Pointwise versions of solutions to Cauchy problems in $L^p$-spaces

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 $\left(0,\infty\right)$.