Geodesic
Cauchy problems in weighted Lebesgue spaces
Czechoslovak Mathematical Journal, Tome 54 (2004) no. 4, pp. 991-1013
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 35B40, 35B41, 35K15, 35K55, 37L30, 47H20, 47N20
Keywords: Cauchy problem; parabolic equation; global existence; asymptotic behavior of solutions 
@article{CMJ_2004_54_4_a14,
     author = {Cholewa, Jan W. and Dlotko, Tomasz},
     title = {Cauchy problems in weighted {Lebesgue} spaces},
     journal = {Czechoslovak Mathematical Journal},
     pages = {991--1013},
     year = {2004},
     volume = {54},
     number = {4},
     mrnumber = {2099352},
     zbl = {1080.35033},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2004_54_4_a14/}
}
TY  - JOUR
AU  - Cholewa, Jan W.
AU  - Dlotko, Tomasz
TI  - Cauchy problems in weighted Lebesgue spaces
JO  - Czechoslovak Mathematical Journal
PY  - 2004
SP  - 991
EP  - 1013
VL  - 54
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/CMJ_2004_54_4_a14/
LA  - en
ID  - CMJ_2004_54_4_a14
ER  - 
%0 Journal Article
%A Cholewa, Jan W.
%A Dlotko, Tomasz
%T Cauchy problems in weighted Lebesgue spaces
%J Czechoslovak Mathematical Journal
%D 2004
%P 991-1013
%V 54
%N 4
%U http://geodesic.mathdoc.fr/item/CMJ_2004_54_4_a14/
%G en
%F CMJ_2004_54_4_a14
Cholewa, Jan W.; Dlotko, Tomasz. Cauchy problems in weighted Lebesgue spaces. Czechoslovak Mathematical Journal, Tome 54 (2004) no. 4, pp. 991-1013. http://geodesic.mathdoc.fr/item/CMJ_2004_54_4_a14/

[1] R. A. Adams: Sobolev Spaces. Academic Press, New York, 1975. | MR | Zbl

[2] H. Amann: Linear and Quasilinear Parabolic Problems. Birkhäuser, Basel, 1995. | MR | Zbl

[3] H. Amann, M. Hieber and G. Simonett: Bounded $H_\infty $-calculus for elliptic operators. Diff. Int. Equations 3 (1994), 613–653. | MR

[4] J. M. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodriguez-Bernal: Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains. Nonlinear Anal. TMA56 (2004), 515–554. | MR

[5] A. V. Babin and M. I. Vishik: Attractors of partial differential evolution equations in unbounded domain. Proc. Roy. Soc. Edinburgh 116 (1990), 221–243. | MR

[6] A. V. Babin and M. I. Vishik: Attractors of Evolution Equations. North-Holland, Amsterdam, 1992. | MR

[7] J. W. Cholewa and T. Dlotko: Cauchy problem with subcritical nonlinearity. J. Math. Anal. Appl. 210 (1997), 531–548. | DOI | MR

[8] J. W. Cholewa and T. Dlotko: Global Attractors in Abstract Parabolic Problems. Cambridge University Press, Cambridge, 2000. | MR

[9] M. A. Efendiev and S. V. Zelik: The attractor for a nonlinear reaction-diffusion system in an unbounded domain. Comm. Pure Appl. Math. 54 (2001), 625–688. | DOI | MR

[10] E. Feireisl: Bounded, locally compact global attractors for semilinear damped wave equations on $R^N$. Differential Integral Equations 9 (1996), 1147–1156. | MR

[11] J. K. Hale: Asymptotic Behavior of Dissipative Systems. AMS, Providence, RI, 1988. | MR | Zbl

[12] D. Henry: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics 840. Springer, Berlin, 1981. | MR

[13] A. Lunardi: Analytic Semigroup and Optimal Regularity in Parabolic Problems. Birkhäuser, Berlin, 1995. | MR

[14] S. Merino: On the existence of the compact global attractor for semilinear reaction diffusion systems on $R^N$. J. Differential Equations 132 (1996), 87–106. | DOI | MR

[15] A. Mielke: The complex Ginzburg-Landau equation on large and unbounded domains: sharper bounds and attractors. Nonlinearity 10 (1997), 199–222. | MR | Zbl

[16] A. Mielke and G. Schneider: Attractors for modulation equations on unbounded domains–existence and comparison. Nonlinearity 8 (1995), 743–768. | DOI | MR

[17] S. M. Nikolsky: A Course of Mathematical Analysis. Mir Publishers, Moscow, 1987.

[18] R. Temam: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York, 1988. | MR | Zbl

[19] H. Triebel: Interpolation Theory, Function Spaces, Differential Operators. VEB Deutscher, Berlin, 1978. | MR | Zbl