Geodesic
Cauchy problems in weighted Lebesgue spaces
Czechoslovak Mathematical Journal, Tome 54 (2004) no. 4, pp. 991-1013.

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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},
     publisher = {mathdoc},
     volume = {54},
     number = {4},
     year = {2004},
     mrnumber = {2099352},
     zbl = {1080.35033},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2004__54_4_a14/}
}
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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/