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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

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

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