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For $p > 1$, consider the equation $$\langle u_t, v \rangle + \int_\Omega |\nabla u|^{p-2}\nabla u \nabla v = \langle f, v \rangle$$ $$u(0) = u_0$$ $$u|_{\partial\Omega} =0$$ for all $v \in W^{1,p}(\Omega)$, where $u_0 \in W_0^{1,p}\cap L^2$ and $f=f(x) \in W^{-1,q}(\Omega)$. This has a solution $u \in L^p(0,T;W^{1,p}_0)$ with $u_t \in L^q(0,T;W^{-1,q})$

If I assume additional smoothness on the data, can I get $u_t$ in a nicer space? Eg. $u_t \in L^s(0,T;L^r)$ for positive $s$ and $r$? I would like $u_t -\Delta_pu =f$ to hold pointwise a.e., basically.

Eg. take $p=q=2$. Then if $f \in L^2(0,T;L^2)$ and $u_0 \in H^1$ then $u_t \in L^2(0,T;L^2)$.

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  • $\begingroup$ I am not familiar with the p-heat equation, but for the stationary p-Laplace equation the best classical regularity for solutions is $C^{1,\alpha}$. The E-L equation of the energy $\int (|\nabla u|^2+\varepsilon)^{p/2}$ does hold point-wise and its solutions converge in $C^1$ to solutions of the original equation. Maybe something like this could be useful for you? $\endgroup$
    – Tommi
    Commented Nov 25, 2014 at 11:11
  • $\begingroup$ @TommiBrander thanks for the comment. I am not really familiar with optimal regularity but maybe that is an alternative approach. $\endgroup$
    – jamesC
    Commented Nov 25, 2014 at 17:52
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    $\begingroup$ I assume you mean the test functions to be time-dependent. If $f \equiv 0$, then $\partial_t u \in L^p$, if I remember correctly. I suspect something similar is true even more generally with non-zero, but smooth enough, $f$. The reference for this result is, however, very difficult to find. I think there is a note by Peter Lindqvist (NTNU) in some journal of the Norwegian Academy or something like that. $\endgroup$
    – Juhana Siljander
    Commented Jan 12, 2015 at 9:36

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