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Consider on $(0,T)\times \Omega$, $\Omega$ a bounded domain $$u_t(t,x) - a(u(t,x))\Delta u(t,x) = f(t,x)$$ $$u|_{\partial\Omega} = 0$$ where $a$ is real-valued and satisfies $C_1 \leq a(r) \leq C_2$ for all $r$ where both constants are strictly positive. Assume $f$ is as smooth as needed.

We know that $u \in L^2(0,T;H^1)$ with $u_t \in L^2(0,T;H^{-1})$.

Is it also true that $u_t \in L^2(0,T;L^2) = L^2(Q)$ (and thus also $\Delta u \in L^2(Q)$)?

I was under the impression that this is true only when one knows something about the differentiability of $t \mapsto a(u(t,x))$. However, I am informed this is not necessary an assumption. Could someone sketch a proof or refer me to a source? Thanks.

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  • $\begingroup$ What conditions do you put on your initial data ? If you allow Dirac measures, your conclusion does not hold. $\endgroup$
    – Hachino
    Feb 23, 2015 at 15:26
  • $\begingroup$ @Hachino Definitely more smooth than that, let's say $u_0 \in H^1(\Omega)$. $\endgroup$
    – C_Al
    Feb 23, 2015 at 15:28
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    $\begingroup$ There is a rather extensive literature on parabolic equations with nonsmooth coefficients. The work of Ladyzhenskaya and Uraltseva is a good place to start searching. $\endgroup$
    – Michael Renardy
    Feb 23, 2015 at 15:40

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