On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type

Abstract

<p style='text-indent:20px;'>We study for nonlinear Kirchhoff's model of pseudo parabolic type by considering its two different problems. <p style='text-indent:20px;'><inline-formula><tex-math id="M1">\begin{document}$ \bullet $\end{document}</tex-math></inline-formula> For initial value problem, we obtain the results on the existence and regularity of solutions. Moreover, we also prove that the solutions <inline-formula><tex-math id="M2">\begin{document}$ u $\end{document}</tex-math></inline-formula> corresponding with <inline-formula><tex-math id="M3">\begin{document}$ \beta &lt; 1 $\end{document}</tex-math></inline-formula> of the problem convergence to <inline-formula><tex-math id="M4">\begin{document}$ u $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M5">\begin{document}$ \beta = 1 $\end{document}</tex-math></inline-formula>. <p style='text-indent:20px;'><inline-formula><tex-math id="M6">\begin{document}$ \bullet $\end{document}</tex-math></inline-formula> For final value problem, we show that the ill-posed property in the sense of Hadamard is occurring. Using the Fourier truncation method to regularize the problem. We establish some stability estimates in the <inline-formula><tex-math id="M7">\begin{document}$ H^1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ L^p $\end{document}</tex-math></inline-formula> norms under some <i>a-priori</i> conditions on the sought solution.