# Article

Full entry | PDF   (0.4 MB)
Keywords:
parabolic initial-boundary value problem; inhomogeneous Robin boundary conditions; existence of weak solution; continuity up to the boundary; asymptotic behavior; asymptotically almost periodic solution
Summary:
Second order parabolic equations on Lipschitz domains subject to inhomogeneous Neumann (or, more generally, Robin) boundary conditions are studied. Existence and uniqueness of weak solutions and their continuity up to the boundary of the parabolic cylinder are proved using methods from the theory of integrated semigroups, showing in particular the well-posedness of the abstract Cauchy problem in spaces of continuous functions. Under natural assumptions on the coefficients and the inhomogeneity the solutions are shown to converge to an equilibrium or to be asymptotically almost periodic.
References:
[1] Arendt, W.: Resolvent positive operators and inhomogeneous boundary conditions. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 29 (2000), 639-670. MR 1817713 | Zbl 1072.35077
[2] Arendt, W., Batty, C. J. K., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics 96 Birkhäuser, Basel (2001). MR 1886588 | Zbl 0978.34001
[3] Arendt, W., Chovanec, M.: Dirichlet regularity and degenerate diffusion. Trans. Am. Math. Soc. 362 (2010), 5861-5878. DOI 10.1090/S0002-9947-2010-05077-9 | MR 2661499 | Zbl 1205.35139
[4] Arendt, W., Nittka, R.: Equivalent complete norms and positivity. Arch. Math. 92 (2009), 414-427. DOI 10.1007/s00013-009-3190-6 | MR 2506943 | Zbl 1182.46006
[5] Arendt, W., Schätzle, R.: Semigroups generated by elliptic operators in non-divergence form on $C_0(\Omega)$. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 13 (2014), 417-434. MR 3235521
[6] Bohr, H.: Almost Periodic Functions. Chelsea Publishing Company, New York (1947). MR 0020163
[7] Bošković, D. M., Krstić, M., Liu, W.: Boundary control of an unstable heat equation via measurement of domain-averaged temperature. IEEE Trans. Autom. Control 46 (2001), 2022-2028. DOI 10.1109/9.975513 | MR 1878234 | Zbl 1006.93039
[8] Cannarsa, P., Gozzi, F., Soner, H. M.: A dynamic programming approach to nonlinear boundary control problems of parabolic type. J. Funct. Anal. 117 (1993), 25-61. DOI 10.1006/jfan.1993.1122 | MR 1240261 | Zbl 0823.49017
[9] Chapko, R., Kress, R., Yoon, J.-R.: An inverse boundary value problem for the heat equation: the Neumann condition. Inverse Probl. 15 (1999), 1033-1046. MR 1710605 | Zbl 1044.35527
[10] Daners, D.: Heat kernel estimates for operators with boundary conditions. Math. Nachr. 217 (2000), 13-41. DOI 10.1002/1522-2616(200009)217:1<13::AID-MANA13>3.0.CO;2-6 | MR 1780769 | Zbl 0973.35087
[11] Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 5: Evolution Problems I. Springer, Berlin (1992). MR 1156075 | Zbl 0755.35001
[12] Denk, R., Hieber, M., Prüss, J.: Optimal {$L^p$}-{$L^q$}-estimates for parabolic boundary value problems with inhomogeneous data. Math. Z. 257 (2007), 193-224. DOI 10.1007/s00209-007-0120-9 | MR 2318575 | Zbl 1210.35066
[13] Dore, G.: {$L^p$} regularity for abstract differential equations. Functional Analysis and Related Topics Proc. Int. Conference, Kyoto University, 1991. Lect. Notes Math. 1540 Springer, Berlin H. Komatsu 25-38 (1993). MR 1225809
[14] Engel, K.-J.: The Laplacian on {$C(\overline\Omega)$} with generalized Wentzell boundary conditions. Arch. Math. 81 (2003), 548-558. DOI 10.1007/s00013-003-0557-y | MR 2029716 | Zbl 1064.47043
[15] Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics 194 Springer, Berlin (2000). MR 1721989 | Zbl 0952.47036
[16] Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 ed. Classics in Mathematics Springer, Berlin (2001). MR 1814364 | Zbl 1042.35002
[17] Griepentrog, J. A.: Maximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces. Adv. Differ. Equ. 12 (2007), 1031-1078. MR 2351837 | Zbl 1157.35023
[18] Haller-Dintelmann, R., Rehberg, J.: Maximal parabolic regularity for divergence operators including mixed boundary conditions. J. Differ. Equations 247 (2009), 1354-1396. DOI 10.1016/j.jde.2009.06.001 | MR 2541414 | Zbl 1178.35210
[19] Ladyženskaja, O. A., Solonnikov, V. A., Ural'ceva, N. N.: Linear and Quasi-Linear Equations of Parabolic Type. Translations of Mathematical Monographs 23 American Mathematical Society, Providence (1968). MR 0241822
[20] Lieberman, G. M.: Second Order Parabolic Differential Equations. World Scientific Singapore (1996). MR 1465184 | Zbl 0884.35001
[21] Lions, J.-L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications. Vol. II. Die Grundlehren der mathematischen Wissenschaften, Band 182 Springer, Berlin (1972). MR 0350178 | Zbl 0227.35001
[22] Nittka, R.: Quasilinear elliptic and parabolic Robin problems on Lipschitz domains. NoDEA, Nonlinear Differ. Equ. Appl. 20 (2013), 1125-1155. DOI 10.1007/s00030-012-0201-2 | MR 3057169 | Zbl 1268.35021
[23] Nittka, R.: Regularity of solutions of linear second order elliptic and parabolic boundary value problems on Lipschitz domains. J. Differ. Equations 251 (2011), 860-880. DOI 10.1016/j.jde.2011.05.019 | MR 2812574 | Zbl 1225.35077
[24] Nittka, R.: Elliptic and parabolic problems with Robin boundary conditions on Lipschitz domains. PhD Thesis, University of Ulm (2010).
[25] Stepanoff, W.: Über einige Verallgemeinerungen der fast periodischen Funktionen. Math. Ann. 95 (1926), 473-498 German. DOI 10.1007/BF01206623 | MR 1512290
[26] Warma, M.: The Robin and Wentzell-Robin Laplacians on Lipschitz domains. Semigroup Forum 73 (2006), 10-30. DOI 10.1007/s00233-006-0617-2 | MR 2277314 | Zbl 1168.35340

Partner of