Strongly quasilinear parabolic systems

Abstract

Using the theory of Young measures, we prove the existence of solutions to a strongly quasilinear parabolic system \[\frac{\partial u}{\partial t}+A(u)=f,\]where \(A(u)=-\text{div}\,\sigma(x,t,u,Du)+\sigma_0(x,t,u,Du)\), \(\sigma(x,t,u,Du)\) and \(\sigma_0(x,t,u,Du)\)satisfy some conditions and \(f\in L^{p'}(0,T;W^{-1,p'}(\Omega;\mathbb{R}^m))\).