A note on critical point and blow-up rates for singular and degenerate parabolic equations

‎In this paper‎, ‎we consider singular and degenerate parabolic‎ ‎equations‎ ‎$$‎ ‎u_t =(x^\alpha u_x)_x‎ +‎u^m (x_0,t)v^{n} (x_0,t),\quad‎ ‎v_t =(x^\beta v_x)_x‎ +‎u^q (x_0,t)v^{p} (x_0,t),$$ in $(0,a)\times (0,T)$‎, ‎subject to null‎ ‎Dirichlet boundary conditions‎, ‎where $m,n‎, ‎p,q\ge 0$‎, ‎$\alpha‎, ‎\beta\in [0,2)$ and $x_0\in (0,a)$‎. ‎The optimal classification of‎ ‎non-simultaneous and simultaneous blow-up solutions is determined‎. ‎Additionally‎, ‎we obtain blow-up rates and sets for the solutions‎. ‎The singular rates for the derivation of the solutions are given.