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

Abstract

Abstract In this paper, we study the Cauchy–Dirichlet problem $$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u - {\text {div}} \left( D_\xi f(t, Du)\right) = 0 &{} \quad \hbox {in} \ \Omega _T, \\ u = u_o &{} \quad \hbox { on} \ \partial _{\mathcal {P}} \Omega _T,\\ \end{array} \right. \end{aligned}$$ ∂ t u - div D ξ f ( t , D u ) = 0 in Ω T , u = u o on ∂ P Ω T , where $$\Omega \subset \mathbb {R}^n$$ Ω ⊂ R n is a convex and bounded domain, $$f:[0,T]\times {\mathbb {R}}^n \rightarrow {\mathbb {R}}$$ f : [ 0 , T ] × R n → R is $$L^1$$ L 1 -integrable in time and convex in the second variable. Assuming that the initial and boundary datum $$u_o:{\overline{\Omega }}\rightarrow {\mathbb {R}}$$ u o : Ω ¯ → R satisfies the bounded slope condition, we prove the existence of a unique variational solution that is Lipschitz continuous in the space variable.