# American Institute of Mathematical Sciences

December  2020, 13(12): 3461-3471. doi: 10.3934/dcdss.2020239

## 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, 2020, 13 (12) : 3461-3471. 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
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