Degenerate flux for dynamic boundary conditions in parabolic and hyperbolic equations

Previous Article
The problem of detecting corrosion by an electric measurement revisited
DCDS-S Home
This Issue
Next Article
A singular limit problem for the Ibragimov-Shabat equation

June 2016, 9(3): 651-660. doi: 10.3934/dcdss.2016019

Degenerate flux for dynamic boundary conditions in parabolic and hyperbolic equations

Raluca Clendenen ^1, , Gisèle Ruiz Goldstein ^2, and Jerome A. Goldstein ^3,

1.	Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152, United States
2.	The University of Memphis, Department of Mathematical Sciences, Memphis, TN 38152
3.	Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, United States

Received April 2015 Revised October 2015 Published April 2016

Abstract
Related Papers

In the dynamic or Wentzell boundary condition for elliptic, parabolic and hyperbolic partial differential equations, the positive flux coefficient $% \beta $ determines the weighted surface measure $dS/\beta $ on the boundary of the given spatial domain, in the appropriate Hilbert space that makes the generator for the problem selfadjoint. Usually, $\beta $ is continuous and bounded away from both zero and infinity, and thus $L^{2}\left( \partial \Omega ,dS\right) $ and $L^{2}\left( \partial \Omega ,dS/\beta \right) $ are equal as sets. In this paper this restriction is eliminated, so that both zero and infinity are allowed to be limiting values for $\beta $. An application includes the parabolic asymptotics for the Wentzell telegraph equation and strongly damped Wentzell wave equation with general $\beta $.

Keywords: parabolic asymptotics, degenerate flux., Dynamic boundary conditions, telegraph equation, Wentzell boundary conditions.

Mathematics Subject Classification: Primary: 34G10; Secondary: 35L10, 35L35, 35Q3.

Citation: Raluca Clendenen, Gisèle Ruiz Goldstein, Jerome A. Goldstein. Degenerate flux for dynamic boundary conditions in parabolic and hyperbolic equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 651-660. doi: 10.3934/dcdss.2016019

References:

[1]	T. Clarke, E. C. Eckstein and J. A. Goldstein, Asymptotics analysis of the abstract telegraph equation,, Differential Integral Equations, 21 (2008), 433. Google Scholar
[2]	T. Clarke, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The Wentzell telegraph equation: Asymptotics and continuous dependence on the boundary conditions,, Commun. Appl. Anal., 15 (2011), 313. Google Scholar
[3]	R. Clendenen, Wentzell Boundary Conditions with General Weights and Asymptotic Parabolicity for Strongly Damped Waves,, Thesis (Ph.D.)-The University of Memphis, (2014). Google Scholar
[4]	G. M. Coclite, A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Continuous dependence on the boundary conditions for the Wentzell Laplacian,, Semigroup Forum, 77 (2008), 101. doi: 10.1007/s00233-008-9068-2. Google Scholar
[5]	G. M. Coclite, A. Favini, C. G. Gal, G. R. Goldstein, J. A. Goldstein, E. Obrecht and S. Romanelli, The role of Wentzell boundary conditions in linear and nonlinear analysis,, in Advances in nonlinear analysis: Theory, 3 (2009), 277. Google Scholar
[6]	G. M. Coclite, A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Continuous dependence in hyperbolic problems with Wentzell boundary conditions,, Commun. Pure Appl. Anal., 13 (2014), 419. doi: 10.3934/cpaa.2014.13.419. Google Scholar
[7]	A. Favini, G. R. Goldstein, J. A. Goldstein, E. Obrecht and S. Romanelli, Elliptic operators with general Wentzell boundary conditions, analytic semigroups, and the angle concavity theorem,, Math. Nachr., 283 (2010), 504. doi: 10.1002/mana.200910086. Google Scholar
[8]	A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with generalized Wentzell boundary conditions,, J. Evol. Equ., 2 (2002), 1. doi: 10.1007/s00028-002-8077-y. Google Scholar
[9]	G. Fragnelli, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Asymptotic parabolicity for strongly damped wave equations,, in Spectral Analysis, 87 (2013), 119. doi: 10.1090/pspum/087/01432. Google Scholar
[10]	J. A. Goldstein, Semigroups of Linear Operators and Applications,, Oxford University Press, (1985). Google Scholar
[11]	P. D. Lax, Functional Analysis,, Wiley-Interscience, (2002). Google Scholar

show all references

References:

