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On solutions to the heat equation with the initial condition in the Orlicz—Slobodetskii space

Published online by Cambridge University Press:  24 July 2014

Agnieszka Kałamajska  and
Mirosłav Krbec
Agnieszka Kałamajska
Affiliation:
Institute of Mathematics, Warsaw University, ul. Banacha 2, 02-097 Warsaw, Poland, [email protected]
Mirosłav Krbec
Affiliation:
Institute of Mathematics, Academy of Sciences of the Czech Republic, Žitná 25, CZ-115 67 Prague 1, Czech Republic
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Abstract

We study the boundary-value problem ũt = Δxũ(x,t), ũ(x, 0) = u(x), where x ∈ Ω, t ∈ (0,T), Ω ⊆ ℝn−1 is a bounded Lipschitz boundary domain, u belongs to a certain Orlicz–Slobodetskii space YR,R(Ω). Under certain assumptions on the Orlicz function R, we prove that the solution u belongs to the Orlicz–Sobolev space W1,R(Ω × (0,T)).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 144 , Issue 4 , August 2014 , pp. 787 - 807
Copyright
Copyright © Royal Society of Edinburgh 2014 

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