Stability Results for Evolution Equations Involving Superposition Operators of Mixed Fractional Orders

The present investigation examines some forward and backward problems for the evolution equations involving superposition operators of mixed fractional orders which was introduced recently by Dipierro et al. in a series of works. This potentially marks the first instance of such an investigation. We present a result concerning the existence, uniqueness, and decay estimate of the solution for the forward problem. For the backward problem, we provide some stability estimates within the Sobolev space HpRd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H<^>p}\left( {{\mathbb {R}<^>d}} \right) $$\end{document} and prove that the problem is ill-posed at the initial time t=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=0$$\end{document}. A general regularization procedure is further formulated to conquer this ill-posedness. These works have been done by examining certain essential characteristics of the Kilbas-Saigo function together with using appropriate estimates