# American Institute of Mathematical Sciences

## On hyperbolic mixed problems with dynamic and Wentzell boundary conditions

 Dipartimento di matematica, Piazza di Porta S. Donato 5, 40126 Bologna, Italy

* Corresponding author

Dedicated to Gisele Ruiz Goldstein in occasion of her sixtieth birthday
The author is member of GNAMPA of Istituto Nazionale di Alta Matematica

Received  December 2018 Revised  August 2019 Published  January 2020

We study mixed hyperbolic systems with dynamic and Wentzell boundary conditions. The boundary condition contains a tangential operator which is strongly elliptic on the boundary. We prove results of generation of strongly continuous groups and well-posedness.

Citation: Davide Guidetti. On hyperbolic mixed problems with dynamic and Wentzell boundary conditions. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020239
##### References:
 [1] M. Cavalcanti, A. Khemmoudj and M. Medjden, Uniform stabilisation of the damped Cauchy-Ventcel problem with variable coefficients and dynamic boundary conditions, J. Math. Anal. Appl., 328 (2007), 900-930.  doi: 10.1016/j.jmaa.2006.05.070.  Google Scholar [2] R. Clendenen, G. R. Goldstein and J. A. Goldstein, Degenerate flux for dynamic boundary conditions in parabolic and hyperbolic equations, Discr. Cont. Dynam. Syst. Ser. S, 9 (2016), 651-660.  doi: 10.3934/dcdss.2016019.  Google Scholar [3] 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 Applied Anal., 13 (2004), 419-433.  doi: 10.3934/cpaa.2014.13.419.  Google Scholar [4] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer, 2000.  Google Scholar [5] C. Giorgi and D. Guidetti, Reconstruction of kernel depending also on a space variable, ath. Methods Appl. Sci., 41 (2018), 4560-4588.  doi: 10.1002/mma.4914.  Google Scholar [6] G. R. Goldstein, J. A. Goldstein, D. Guidetti and S. Romanelli, Maximal regularity, analytic semigroups, and dynamic and general Wentzell boundary conditions with a diffusion term on the boundary, Annali di Matematica Pura ed Applicata, 2019. doi: 10.1007/s10231-019-00868-3.  Google Scholar [7] J. A. Goldstein, Semigroups of Linear Operators & Applications, Dover Publications, Inc. (Second Edition), 2017.  Google Scholar [8] I. Lasiecka, J. L. Lions and R. Triggiani, Non homogeneous boundary value problems for second order hyperbolic operators, J Math Pures et Appl., 65 (1986), 149-192.   Google Scholar [9] S. Nicaise and K. Laoubi, Polynomial stabilization of the wave equation with Ventcel's boundary conditions, Math. Nachr., 283 (2010), 1428-1438.  doi: 10.1002/mana.200710162.  Google Scholar [10] G. Ruiz Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Diff. Eq., 11 (2006), 457-480.   Google Scholar [11] H. Tanabe, Equations of Evolution, Monographs and Studies in Mathematics, 6. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979.  Google Scholar [12] E. Vitillaro, On the wave equation with hyperbolic dynamical boundary conditions, interior and boundary damping and supercritical sources, Journ. Diff. Eq., 265 (2018), 4873-4941.  doi: 10.1016/j.jde.2018.06.022.  Google Scholar

