Separate variable blow-up patterns for a reaction–diffusion equation with critical weighted reaction

We study the separate variable blow-up patterns associated to following second order reaction–diffusion equation: ?tu=?um+|x|?um,posed for x?RN, t?0, where m>1, dimension N?2 and ?>0. A new explicit critical exponent ?c=2(m?1)(N?1)3m+1is introduced a classification of profiles is given. The most interesting contribution paper showing that existence behavior split into different regimes by ?c also depends strongly on whether N?4 or N?{2,3}. These results extend previous works authors in N=1.