Instantaneous blow-up for heat equations

Jerry Goldstein U of Memphis

Of concern is the existence of nontrivial positive solutions of du/dt = Lu+V (x)u, where L is a second order symmetric elliptic operator and the potential function V has singularities. In many cases there are sharp borderlines between good and bad behavior. The main cases are: x in R^{n} and L=Laplacian [Baras-Goldstein], the analogous problem on the Heisenberg group [Goldstein-Zhang], L uniformly elliptic but with bad coefficients [Goldstein-Zhang], and V highly oscillatory [Goldstein-Kombe-Zhang].