Please use this identifier to cite or link to this item: http://hdl.handle.net/10077/4200
Title: Pointwise versions of solutions to Cauchy problems in $L^p$-spaces
Authors: Desch, Wolfgang
Homan, Krista W.
Issue Date: 2002
Publisher: Università degli Studi di Trieste. Dipartimento di Scienze Matematiche
Source: W. Desch and K. W. Homan, "Pointwise versions of solutions to Cauchy problems in $L^p$-spaces", in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 34 (2002), pp. 121-142.
Series/Report no.: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics
34 (2002)
Abstract: 
We consider a cauchy problem
\[
\begin{array}{cc}
\frac{\partial}{\partial t}\varphi\left(t,\omega\right)=\left(\mathcal{A\varphi\left(\mathit{t,\cdot}\right)}\right)\left(\omega\right),t>0 & \omega\epsilon\Omega\\
\varphi\left(0,\omega\right)=\varphi_{0}\left(\omega\right), & \omega\epsilon\Omega
\end{array}
\]
and assume that it can be solved by a strongly continuous semigroup
on a Banach space valued function space $L^{p}\left(\Omega,X\right)$.
For fixed t > 0 the solution $\varphi\left(t,\omega\right)$ is only
defined almost everywhere on $\Omega$. Therefore it is not obvious
what kind of regularity of $t\mapsto\varphi\left(t,\omega\right)$
has for fixed $\omega\;\epsilon\;\Omega$. We show that if the semigroup
is analityc, then there exists a version of $\varphi\left(t,\cdot\right)$
such that for almost every $\omega\;\epsilon\;\Omega$, $t\mapsto\varphi\left(t,\omega\right)$
is analityc in $\left(0,\infty\right)$.
Type: Article
URI: http://hdl.handle.net/10077/4200
ISSN: 0049-4704
Appears in Collections:Rendiconti dell'Istituto di matematica dell'Università di Trieste: an International Journal of Mathematics vol.34 (2002)

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