# American Institute of Mathematical Sciences

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

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

 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 $\beta$.
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
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