Degenerate flux for dynamic boundary conditions in parabolic and hyperbolic equations

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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

Abstract
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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 and 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), 433-442.
[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), 313-324.
[3]	R. Clendenen, Wentzell Boundary Conditions with General Weights and Asymptotic Parabolicity for Strongly Damped Waves, Thesis (Ph.D.)-The University of Memphis, 2014.
[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), 101-108. doi: 10.1007/s00233-008-9068-2.
[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, methods and applications, Math. Probl. Eng. Aerosp. Sci. 3, Camb. Sci. Publ., Cambridge, (2009), 277-289.
[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), 419-433. doi: 10.3934/cpaa.2014.13.419.
[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), 504-521. doi: 10.1002/mana.200910086.
[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), 1-19. doi: 10.1007/s00028-002-8077-y.
[9]	G. Fragnelli, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Asymptotic parabolicity for strongly damped wave equations, in Spectral Analysis, Differential Equations and Mathematical Physics: A festschrift in honor of Fritz Gesztesy's 60th birthday, Proc. Sympos. in Pure Math. 87 Amer. Math. Soc., Providence, RI, (2013), 119-131. doi: 10.1090/pspum/087/01432.
[10]	J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, Oxford, New York, 1985.
[11]	P. D. Lax, Functional Analysis, Wiley-Interscience, New York, 2002.

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), 433-442.
[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), 313-324.
[3]	R. Clendenen, Wentzell Boundary Conditions with General Weights and Asymptotic Parabolicity for Strongly Damped Waves, Thesis (Ph.D.)-The University of Memphis, 2014.
[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), 101-108. doi: 10.1007/s00233-008-9068-2.
[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, methods and applications, Math. Probl. Eng. Aerosp. Sci. 3, Camb. Sci. Publ., Cambridge, (2009), 277-289.
[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), 419-433. doi: 10.3934/cpaa.2014.13.419.
[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), 504-521. doi: 10.1002/mana.200910086.
[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), 1-19. doi: 10.1007/s00028-002-8077-y.
[9]	G. Fragnelli, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Asymptotic parabolicity for strongly damped wave equations, in Spectral Analysis, Differential Equations and Mathematical Physics: A festschrift in honor of Fritz Gesztesy's 60th birthday, Proc. Sympos. in Pure Math. 87 Amer. Math. Soc., Providence, RI, (2013), 119-131. doi: 10.1090/pspum/087/01432.
[10]	J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, Oxford, New York, 1985.
[11]	P. D. Lax, Functional Analysis, Wiley-Interscience, New York, 2002.

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