Options
Pointwise versions of solutions to Cauchy problems in $L^p$-spaces
Desch, Wolfgang
Homan, Krista W.
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 $\left(0,\infty\right)$.
Series
Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics
34 (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.
Languages
en
File(s)