The bounded slope condition for parabolic equations with time-dependent integrands

Abstract

In this paper, we study the Cauchy–Dirichlet problem {∂tu-div(Dξf(t,Du))=0inΩT,u=uoon∂PΩT, where Ω ⊂ Rn is a convex and bounded domain, f: [0 , T] × Rn→ R is L1 -integrable in time and convex in the second variable. Assuming that the initial and boundary datum uo: Ω ¯ → R satisfies the bounded slope condition, we prove the existence of a unique variational solution that is Lipschitz continuous in the space variable.