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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.