Study on the diffusion fractional m-Laplacian with singular potential term

This paper addresses the questions of well-posedness to fractional m-Laplacian reaction diffusion equation with singular potential term and logarithmic nonlinearity: \(\begin{aligned} \left| x\right| ^{-2s}\partial _t u+(-\varDelta )_{m}^{s} u+ (-\varDelta )^{s} \partial _t u\!=\!u|u|^{-2} R(u), \end{aligned}\) $$\begin{array}{c}\hfill {\left|x\right|}^{-2s}{\partial }_{t}u+{(-\Delta )}_m^{s}u+{(-\Delta )}^s{\partial }_{t}u=u{|u|}^{-2}R(u),\end{array}$$ where \(R(u)=\left| u\right| ^{r}\ln (|u|)\) $R(u)={\left|u\right|}^{r}ln(|u|)$ . Guided by the made assumptions, we arrive at the conclusions of the local and global solvability of solutions within the framework of Galerkin approximation. In addition, this study considers weak solutions’ asymptotic stability and explosion in finite time. Significantly, we not only figure out the relationship between the non-local fractional operator and singular potential term, but generalize and improve earlier results in the literature.