[1]	T. Clarke, E. C. Eckstein and J. A. Goldstein, Asymptotics analysis of the abstract telegraph equation,, Differential Integral Equations, 21 (2008), 433. Google Scholar
[2]	T. Clarke, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The Wentzell telegraph equation: Asymptotics and continuous dependence on the boundary conditions,, Commun. Appl. Anal., 15 (2011), 313. Google Scholar
[3]	R. Clendenen, Wentzell Boundary Conditions with General Weights and Asymptotic Parabolicity for Strongly Damped Waves,, Thesis (Ph.D.)-The University of Memphis, (2014). Google Scholar
[4]	G. M. Coclite, A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Continuous dependence on the boundary conditions for the Wentzell Laplacian,, Semigroup Forum, 77 (2008), 101. doi: 10.1007/s00233-008-9068-2. Google Scholar
[5]	G. M. Coclite, A. Favini, C. G. Gal, G. R. Goldstein, J. A. Goldstein, E. Obrecht and S. Romanelli, The role of Wentzell boundary conditions in linear and nonlinear analysis,, in Advances in nonlinear analysis: Theory, 3 (2009), 277. Google Scholar
[6]	G. M. Coclite, A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Continuous dependence in hyperbolic problems with Wentzell boundary conditions,, Commun. Pure Appl. Anal., 13 (2014), 419. doi: 10.3934/cpaa.2014.13.419. Google Scholar
[7]	A. Favini, G. R. Goldstein, J. A. Goldstein, E. Obrecht and S. Romanelli, Elliptic operators with general Wentzell boundary conditions, analytic semigroups, and the angle concavity theorem,, Math. Nachr., 283 (2010), 504. doi: 10.1002/mana.200910086. Google Scholar
[8]	A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with generalized Wentzell boundary conditions,, J. Evol. Equ., 2 (2002), 1. doi: 10.1007/s00028-002-8077-y. Google Scholar
[9]	G. Fragnelli, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Asymptotic parabolicity for strongly damped wave equations,, in Spectral Analysis, 87 (2013), 119. doi: 10.1090/pspum/087/01432. Google Scholar
[10]	J. A. Goldstein, Semigroups of Linear Operators and Applications,, Oxford University Press, (1985). Google Scholar
[11]	P. D. Lax, Functional Analysis,, Wiley-Interscience, (2002). Google Scholar

[1]	Davide Guidetti. Parabolic problems with general Wentzell boundary conditions and diffusion on the boundary. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1401-1417. doi: 10.3934/cpaa.2016.15.1401
[2]	Mahamadi Warma. Semi linear parabolic equations with nonlinear general Wentzell boundary conditions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5493-5506. doi: 10.3934/dcds.2013.33.5493
[3]	Lahcen Maniar, Martin Meyries, Roland Schnaubelt. Null controllability for parabolic equations with dynamic boundary conditions. Evolution Equations & Control Theory, 2017, 6 (3) : 381-407. doi: 10.3934/eect.2017020
[4]	Mustapha Mokhtar-Kharroubi, Quentin Richard. Time asymptotics of structured populations with diffusion and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4087-4116. doi: 10.3934/dcdsb.2018127
[5]	Giuseppe Maria Coclite, Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Continuous dependence in hyperbolic problems with Wentzell boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (1) : 419-433. doi: 10.3934/cpaa.2014.13.419
[6]	Antonio Suárez. A logistic equation with degenerate diffusion and Robin boundary conditions. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1255-1267. doi: 10.3934/cpaa.2008.7.1255
[7]	Alassane Niang. Boundary regularity for a degenerate elliptic equation with mixed boundary conditions. Communications on Pure & Applied Analysis, 2019, 18 (1) : 107-128. doi: 10.3934/cpaa.2019007
[8]	Paul Sacks, Mahamadi Warma. Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1$-data. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 761-787. doi: 10.3934/dcds.2014.34.761
[9]	Davide Guidetti. Classical solutions to quasilinear parabolic problems with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 717-736. doi: 10.3934/dcdss.2016024
[10]	Ciprian G. Gal, Mahamadi Warma. Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions. Evolution Equations & Control Theory, 2016, 5 (1) : 61-103. doi: 10.3934/eect.2016.5.61
[11]	Ángela Jiménez-Casas, Aníbal Rodríguez-Bernal. Dynamic boundary conditions as limit of singularly perturbed parabolic problems. Conference Publications, 2011, 2011 (Special) : 737-746. doi: 10.3934/proc.2011.2011.737
[12]	Ciprian G. Gal, Hao Wu. Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 1041-1063. doi: 10.3934/dcds.2008.22.1041
[13]	Yoshifumi Mimura. Critical mass of degenerate Keller-Segel system with no-flux and Neumann boundary conditions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1603-1630. doi: 10.3934/dcds.2017066
[14]	Genni Fragnelli, Gisèle Ruiz Goldstein, Jerome Goldstein, Rosa Maria Mininni, Silvia Romanelli. Generalized Wentzell boundary conditions for second order operators with interior degeneracy. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 697-715. doi: 10.3934/dcdss.2016023
[15]	Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Enrico Obrecht, Silvia Romanelli. Nonsymmetric elliptic operators with Wentzell boundary conditions in general domains. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2475-2487. doi: 10.3934/cpaa.2016045
[16]	Nicolas Fourrier, Irena Lasiecka. Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions. Evolution Equations & Control Theory, 2013, 2 (4) : 631-667. doi: 10.3934/eect.2013.2.631
[17]	Alain Miranville, Sergey Zelik. The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 275-310. doi: 10.3934/dcds.2010.28.275
[18]	Ciprian G. Gal, Maurizio Grasselli. The non-isothermal Allen-Cahn equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 1009-1040. doi: 10.3934/dcds.2008.22.1009
[19]	Laurence Cherfils, Madalina Petcu, Morgan Pierre. A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1511-1533. doi: 10.3934/dcds.2010.27.1511
[20]	Gianni Gilardi, A. Miranville, Giulio Schimperna. On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (3) : 881-912. doi: 10.3934/cpaa.2009.8.881

2018 Impact Factor: 0.545

American Institute of Mathematical Sciences