The balance between diffusion and absorption in semilinear parabolic equations

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Let h:[0,∞) [0,∞) be continuous and nondecreasing, h(t)>0 if t>0, and m,q be positive real numbers. We investigate the behavior when k→∞ of the fundamental solutions u=uₖ of ₜ u-Δuᵐ+h(t)uq=0 in Ω (0,T) satisfying uₖ(x,0)=kδ₀. The main question is wether the limit is still a solution of the above equation with an isolated singularity at (0,0), or a solution of the associated ordinary differential equation u'+h(t)uq=0 which blows-up at t=0.