Heat escape

We study non-uniqueness of nonnegative solutions of the Cauchy or Dirichlet problem for parabolic equations on domains which are not relatively compact in Riemannian manifolds. By introducing the notion of heat escape, we give a general and sharp sufficient condition for the non-uniqueness: If there is a heat escape, then the Cauchy problem (or Dirichlet problem) has a positive solution with zero initial value (or zero initial-boundary value). We also show, under a general and sharp condition, uniqueness of nonnegative solutions to the Dirichlet problem for parabolic equations. Mathematics Subject Classification (1991):31C12, 35K20, 35J25, 35K15, 58G11, 58G03, 31C05