Non-uniqueness and Uniqueness in the Cauchy Problem of Elliptic and Backward-Parabolic Equations

In this paper we consider the non-uniqueness and the uniqueness property for the solutions to the Cauchy problem for the operators Eu=∂2tu+∑k,l=1n∂xk(akl(t,x)∂xlu)+β(t,x)∂tu+∑m=1nbm(t,x)∂xmu+c(t,x)u and Pu=∂tu+∑k,l=1n∂xk(akl(t,x)∂xlu)+∑m=1nbm(t,x)∂xmu+c(t,x)u, where ∑nk,l=1akl(t,x)ξkξl|ξ|−2≥a0>0 . We study non-uniqueness and uniqueness in dependence of global and local regularity properties of the coefficients of the principal part. The global regularity will be ruled by the modulus of continuity of a kl on [0,T] while the local regularity will concern a bound on |∂ t a kl (t,x)| on every interval [ε,T]⊆(0,T]. By suitable counterexamples we show that our conditions seem to be sharp in many cases and we compare our statements with known results in the theory of hyperbolic Cauchy problems. We make also some remarks on continuous dependence for P .