Existence for evolutionary Neumann problems with linear growth by stability results

Abstract. We are concerned with the Neumann type boundary value problem to parabolic systems ∂tu− div(Dξf(x,Du)) = −Dug(x, u), where u is vector-valued, f satisfies a linear growth condition and ξ 7→ f(x, ξ) is convex. We prove that variational solutions of such systems can be approximated by variational solutions to ∂tu− div(Dξf(x,Du)) = −Dug(x, u) with p > 1. This can be interpreted both as a stability and existence result for general flows with linear growth.