Let $\Omega$ be a bounded Lipschitz domain, and let $u\colon[-T,0]\times \Omega \to \mathbb{R}$ be a function with $u(-T)=0$.
Suppose that $$\sup_{-T \leq t \leq 0} \int_\Omega |(u(t)-k)^+|^2 + \int_{-T}^0 \int_\Omega |\nabla (u-k)^+|^2 \leq Ck^2\int_{-T}^0 \mu(\{u(t)|_{\partial\Omega} \geq k\})$$ holds for all $k \geq k_0$, where $\mu$ is the measure on $\partial\Omega$.
Does this imply that $$\operatorname{ess sup}_{t \in [-T,0]}u(t,x) \leq C\left(\int_0^T \int_\Omega |u(t)|^2 + C_1\right)^{1/2}$$ for a constant $C_1$ that may depend on $k_0$?
We know that $u \in L^\infty(-T,0);L^2(\Omega)) \cap L^2(-T,0;H^1(\Omega))$.
Basically, I am trying to show that the solution of a PDE stays bounded. There's probably some iterative argument to use here but I can't see it. Thanks for any help.
