Some parabolic problems with unbounded coefficients of nonhomogeneous rates

Abstract

We consider a 3--dimensional Cauchy problem for a parabolic equation where the diffusion matrix has two eigenvalues which diverge with order larger than 2 and one eigenvalue which diverges with order less than 2, with respect to $|x|$, as $|x|\to \infty$. Order 2 of divergence is the critical value below which uniqueness and above which non--uniqueness results are known to hold in the set of bounded functions. Hence we are in an intermediate case. However we prove a uniqueness result, in which the presence of first order terms is crucial. Shauder type estimates of solutions are given too. The problem is of interest in the study of plasma physics.