A new second critical exponent and life span for a quasilinear degenerate parabolic equation with weighted nonlocal sources

In this paper, we consider positive solutions of a Cauchy problem for the following quasilinear degenerate parabolic equation with weighted nonlocal sources: $u_{t}=\Delta_{p}u+ \left(\int_{\mathbb{R}^{N}}K(x)u^{q}(x, t)dx\right)^{\frac{r-1}{q}}u^{s+1}, (x, t) \in \mathbb{R}^{N} \times(0, T), $ where $N≥1$, $p>2$, $q$, $r≥1$, $s≥0$, and $r+s>1$. We classify global and non-global solutions of the equation in the coexistence region by finding a new second critical exponent via the slow decay asymptotic behavior of an initial value at spatial infinity, and the life span of non-global solution is studied.