In the present paper, we are concerned with the analysis of the asymptotic characteristics of solutions within a new class of nonlinear fractional \(p(\textrm{x},.)-\) Kirchhoff parabolic equations as follows: \(\begin{aligned} \left\{ \begin{array}{lll} \displaystyle {\mathfrak {u}}_{t}+M \left( \left[ {\mathfrak {u}} \right] _{s}^{p(\textrm{x}, \textrm{y})}\right) {\mathfrak {L}}_{s}^{p(\textrm{x}, \textrm{y})} {\mathfrak {u}}= \vert {\mathfrak {u}}\vert ^{\sigma (\textrm{x})-2} {\mathfrak {u}} \log \vert {\mathfrak {u}}\vert ,\, \, & \text{ in }\quad & {\mathcal {U}} \times (0, T),\\ \displaystyle {\mathfrak {u}}(\textrm{x}, t)=0,\, \, & \text{ in }\quad & \partial {\mathcal {U}} \times (0, T),\\ \displaystyle {\mathfrak {u}}(\textrm{x}, 0)={\mathfrak {u}}_{0}(\textrm{x}), & \text{ in }& {\mathcal {U}}. \end{array} \right. \end{aligned}\) Using potential well theory, Gronwall’s inequality, and Komornik’s integral inequality, we give a classification of global existence and blow-up in finite time of solutions when the initial data satisfies different conditions. Moreover, we give two-sided estimates of asymptotic behavior when the diffusion term dominates the source.
