo Gianazza ◽  
Naian Liao
Keyword(s):  
Modulus Of Continuity ◽  
Weak Solutions ◽  
Parabolic Equations ◽  
Boundary Point ◽  
Cylindrical Domain ◽  
Boundary Estimate ◽  
Critical Range ◽  
Type Integral ◽  
Wiener Type ◽  
Singular Parabolic Equations

Abstract We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to singular parabolic equations of $p$-Laplacian type, with $p$ in the sub-critical range $\big(1,\frac{2N}{N+1}\big]$. The estimate is given in terms of a Wiener-type integral, defined by a proper elliptic $p$-capacity.

Nonlinear Singular Parabolic Equations

2020 ◽  
pp. 21-36
Author(s):  
Piotr Biler ◽  
Tadeusz Nadzieja ◽  
Andrzej Raczyński
Keyword(s):  
Parabolic Equations ◽  
Singular Parabolic Equations

scholarly journals Carleman estimates and null controllability of a class of singular parabolic equations

2018 ◽  
Vol 8 (1) ◽  
pp. 1057-1082
Author(s):  
Runmei Du ◽  
Jürgen Eichhorn ◽  
Qiang Liu ◽  
Chunpeng Wang
Keyword(s):  
Control Systems ◽  
Parabolic Equations ◽  
Null Controllability ◽  
Carleman Estimates ◽  
Energy Estimates ◽  
Semilinear Parabolic Equations ◽  
Linearized Equations ◽  
Singular Parabolic Equations ◽  
Semilinear Parabolic

Abstract In this paper, we consider control systems governed by a class of semilinear parabolic equations, which are singular at the boundary and possess singular convection and reaction terms. The systems are shown to be null controllable by establishing Carleman estimates, observability inequalities and energy estimates for solutions to linearized equations.

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