On uniqueness and nonuniqueness of the positive Cauchy problem for parabolic equations with unbounded coefficients

Abstract

where ∂i = ∂/∂xi. A nonnegative solution u(x, t) of the equation Lu = 0 in XT = X × [0, T ) with a nonnegative initial data u(x, 0) = u0(x) is called a solution of the positive Cauchy problem. Such a solution can be thought of as the temperature field at the time t in an n-dimensional body which occupies the domain X . The coefficients aij(x) are the components of the matrix of the thermal conductivities while the function 1/ρ(x) is the heat capacity of a unit hypervolume. Hence nonnegative solutions of the Cauchy problem are natural objects to study. In particular, it is of interest to know whether the