Article Contents
Article Contents

# Degenerate flux for dynamic boundary conditions in parabolic and hyperbolic equations

• 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 $\beta$.
Mathematics Subject Classification: Primary: 34G10; Secondary: 35L10, 35L35, 35Q35.

 Citation:

•  [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.