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 & 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.  doi: 10.1016/j.jde.2008.10.004.  Google Scholar

[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.   Google Scholar

[3]

J. Escher, Quasilinear parabolic systems with dynamical boundary conditions,, Comm. Partial Differential Equations, 18 (1993), 1309.  doi: 10.1080/03605309308820976.  Google Scholar

[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.  doi: 10.1007/PL00000449.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s10231-014-0457-8.  Google Scholar

[7]

T. Hintermann, Evolution equations with dynamic boundary conditions,, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 43.  doi: 10.1017/S0308210500023945.  Google Scholar

[8]

J. Kakŭr, Nonlinear parabolic equations with the mixed nonlinear and nonstationary boundary conditions,, Math. Slovaca, 30 (1980), 213.   Google Scholar

[9]

M. Warma, The Robin and Wentzell-Robin Laplacians on Lipschitz domains,, Semigroup Forum, 73 (2006), 10.  doi: 10.1007/s00233-006-0617-2.  Google Scholar

[10]

M. Warma, Quasilinear parabolic equations with nonlinear Wentzell-Robin type boundary conditions,, J. Math. Anal. Appl., 336 (2007), 1132.  doi: 10.1016/j.jmaa.2007.03.050.  Google Scholar

show all references

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.  doi: 10.1016/j.jde.2008.10.004.  Google Scholar

[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.   Google Scholar

[3]

J. Escher, Quasilinear parabolic systems with dynamical boundary conditions,, Comm. Partial Differential Equations, 18 (1993), 1309.  doi: 10.1080/03605309308820976.  Google Scholar

[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.  doi: 10.1007/PL00000449.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s10231-014-0457-8.  Google Scholar

[7]

T. Hintermann, Evolution equations with dynamic boundary conditions,, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 43.  doi: 10.1017/S0308210500023945.  Google Scholar

[8]

J. Kakŭr, Nonlinear parabolic equations with the mixed nonlinear and nonstationary boundary conditions,, Math. Slovaca, 30 (1980), 213.   Google Scholar

[9]

M. Warma, The Robin and Wentzell-Robin Laplacians on Lipschitz domains,, Semigroup Forum, 73 (2006), 10.  doi: 10.1007/s00233-006-0617-2.  Google Scholar

[10]

M. Warma, Quasilinear parabolic equations with nonlinear Wentzell-Robin type boundary conditions,, J. Math. Anal. Appl., 336 (2007), 1132.  doi: 10.1016/j.jmaa.2007.03.050.  Google Scholar

[1]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267

[2]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

[3]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[4]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[5]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[6]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[7]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

[8]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453

[9]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379

[10]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[11]

Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHum approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055

[12]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375

[13]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433

[14]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[15]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052

[16]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[17]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115

[18]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297

[19]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[20]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

2019 Impact Factor: 1.233

Metrics

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

Other articles
by authors

[Back to Top]