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  5. Rendiconti dell'Istituto di Matematica dell'Università di Trieste: an International Journal of Mathematics vol.34 (2002)
  6. Pointwise versions of solutions to Cauchy problems in $L^p$-spaces
 
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Pointwise versions of solutions to Cauchy problems in $L^p$-spaces

Desch, Wolfgang
•
Homan, Krista W.
2002
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ISSN
0049-4704
http://hdl.handle.net/10077/4200
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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.
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