Degenerate flux for dynamic boundary conditions in parabolic and hyperbolic equations

Raluca  Clendenen; Gisèle Ruiz Goldstein; Jerome A. Goldstein

doi:10.3934/dcdss.2016019

Article Contents

2016, Volume 9, Issue 3: 651-660. Doi: 10.3934/dcdss.2016019

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Degenerate flux for dynamic boundary conditions in parabolic and hyperbolic equations

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

Received: March 31, 2015

Revised: September 30, 2015

Published: May 2016

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Abstract

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 $.

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Mathematics Subject Classification: Primary: 34G10; Secondary: 35L10, 35L35, 35Q35.

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References

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