show all references

##### References:
 [1] M. Cavalcanti, A. Khemmoudj and M. Medjden, Uniform stabilisation of the damped Cauchy-Ventcel problem with variable coefficients and dynamic boundary conditions, J. Math. Anal. Appl., 328 (2007), 900-930.  doi: 10.1016/j.jmaa.2006.05.070.  Google Scholar [2] R. Clendenen, G. R. Goldstein and J. A. Goldstein, Degenerate flux for dynamic boundary conditions in parabolic and hyperbolic equations, Discr. Cont. Dynam. Syst. Ser. S, 9 (2016), 651-660.  doi: 10.3934/dcdss.2016019.  Google Scholar [3] 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 Applied Anal., 13 (2004), 419-433.  doi: 10.3934/cpaa.2014.13.419.  Google Scholar [4] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer, 2000.  Google Scholar [5] C. Giorgi and D. Guidetti, Reconstruction of kernel depending also on a space variable, ath. Methods Appl. Sci., 41 (2018), 4560-4588.  doi: 10.1002/mma.4914.  Google Scholar [6] G. R. Goldstein, J. A. Goldstein, D. Guidetti and S. Romanelli, Maximal regularity, analytic semigroups, and dynamic and general Wentzell boundary conditions with a diffusion term on the boundary, Annali di Matematica Pura ed Applicata, 2019. doi: 10.1007/s10231-019-00868-3.  Google Scholar [7] J. A. Goldstein, Semigroups of Linear Operators & Applications, Dover Publications, Inc. (Second Edition), 2017.  Google Scholar [8] I. Lasiecka, J. L. Lions and R. Triggiani, Non homogeneous boundary value problems for second order hyperbolic operators, J Math Pures et Appl., 65 (1986), 149-192.   Google Scholar [9] S. Nicaise and K. Laoubi, Polynomial stabilization of the wave equation with Ventcel's boundary conditions, Math. Nachr., 283 (2010), 1428-1438.  doi: 10.1002/mana.200710162.  Google Scholar [10] G. Ruiz Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Diff. Eq., 11 (2006), 457-480.   Google Scholar [11] H. Tanabe, Equations of Evolution, Monographs and Studies in Mathematics, 6. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979.  Google Scholar [12] E. Vitillaro, On the wave equation with hyperbolic dynamical boundary conditions, interior and boundary damping and supercritical sources, Journ. Diff. Eq., 265 (2018), 4873-4941.  doi: 10.1016/j.jde.2018.06.022.  Google Scholar
 [1] Giuseppe Maria Coclite, Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Continuous dependence in hyperbolic problems with Wentzell boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (1) : 419-433. doi: 10.3934/cpaa.2014.13.419 [2] Davide Guidetti. Parabolic problems with general Wentzell boundary conditions and diffusion on the boundary. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1401-1417. doi: 10.3934/cpaa.2016.15.1401 [3] Xiaoyu Fu. Stabilization of hyperbolic equations with mixed boundary conditions. Mathematical Control & Related Fields, 2015, 5 (4) : 761-780. doi: 10.3934/mcrf.2015.5.761 [4] 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 [5] Davide Guidetti. Classical solutions to quasilinear parabolic problems with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 717-736. doi: 10.3934/dcdss.2016024 [6] Ángela Jiménez-Casas, Aníbal Rodríguez-Bernal. Dynamic boundary conditions as limit of singularly perturbed parabolic problems. Conference Publications, 2011, 2011 (Special) : 737-746. doi: 10.3934/proc.2011.2011.737 [7] Genni Fragnelli, Gisèle Ruiz Goldstein, Jerome Goldstein, Rosa Maria Mininni, Silvia Romanelli. Generalized Wentzell boundary conditions for second order operators with interior degeneracy. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 697-715. doi: 10.3934/dcdss.2016023 [8] Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Enrico Obrecht, Silvia Romanelli. Nonsymmetric elliptic operators with Wentzell boundary conditions in general domains. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2475-2487. doi: 10.3934/cpaa.2016045 [9] Mahamadi Warma. Semi linear parabolic equations with nonlinear general Wentzell boundary conditions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5493-5506. doi: 10.3934/dcds.2013.33.5493 [10] Wen-Qing Xu. Boundary conditions for multi-dimensional hyperbolic relaxation problems. Conference Publications, 2003, 2003 (Special) : 916-925. doi: 10.3934/proc.2003.2003.916 [11] Ciprian G. Gal, M. Grasselli. On the asymptotic behavior of the Caginalp system with dynamic boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 689-710. doi: 10.3934/cpaa.2009.8.689 [12] Lahcen Maniar, Martin Meyries, Roland Schnaubelt. Null controllability for parabolic equations with dynamic boundary conditions. Evolution Equations & Control Theory, 2017, 6 (3) : 381-407. doi: 10.3934/eect.2017020 [13] V. Casarino, K.-J. Engel, G. Nickel, S. Piazzera. Decoupling techniques for wave equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 761-772. doi: 10.3934/dcds.2005.12.761 [14] József Z. Farkas, Peter Hinow. Physiologically structured populations with diffusion and dynamic boundary conditions. Mathematical Biosciences & Engineering, 2011, 8 (2) : 503-513. doi: 10.3934/mbe.2011.8.503 [15] Robert Denk, Yoshihiro Shibata. Generation of semigroups for the thermoelastic plate equation with free boundary conditions. Evolution Equations & Control Theory, 2019, 8 (2) : 301-313. doi: 10.3934/eect.2019016 [16] Alassane Niang. Boundary regularity for a degenerate elliptic equation with mixed boundary conditions. Communications on Pure & Applied Analysis, 2019, 18 (1) : 107-128. doi: 10.3934/cpaa.2019007 [17] Gabriele Bonanno, Giuseppina D'Aguì. Mixed elliptic problems involving the $p-$Laplacian with nonhomogeneous boundary conditions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5797-5817. doi: 10.3934/dcds.2017252 [18] Paul Sacks, Mahamadi Warma. Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1$-data. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 761-787. doi: 10.3934/dcds.2014.34.761 [19] Hung Le. Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3357-3385. doi: 10.3934/dcds.2018144 [20] Silvia Romanelli. Goldstein-Wentzell boundary conditions: Recent results with Jerry and Gisèle Goldstein. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 749-760. doi: 10.3934/dcds.2014.34.749

2019 Impact Factor: 1.233

## Metrics

• PDF downloads (47)
• HTML views (215)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]