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Let $\Phi(x,s)$ be a smooth, non-negative function with $\Phi(\cdot,0)=0$ and $\Phi$ increasing in $s$ and $\partial_s \Phi(x,0)=0$ and $\Omega$ be a bounded smooth domain.

Now assume there is a weak solution $s\in L^\infty((0,T)\times \Omega,[0,1])$ such that $\Phi(x,s)\in L^2(0,T;H^1_0(\Omega))$ and $\partial_t s \in L^2(0,T;H^ {-1}(\Omega))$ for the equation (GPME) $$\partial_t s =\Delta \Phi(x,s)$$ and some initial data $s_0\in L^\infty(\Omega,[0,1])$.

Is there an elementary/easy way to show that in fact $s\in C(0,T;L^2(\Omega))$.

Several papers are using this, though it is nowhere explicitly shown in their references. However, what is easy is that $s\in C(0,T;H^{-1}(\Omega))$. It is important so state that $s\not\in L^2(0,T;H^1_0(\Omega))$. In that case the proof can be found e.g. in Evans book. However in the case here, I don't see how to adapt the proof since it exploits the pairing of $H^{-1}$ and $H^1$.

Any suggestions?

Edit: If it is more common, a proof where $\Phi$ is independent of $x$ and $\Phi'$ Lipschitz would also suffice.

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