Cauchy problems in weighted Lebesgue spaces

Jan W. Cholewa; Tomasz Dłotko

Czechoslovak Mathematical Journal (2004)

  • Volume: 54, Issue: 4, page 991-1013
  • ISSN: 0011-4642

Abstract

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Global solvability and asymptotics of semilinear parabolic Cauchy problems in ℝ n are considered. Following the approach of A. Mielke [15] these problems are investigated in weighted Sobolev spaces. The paper provides also a theory of second order elliptic operators in such spaces considered over ℝ n , n ∈ ℕ . In particular, the generation of analytic semigroups and the embeddings for the domains of fractional powers of elliptic operators are discussed.

How to cite

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Cholewa, Jan W., and Dłotko, Tomasz. "Cauchy problems in weighted Lebesgue spaces." Czechoslovak Mathematical Journal 54.4 (2004): 991-1013. <http://eudml.org/doc/30917>.

@article{Cholewa2004,
abstract = {Global solvability and asymptotics of semilinear parabolic Cauchy problems in $\mathbb \{R\}^n$ are considered. Following the approach of A. Mielke [15] these problems are investigated in weighted Sobolev spaces. The paper provides also a theory of second order elliptic operators in such spaces considered over $\mathbb \{R\}^n$, $n\in \mathbb \{N\}$. In particular, the generation of analytic semigroups and the embeddings for the domains of fractional powers of elliptic operators are discussed.},
author = {Cholewa, Jan W., Dłotko, Tomasz},
journal = {Czechoslovak Mathematical Journal},
keywords = {Cauchy problem; parabolic equation; global existence; asymptotic behavior of solutions; Cauchy problem; parabolic equation; global existence; asymptotic behavior of solutions},
language = {eng},
number = {4},
pages = {991-1013},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Cauchy problems in weighted Lebesgue spaces},
url = {http://eudml.org/doc/30917},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Cholewa, Jan W.
AU - Dłotko, Tomasz
TI - Cauchy problems in weighted Lebesgue spaces
JO - Czechoslovak Mathematical Journal
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 4
SP - 991
EP - 1013
AB - Global solvability and asymptotics of semilinear parabolic Cauchy problems in $\mathbb {R}^n$ are considered. Following the approach of A. Mielke [15] these problems are investigated in weighted Sobolev spaces. The paper provides also a theory of second order elliptic operators in such spaces considered over $\mathbb {R}^n$, $n\in \mathbb {N}$. In particular, the generation of analytic semigroups and the embeddings for the domains of fractional powers of elliptic operators are discussed.
LA - eng
KW - Cauchy problem; parabolic equation; global existence; asymptotic behavior of solutions; Cauchy problem; parabolic equation; global existence; asymptotic behavior of solutions
UR - http://eudml.org/doc/30917
ER -

References

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  1. Sobolev Spaces, Academic Press, New York, 1975. (1975) Zbl0314.46030MR0450957
  2. Linear and Quasilinear Parabolic Problems, Birkhäuser, Basel, 1995. (1995) Zbl0819.35001MR1345385
  3. Bounded H ∞ -calculus for elliptic operators, Diff. Int. Equations 3 (1994), 613–653. (1994) MR1270095
  4. Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains, Nonlinear Anal. TMA56 (2004), 515–554. (2004) MR2035325
  5. Attractors of partial differential evolution equations in unbounded domain, Proc. Roy. Soc. Edinburgh 116 (1990), 221–243. (1990) MR1084733
  6. Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. (1992) MR1156492
  7. 10.1006/jmaa.1997.5409, J. Math. Anal. Appl. 210 (1997), 531–548. (1997) MR1453190DOI10.1006/jmaa.1997.5409
  8. Global Attractors in Abstract Parabolic Problems, Cambridge University Press, Cambridge, 2000. (2000) MR1778284
  9. 10.1002/cpa.1011, Comm. Pure Appl. Math. 54 (2001), 625–688. (2001) MR1815444DOI10.1002/cpa.1011
  10. Bounded, locally compact global attractors for semilinear damped wave equations on R N , Differential Integral Equations 9 (1996), 1147–1156. (1996) MR1392099
  11. Asymptotic Behavior of Dissipative Systems, AMS, Providence, RI, 1988. (1988) Zbl0642.58013MR0941371
  12. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics 840, Springer, Berlin, 1981. (1981) MR0610244
  13. Analytic Semigroup and Optimal Regularity in Parabolic Problems, Birkhäuser, Berlin, 1995. (1995) MR1329547
  14. 10.1006/jdeq.1996.0172, J. Differential Equations 132 (1996), 87–106. (1996) MR1418501DOI10.1006/jdeq.1996.0172
  15. The complex Ginzburg-Landau equation on large and unbounded domains: sharper bounds and attractors, Nonlinearity 10 (1997), 199–222. (1997) Zbl0905.35043MR1430749
  16. 10.1088/0951-7715/8/5/006, Nonlinearity 8 (1995), 743–768. (1995) MR1355041DOI10.1088/0951-7715/8/5/006
  17. A Course of Mathematical Analysis, Mir Publishers, Moscow, 1987. (1987) 
  18. Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1988. (1988) Zbl0662.35001MR0953967
  19. Interpolation Theory, Function Spaces, Differential Operators, VEB Deutscher, Berlin, 1978. (1978) Zbl0387.46033MR0503903

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