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.