The Quantizing Effect of Potentials on the Critical Number of Reaction–Diffusion Equations

We give an almost complete characterization of the relation between the critical Fujita number of the equation Δu−Vu+up−ut=0 and the potentials V behaving like ±a/(1+d(x)b). Here a>0 and b is any real number. The only type of V not covered is when V(x) −a/(1+d(x)b) with b>2 and a is not small. It is interesting to note that when V=±a/(1+d(x)2) we are at a border line case where the critical exponent (depending on a) can vary from 1 to ∞. For all other V (except the ones the theorem does not cover), the critical Fujita number takes only three discrete values: 1, ∞, and 1+ in the Euclidean case. We also obtain global estimates of Schrödinger heat kernels and a Liouville theorem for a semilinear elliptic equation.