Admissible conditions for parabolic equations degenerating at infinity

Well-posedness in $L^\infty(\mathbb{R}^n)$ $(n\ge3)$ of the Cauchy problem is studied for a class of linear parabolic equations with variable density. In view of degeneracy at infinity, some conditions at infinity are possibly needed to make the problem well-posed. Existence and uniqueness results are proved for bounded solutions that satisfy either Dirichlet or Neumann conditions at infinity.