Inhomogeneous parabolic Neumann problems

Robin Nittka

Czechoslovak Mathematical Journal (2014)

  • Volume: 64, Issue: 3, page 703-742
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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Nittka, Robin. "Inhomogeneous parabolic Neumann problems." Czechoslovak Mathematical Journal 64.3 (2014): 703-742. <http://eudml.org/doc/262147>.

@article{Nittka2014,
abstract = {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.},
author = {Nittka, Robin},
journal = {Czechoslovak Mathematical Journal},
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; parabolic initial-boundary value problem; inhomogeneous Neumann boundary conditions; Robin boundary conditions; existence of weak solutions; continuity up to the boundary; asymptotic behavior; asymptotically almost periodic solutions},
language = {eng},
number = {3},
pages = {703-742},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Inhomogeneous parabolic Neumann problems},
url = {http://eudml.org/doc/262147},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Nittka, Robin
TI - Inhomogeneous parabolic Neumann problems
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 3
SP - 703
EP - 742
AB - 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.
LA - eng
KW - parabolic initial-boundary value problem; inhomogeneous Robin boundary conditions; existence of weak solution; continuity up to the boundary; asymptotic behavior; asymptotically almost periodic solution; parabolic initial-boundary value problem; inhomogeneous Neumann boundary conditions; Robin boundary conditions; existence of weak solutions; continuity up to the boundary; asymptotic behavior; asymptotically almost periodic solutions
UR - http://eudml.org/doc/262147
ER -

References

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  1. Arendt, W., Resolvent positive operators and inhomogeneous boundary conditions, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 29 (2000), 639-670. (2000) Zbl1072.35077MR1817713
  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). (2001) Zbl0978.34001MR1886588
  3. Arendt, W., Chovanec, M., 10.1090/S0002-9947-2010-05077-9, Trans. Am. Math. Soc. 362 (2010), 5861-5878. (2010) Zbl1205.35139MR2661499DOI10.1090/S0002-9947-2010-05077-9
  4. Arendt, W., Nittka, R., 10.1007/s00013-009-3190-6, Arch. Math. 92 (2009), 414-427. (2009) Zbl1182.46006MR2506943DOI10.1007/s00013-009-3190-6
  5. Arendt, W., Schätzle, R., Semigroups generated by elliptic operators in non-divergence form on C 0 ( Ω ) , Ann. Sc. Norm. Super. Pisa, Cl. Sci. 13 (2014), 417-434. (2014) MR3235521
  6. Bohr, H., Almost Periodic Functions, Chelsea Publishing Company, New York (1947). (1947) MR0020163
  7. Bošković, D. M., Krstić, M., Liu, W., 10.1109/9.975513, IEEE Trans. Autom. Control 46 (2001), 2022-2028. (2001) Zbl1006.93039MR1878234DOI10.1109/9.975513
  8. Cannarsa, P., Gozzi, F., Soner, H. M., 10.1006/jfan.1993.1122, J. Funct. Anal. 117 (1993), 25-61. (1993) Zbl0823.49017MR1240261DOI10.1006/jfan.1993.1122
  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. (1999) Zbl1044.35527MR1710605
  10. Daners, D., 10.1002/1522-2616(200009)217:1<13::AID-MANA13>3.0.CO;2-6, Math. Nachr. 217 (2000), 13-41. (2000) Zbl0973.35087MR1780769DOI10.1002/1522-2616(200009)217:1<13::AID-MANA13>3.0.CO;2-6
  11. Dautray, R., Lions, J.-L., Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 5: Evolution Problems I, Springer, Berlin (1992). (1992) Zbl0755.35001MR1156075
  12. Denk, R., Hieber, M., Prüss, J., 10.1007/s00209-007-0120-9, Math. Z. 257 (2007), 193-224. (2007) Zbl1210.35066MR2318575DOI10.1007/s00209-007-0120-9
  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). (1993) MR1225809
  14. Engel, K.-J., 10.1007/s00013-003-0557-y, Arch. Math. 81 (2003), 548-558. (2003) Zbl1064.47043MR2029716DOI10.1007/s00013-003-0557-y
  15. Engel, K.-J., Nagel, R., One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics 194 Springer, Berlin (2000). (2000) Zbl0952.47036MR1721989
  16. Gilbarg, D., Trudinger, N. S., Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 ed, Classics in Mathematics Springer, Berlin (2001). (2001) Zbl1042.35002MR1814364
  17. Griepentrog, J. A., Maximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces, Adv. Differ. Equ. 12 (2007), 1031-1078. (2007) Zbl1157.35023MR2351837
  18. Haller-Dintelmann, R., Rehberg, J., 10.1016/j.jde.2009.06.001, J. Differ. Equations 247 (2009), 1354-1396. (2009) Zbl1178.35210MR2541414DOI10.1016/j.jde.2009.06.001
  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). (1968) MR0241822
  20. Lieberman, G. M., Second Order Parabolic Differential Equations, World Scientific Singapore (1996). (1996) Zbl0884.35001MR1465184
  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). (1972) Zbl0227.35001MR0350178
  22. Nittka, R., 10.1007/s00030-012-0201-2, NoDEA, Nonlinear Differ. Equ. Appl. 20 (2013), 1125-1155. (2013) Zbl1268.35021MR3057169DOI10.1007/s00030-012-0201-2
  23. Nittka, R., 10.1016/j.jde.2011.05.019, J. Differ. Equations 251 (2011), 860-880. (2011) Zbl1225.35077MR2812574DOI10.1016/j.jde.2011.05.019
  24. Nittka, R., Elliptic and parabolic problems with Robin boundary conditions on Lipschitz domains, PhD Thesis, University of Ulm (2010). (2010) 
  25. Stepanoff, W., 10.1007/BF01206623, Math. Ann. 95 (1926), 473-498 German. (1926) MR1512290DOI10.1007/BF01206623
  26. Warma, M., 10.1007/s00233-006-0617-2, Semigroup Forum 73 (2006), 10-30. (2006) Zbl1168.35340MR2277314DOI10.1007/s00233-006-0617-2

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