# American Institute of Mathematical Sciences

June  2016, 9(3): 717-736. doi: 10.3934/dcdss.2016024

## Classical solutions to quasilinear parabolic problems with dynamic boundary conditions

 1 Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy

Received  November 2014 Published  April 2016

We study linear nonautonomous parabolic systems with dynamic boundary conditions. Next, we apply these results to show a theorem of local existence and uniqueness of a classical solution to a second order quasilinear system with nonlinear dynamic boundary conditions.
Citation: Davide Guidetti. Classical solutions to quasilinear parabolic problems with dynamic boundary conditions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 717-736. doi: 10.3934/dcdss.2016024
##### References:
 [1] G. Coclite, G. Ruiz Goldstein and J. A. Goldstein, Stability of parabolic problems with nonlinear Wentzell boundary conditions, J. Differential Equations, 246 (2009), 2434-2447. doi: 10.1016/j.jde.2008.10.004. [2] G. Coclite, G. Ruiz Goldstein and J. A. Goldstein, Well-posedness of nonlinear parabolic problems with nonlinear Wentzell boundary conditions, Adv. Differential Equations, 16 (2011), 895-916. [3] J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations, 18 (1993), 1309-1364. doi: 10.1080/03605309308820976. [4] A. Favini, G. Ruiz Goldstein, J. Goldstein and S. Romanelli, Nonlinear boundary conditions for nonlinear second order differential operators on $C[0, 1]$, Arch. Math. (Basel), 76 (2001), 391-400. doi: 10.1007/PL00000449. [5] C. Gal and M. Warma, Well posedness and the global attractor of some quasi-linear parabolic equations with nonlinear dynamic boundary conditions, Differential Integral Equations, 23 (2010), 327-358. [6] D. Guidetti, Linear parabolic problems with dynamic boundary conditions in spaces of H\"older continuous functions, Ann. Mat. Pura Appl. (4), 195 (2016), 167-198. doi: 10.1007/s10231-014-0457-8. [7] T. Hintermann, Evolution equations with dynamic boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 43-60. doi: 10.1017/S0308210500023945. [8] J. Kakŭr, Nonlinear parabolic equations with the mixed nonlinear and nonstationary boundary conditions, Math. Slovaca, 30 (1980), 213-237. [9] M. Warma, The Robin and Wentzell-Robin Laplacians on Lipschitz domains, Semigroup Forum, 73 (2006), 10-30. doi: 10.1007/s00233-006-0617-2. [10] M. Warma, Quasilinear parabolic equations with nonlinear Wentzell-Robin type boundary conditions, J. Math. Anal. Appl., 336 (2007), 1132-1148. doi: 10.1016/j.jmaa.2007.03.050.

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##### References:
 [1] G. Coclite, G. Ruiz Goldstein and J. A. Goldstein, Stability of parabolic problems with nonlinear Wentzell boundary conditions, J. Differential Equations, 246 (2009), 2434-2447. doi: 10.1016/j.jde.2008.10.004. [2] G. Coclite, G. Ruiz Goldstein and J. A. Goldstein, Well-posedness of nonlinear parabolic problems with nonlinear Wentzell boundary conditions, Adv. Differential Equations, 16 (2011), 895-916. [3] J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations, 18 (1993), 1309-1364. doi: 10.1080/03605309308820976. [4] A. Favini, G. Ruiz Goldstein, J. Goldstein and S. Romanelli, Nonlinear boundary conditions for nonlinear second order differential operators on $C[0, 1]$, Arch. Math. (Basel), 76 (2001), 391-400. doi: 10.1007/PL00000449. [5] C. Gal and M. Warma, Well posedness and the global attractor of some quasi-linear parabolic equations with nonlinear dynamic boundary conditions, Differential Integral Equations, 23 (2010), 327-358. [6] D. Guidetti, Linear parabolic problems with dynamic boundary conditions in spaces of H\"older continuous functions, Ann. Mat. Pura Appl. (4), 195 (2016), 167-198. doi: 10.1007/s10231-014-0457-8. [7] T. Hintermann, Evolution equations with dynamic boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 43-60. doi: 10.1017/S0308210500023945. [8] J. Kakŭr, Nonlinear parabolic equations with the mixed nonlinear and nonstationary boundary conditions, Math. Slovaca, 30 (1980), 213-237. [9] M. Warma, The Robin and Wentzell-Robin Laplacians on Lipschitz domains, Semigroup Forum, 73 (2006), 10-30. doi: 10.1007/s00233-006-0617-2. [10] M. Warma, Quasilinear parabolic equations with nonlinear Wentzell-Robin type boundary conditions, J. Math. Anal. Appl., 336 (2007), 1132-1148. doi: 10.1016/j.jmaa.2007.03.050.
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