The main result of the paper is the Aleksandrov–Bakelman–Pucci–Krylov–Tso (ABPKT) maximum principle for L^n+1 -viscosity sub/super-solutions of fully nonlinear uniformly parabolic equations uₜ+F (t, x, u, Du, D²u) =f (t, x) in (0, T], where {R}ⁿ. In this version of the maximum principle, the L^n+1 norm of f is taken over the so-called contact set. Equations have measurable and unbounded terms and we assume that the “drift” term which governs the dependence of F on the gradient variable is unbounded and is a function in L^n+2 (Q). Other versions of the ABPKT maximum principle for Lᵖ -viscosity solutions and its pointwise version are also obtained. We use the maximum principles to prove various properties of Lᵖ -viscosity solutions and build basic theory of Lᵖ -viscosity solutions for uniformly parabolic equations with the unbounded drift term.
Koike et al. (Fri,) studied this question.