November  2013, 33(11&12): 5493-5506. doi: 10.3934/dcds.2013.33.5493

Semi linear parabolic equations with nonlinear general Wentzell boundary conditions

1. 

University of Puerto Rico, Rio Piedras Campus, Department of Mathematics, P.O. Box 70377, San Juan PR 00936-8377, United States

Received  October 2011 Revised  October 2012 Published  May 2013

Let $A$ be a uniformly elliptic operator in divergence form with bounded coefficients. We show that on a bounded domain $Ω ⊂ \mathbb{R}^N$ with Lipschitz continuous boundary $∂Ω$, a realization of $Au-\beta_1(x,u)$ in $C(\bar{Ω})$ with the nonlinear general Wentzell boundary conditions $[Au-\beta_1(x,u)]|_{∂Ω}-\Delta_\Gamma u+\partial_\nu^au+\beta_2(x,u)= 0$ on $∂Ω$ generates a strongly continuous nonlinear semigroup on $C(\bar{Ω})$. Here, $\partial_\nu^au$ is the conormal derivative of $u$, and $\beta_1(x,\cdot)$ ($x \in Ω$), $\beta_2(x,\cdot)$ ($x \in ∂Ω$) are continuous on $\mathbb{R}$ satisfying a certain growth condition.
Citation: Mahamadi Warma. Semi linear parabolic equations with nonlinear general Wentzell boundary conditions. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5493-5506. doi: 10.3934/dcds.2013.33.5493
References:
[1]

R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, 65. Academic Press, New York, 1975.

[2]

M. Biegert and M. Warma, The heat equation with nonlinear generalized Robin boundary conditions, J. Differential Equations, 247 (2009), 1949-1979. doi: 10.1016/j.jde.2009.07.017.

[3]

M. Biegert and M. Warma, Some Quasi-linear elliptic Equations with inhomogeneous generalized Robin boundary conditions on "bad'' domains, Adv. Differential Equations, 15 (2010), 893-924.

[4]

G. M. Coclite, G. R. 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.

[5]

M. G. Crandall and T. M. Liggett, Generation of semigroups of nonlinear transformations on general Banach spaces, Amer. J. Math, 93 (1971), 265-298. doi: 10.2307/2373376.

[6]

P. Drábek and J. Milota, "Methods of Nonlinear Analysis. Applications to Differential Equations," Birkhäuser Advanced Texts, Birkhäuser, Basel, 2007.

[7]

A. Favini, G. R. Goldstein, J. A. Goldstein, E. Obrecht and S. Romanelli, Elliptic operators with general Wentzell boundary conditions, analytic semigroups and the angle concavity theorem, Math. Nachr., 283 (2010), 504-521. doi: 10.1002/mana.200910086.

[8]

A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with Wentzell boundary conditions, J. Evol. Eq., 2 (2002), 1-19. doi: 10.1007/s00028-002-8077-y.

[9]

A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with nonlinear general Wentzell boundary condition, Adv. Differential Equations, 11 (2006), 481-510.

[10]

G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480.

[11]

M. K. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate elliptic operators, Ann. Mat. Pura Appl., 80 (1968), 1-122. doi: 10.1007/BF02413623.

[12]

M. M. Rao and Z. D. Ren, "Applications of Orlicz Spaces," Monographs and Textbooks in Pure and Applied Mathematics, 250. Marcel Dekker, Inc., New York, 2002. doi: 10.1201/9780203910863.

[13]

R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations," Amer. Math. Soc., Providence, RI, 1997.

[14]

M. Warma, Wentzell-Robin boundary conditions on $C[0,1]$, Semigroup Forum, 66 (2003), 162-170. doi: 10.1007/s002330010124.

[15]

M. Warma, Regularity and well-posedness of some quasi-linear elliptic and parabolic problems with nonlinear general Wentzell boundary conditions on nonsmooth domains, Nonlinear Anal., 14 (2012), 5561-5588. doi: 10.1016/j.na.2012.05.004.

[16]

M. Warma, Parabolic and elliptic problems with general Wentzell boundary conditions on Lipschitz domains, Commun. Pure Appl. Anal., 12 (2013), 1881-1905. doi: 10.3934/cpaa.2013.12.1881.

show all references

References:
[1]

R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, 65. Academic Press, New York, 1975.

[2]

M. Biegert and M. Warma, The heat equation with nonlinear generalized Robin boundary conditions, J. Differential Equations, 247 (2009), 1949-1979. doi: 10.1016/j.jde.2009.07.017.

[3]

M. Biegert and M. Warma, Some Quasi-linear elliptic Equations with inhomogeneous generalized Robin boundary conditions on "bad'' domains, Adv. Differential Equations, 15 (2010), 893-924.

[4]

G. M. Coclite, G. R. 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.

[5]

M. G. Crandall and T. M. Liggett, Generation of semigroups of nonlinear transformations on general Banach spaces, Amer. J. Math, 93 (1971), 265-298. doi: 10.2307/2373376.

[6]

P. Drábek and J. Milota, "Methods of Nonlinear Analysis. Applications to Differential Equations," Birkhäuser Advanced Texts, Birkhäuser, Basel, 2007.

[7]

A. Favini, G. R. Goldstein, J. A. Goldstein, E. Obrecht and S. Romanelli, Elliptic operators with general Wentzell boundary conditions, analytic semigroups and the angle concavity theorem, Math. Nachr., 283 (2010), 504-521. doi: 10.1002/mana.200910086.

[8]

A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with Wentzell boundary conditions, J. Evol. Eq., 2 (2002), 1-19. doi: 10.1007/s00028-002-8077-y.

[9]

A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with nonlinear general Wentzell boundary condition, Adv. Differential Equations, 11 (2006), 481-510.

[10]

G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480.

[11]

M. K. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate elliptic operators, Ann. Mat. Pura Appl., 80 (1968), 1-122. doi: 10.1007/BF02413623.

[12]

M. M. Rao and Z. D. Ren, "Applications of Orlicz Spaces," Monographs and Textbooks in Pure and Applied Mathematics, 250. Marcel Dekker, Inc., New York, 2002. doi: 10.1201/9780203910863.

[13]

R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations," Amer. Math. Soc., Providence, RI, 1997.

[14]

M. Warma, Wentzell-Robin boundary conditions on $C[0,1]$, Semigroup Forum, 66 (2003), 162-170. doi: 10.1007/s002330010124.

[15]

M. Warma, Regularity and well-posedness of some quasi-linear elliptic and parabolic problems with nonlinear general Wentzell boundary conditions on nonsmooth domains, Nonlinear Anal., 14 (2012), 5561-5588. doi: 10.1016/j.na.2012.05.004.

[16]

M. Warma, Parabolic and elliptic problems with general Wentzell boundary conditions on Lipschitz domains, Commun. Pure Appl. Anal., 12 (2013), 1881-1905. doi: 10.3934/cpaa.2013.12.1881.

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