Strong solution for singularly nonautonomous evolution equation with almost sectorial operators

Abstract

<p style='text-indent:20px;'>In this paper we consider the singularly nonautonomous evolution problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_t +A(t) u = f(t), \mbox{ } \tau<t<\tau+T; \quad u(\tau) = u_0 \in X, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>associated with a family of uniformly almost sectorial linear operators <inline-formula><tex-math id="M1">\begin{document}$ A(t):D\subset X \rightarrow X $\end{document}</tex-math></inline-formula>, that is, a family for which a sector of the complex plane is contained in the resolvent of <inline-formula><tex-math id="M2">\begin{document}$ -A(t) $\end{document}</tex-math></inline-formula> and satisfies <inline-formula><tex-math id="M3">\begin{document}$ \|(\lambda+A(t))^{-1}\|_{\mathcal{L} (X)} \leq \frac{C}{|\lambda|^{\alpha}} $\end{document}</tex-math></inline-formula>, for some <inline-formula><tex-math id="M4">\begin{document}$ \alpha \in (0, 1) $\end{document}</tex-math></inline-formula>, uniformly in <inline-formula><tex-math id="M5">\begin{document}$ t $\end{document}</tex-math></inline-formula>. Under a proper condition on the value of <inline-formula><tex-math id="M6">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> we prove that the linear process associated to the family <inline-formula><tex-math id="M7">\begin{document}$ A(t) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ t\in \mathbb{R} $\end{document}</tex-math></inline-formula>, is strongly differentiable and that the singularly nonautonomous problem has a strong solution. An example of a singularly nonautonomous reaction-diffusion equation in a domain with a handle illustrates the abstracts results obtained.</p>