# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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 (1975). Google Scholar [2] M. Biegert and M. Warma, The heat equation with nonlinear generalized Robin boundary conditions,, J. Differential Equations, 247 (2009), 1949. doi: 10.1016/j.jde.2009.07.017. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.jde.2008.10.004. Google Scholar [5] M. G. Crandall and T. M. Liggett, Generation of semigroups of nonlinear transformations on general Banach spaces,, Amer. J. Math, 93 (1971), 265. doi: 10.2307/2373376. Google Scholar [6] P. Drábek and J. Milota, "Methods of Nonlinear Analysis. Applications to Differential Equations,", Birkhäuser Advanced Texts, (2007). Google Scholar [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. doi: 10.1002/mana.200910086. Google Scholar [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. doi: 10.1007/s00028-002-8077-y. Google Scholar [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. Google Scholar [10] G. R. Goldstein, Derivation and physical interpretation of general boundary conditions,, Adv. Differential Equations, 11 (2006), 457. Google Scholar [11] M. K. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate elliptic operators,, Ann. Mat. Pura Appl., 80 (1968), 1. doi: 10.1007/BF02413623. Google Scholar [12] M. M. Rao and Z. D. Ren, "Applications of Orlicz Spaces,", Monographs and Textbooks in Pure and Applied Mathematics, 250 (2002). doi: 10.1201/9780203910863. Google Scholar [13] R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations,", Amer. Math. Soc., (1997). Google Scholar [14] M. Warma, Wentzell-Robin boundary conditions on $C[0,1]$,, Semigroup Forum, 66 (2003), 162. doi: 10.1007/s002330010124. Google Scholar [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. doi: 10.1016/j.na.2012.05.004. Google Scholar [16] M. Warma, Parabolic and elliptic problems with general Wentzell boundary conditions on Lipschitz domains,, Commun. Pure Appl. Anal., 12 (2013), 1881. doi: 10.3934/cpaa.2013.12.1881. Google Scholar

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##### References:
 [1] R. A. Adams, "Sobolev Spaces,", Pure and Applied Mathematics, 65 (1975). Google Scholar [2] M. Biegert and M. Warma, The heat equation with nonlinear generalized Robin boundary conditions,, J. Differential Equations, 247 (2009), 1949. doi: 10.1016/j.jde.2009.07.017. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.jde.2008.10.004. Google Scholar [5] M. G. Crandall and T. M. Liggett, Generation of semigroups of nonlinear transformations on general Banach spaces,, Amer. J. Math, 93 (1971), 265. doi: 10.2307/2373376. Google Scholar [6] P. Drábek and J. Milota, "Methods of Nonlinear Analysis. Applications to Differential Equations,", Birkhäuser Advanced Texts, (2007). Google Scholar [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. doi: 10.1002/mana.200910086. Google Scholar [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. doi: 10.1007/s00028-002-8077-y. Google Scholar [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. Google Scholar [10] G. R. Goldstein, Derivation and physical interpretation of general boundary conditions,, Adv. Differential Equations, 11 (2006), 457. Google Scholar [11] M. K. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate elliptic operators,, Ann. Mat. Pura Appl., 80 (1968), 1. doi: 10.1007/BF02413623. Google Scholar [12] M. M. Rao and Z. D. Ren, "Applications of Orlicz Spaces,", Monographs and Textbooks in Pure and Applied Mathematics, 250 (2002). doi: 10.1201/9780203910863. Google Scholar [13] R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations,", Amer. Math. Soc., (1997). Google Scholar [14] M. Warma, Wentzell-Robin boundary conditions on $C[0,1]$,, Semigroup Forum, 66 (2003), 162. doi: 10.1007/s002330010124. Google Scholar [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. doi: 10.1016/j.na.2012.05.004. Google Scholar [16] M. Warma, Parabolic and elliptic problems with general Wentzell boundary conditions on Lipschitz domains,, Commun. Pure Appl. Anal., 12 (2013), 1881. doi: 10.3934/cpaa.2013.12.1881. Google Scholar
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