A nonlocal Kirchhoff diffusion problem with singular potential and logarithmic nonlinearity.

In this paper, we investigate the following fractional Kirchhoff‐type pseudo parabolic equation driven by a nonlocal integro‐differential operator ℒK$$ {\mathcal{L}}_K $$: ut|x|2s+M([u]s2)ℒKu+ℒKut=|u|p−2ulog|u|,$$ \frac{u_t}{{\left|x\right|}^{2s}}+M\left({\left[u\right]}_s^2\right){\mathcal{L}}_Ku+{\mathcal{L}}_K{u}_t={\left|u\right|}^{p-2}u\log \mid u\mid, $$ where [u]s$$ {\left[u\right]}_s $$ represents the Gagliardo seminorm of u$$ u $$. Instead of imposing specific assumptions on the Kirchhoff function, we introduce a more general sense to establish the local existence of weak solutions. Moreover, via the sharp fractional Hardy inequality, the decay estimates for global weak solutions, the blow‐up criterion, blow‐up rate, and the upper and lower bounds of the blow‐up time are derived. Lastly, we discuss the global existence and finite time blow‐up results with high initial energy. [ABSTRACT FROM AUTHOR]