On the time-delayed anomalous diffusion equations with nonlocal initial conditions

Abstract

<p style='text-indent:20px;'>In this paper, we are interested in the existence of solutions to the anomalous diffusion equations with delay subjected to nonlocal initial condition:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \label{01} \begin{cases} \partial _t(k*(u-u_0)) +(- \Delta)^\sigma u = f(t,u,u_\rho) \; {\rm {in }}\ \mathbb R^+\times \Omega,\\ u\bigr |_{\partial \Omega} = 0\; {\rm {in }}\ \mathbb R^+\times \partial \Omega,\\ u(s)+g(u)(s) = \phi(s) \;{\rm {in }}\ \Omega, s\in [-h,0]. \end{cases} \notag \tag{1} \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded domain of <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>, the constant <inline-formula><tex-math id="M3">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula> is in <inline-formula><tex-math id="M4">\begin{document}$ (0,1] $\end{document}</tex-math></inline-formula>. Under appropriate assumptions on <inline-formula><tex-math id="M5">\begin{document}$ k $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ f,g $\end{document}</tex-math></inline-formula>, we obtain the existence of global solutions and decay mild solutions for (1). The tools used include theory of completely positive functions, resolvent operators, the technique of measures of noncompactness and some fixed point arguments in suitable function spaces. Two application examples with respect to the specific cases of the term <inline-formula><tex-math id="M7">\begin{document}$ k $\end{document}</tex-math></inline-formula> in (1) are presented.</p>