Let $\Omega$ be a bounded smooth domain and let $p < 0$ be real. Suppose that $u, v \in L^2(0,T;H^1(\Omega) \cap L^2(0,T;L^2(\partial\Omega))$ with $|u|^{p+1} \in L^2(0,T;L^2(\partial\Omega))$ and likewise for $v$.

Suppose $u(0) \leq v(0)$ a.e and
$$-\int_0^T \int_{\partial\Omega} (|u|^pu - |v|^pv)\varphi' + \int_0^T \int_{\Omega}\nabla (u-v)\nabla \varphi \leq \int_{\partial\Omega}(|u(0)|^{p+1}-|v(0)|^{p+1})\varphi(0)$$
for all $\varphi \in C^1(0,T;L^2(\Omega)) \cap L^2(0,T;H^1(\Omega))$ with $\varphi \geq 0$ and $\varphi(T) = 0$.

How do I show that $u \leq v$ a.e?

It seems like one should test with $\varphi = (|u|^pu - |v|^pv)^+$ however this causes problems with the second term on the left..