Uniqueness and nonuniqueness of solutions to parabolic problems with singular coefficients

Uniqueness and nonuniqueness of solutions to the first initial-boundary value problem for degenerate semilinear parabolic equations, with possibly unbounded coefficients, are studied. Sub- and supersolutions of suitable auxiliary problems, such as the first exit time problem, are used to determine on which part of the boundary Dirichlet data must be given. As an application of the general results, we study uniqueness and nonuniqueness of bounded solutions to a semilinear Cauchy problem in the hyperbolic space ℍn using the Poincaré model.