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I'm looking for reference discussing the regularity of the weak solution $u$ to the equation $$u_t - \Delta \beta(t, u) = f$$ $$u(0) = u_0$$ where $\beta(t,\cdot)$ is a nonlinear function depending on time. Typically the solution would be $u \in C([0,T];H)$ with $\beta(u) \in L^2(0,T;H^1)$.

I am interested in regularity in the time derivative; I want $u_t \in L^1$ or $L^2$ (i.e., a function, not just a distribution). Can someone point out some references discussing this?

When $\beta$ is independent of time, one can use $L^1$ contraction argument to get such regularity, at least when $\beta$ is of porous medium type.

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  • $\begingroup$ What goes wrong if you try to use the same $L^1$ contraction argument used for $\beta$ independent of time? $\endgroup$
    – Deane Yang
    Commented Oct 12, 2015 at 20:03
  • $\begingroup$ @DeaneYang Let us fix $\beta$ to be PME nonlinearity. One thing we would need is estimate of the form $|u(t+h)-u(t)|_{L^1} \leq C|h|$. This follows since: if $u$ is a (weak solution) solution with initial data $u_0$, then $v(t):= \lambda u(\lambda^{m-1}t)$ is solution with initial data $\lambda u_0$. We pick $\lambda^{m-1}t$ such that it equals $t+h$, and then a simple argument gives the result, using the $L^1$ cts. dependence. If eg. we have a time-dependence, the function $v$ is no longer solution to the same problem (think of the weak formulation), it would be a sort of "rescaled" problem. $\endgroup$
    – Pace
    Commented Oct 12, 2015 at 20:51
  • $\begingroup$ What are you assuming about $\beta$? $\endgroup$
    – Deane Yang
    Commented Oct 12, 2015 at 21:09
  • $\begingroup$ $\beta(t,\cdot)$ (fixed $t$) can be degenerate nonlinearity. For $t \mapsto \beta(t)$, as smooth as necessary. $\endgroup$
    – Pace
    Commented Oct 13, 2015 at 7:40

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