Quasi-linear parabolic systems in divergence form with weak monotonicity

Abstract

We consider the initial and boundary value problem for the quasi-linear parabolic system

$\begin{alignat*}{2} \frac{\partial u}{\partial t}-\mathrm{div} \sigma\big(x,t,u(x,t),Du(x,t)\big) &=f &\text{on }&\Omega\times(0,T),\\ u(x,t)&=0&\text{on }&\partial\Omega\times(0,T),\\ u(x,0)&=u_0(x)\quad&\text{on }&\Omega \end{alignat*}$

for a function u : Ω×[0,T)→ℝ^m with T>0. Here, f∈L^p′(0,T;W^p′(Ω;ℝ^m)) for some p∈(2n/2n+2),∞, and u₀∈L²(Ω, ℝ^m). We prove existence of a weak solution under classical regularity, growth, and coercivity conditions for σ but with only very mild monotonicity assumptions.