Positive Solutions of Reaction diffusion equations with super-linear absorption: universal bounds, uniqueness for the Cauchy problem, boundedness of stationary solutions

Consider classical solutions u ∈ C2(Rn×(0,∞))∩C(Rn × [0,∞)) to the parabolic reaction diffusion equation ut = Lu+ f(x, u), (x, t) ∈ Rn × (0,∞); u(x, 0) = g(x) ≥ 0, x ∈ Rn; u ≥ 0, where L = n∑ i,j=1 ai,j(x