Consider the problem of finding $u \in L^2(0,T;H^1)$ with $u' \in L^2(0,T;L^2)$ such that
$$\int\_0^T \int\_{\Omega}u'(t)\varphi(t) + \int\_0^T \int\_{\Omega}\nabla (F(u(t)))\nabla \varphi(t) = \int\_0^T \int\_\Omega f(t)\varphi(t)$$
where $F$ is differentiable and $F'$ bounded above and below away from $0$, and $F(0)=0$.


**I want to solve this problem without using a time-discretisation or semigroup approach**.


Now the only issue I have is with showing that $u' \in L^2(0,T;L^2)$. I can get $u' \in L^2(0,T;H^{-1})$ (so the first term of the equation is a duality pairing for this case) by using a fixed point approach.


Does anyone have references to where this is shown? Again I want to avoid time discretisation or semigroup proofs. A fixed point proof would nice for example. Assume whatever smoothness of right hand side $f$ as needed (eg. $L^2(0,T;L^2)$)



Existence of solution with $u' \in L^2(0,T;L^2)$ to a nonlinear parabolic PDE
Given your setting, you cannot hope to get the strong regularity $\partial\_t u\in L^2(0,T;L^2)$ for free. Indeed you're solving the generalized Porous Media Equation
$$
\partial\_tu=\Delta F(u)
$$
with zero Dirichlet boundary conditions. The fundamental reason is that $F(u)\in L^2(0,T;H^1\_0)$ is really the energy space for this type of quasilinear equations, so the corresponding "natural" space for $\partial\_tu$ is the dual $(L^2(0,T;H^1\_0))'=L^2(0,T;H^{-1})$. In other words: since your weak formulation involves $\int\_0^T\int\_\Omega\left<\nabla F(u),\nabla\varphi\right>$ for test functions in $L^2(0,T;H^1\_0)$, your time derivative can only be in the dual. Unless of course you can integrate by parts $\int\_0^T\int\_\Omega\left<\nabla F(u),\nabla\varphi\right>=-\int\_0^T\int\_\Omega (\Delta F(u))\varphi$ in order to act on $\varphi$ via $L^2(\Omega)$ product only, but in order to do so you have to prove that $\Delta F(u)\in L^2(0,T;L^2)$. This is highly non-trivial and also not true in general. For that you need further assumptions on your initial datum and nonlinearity. 


Since you're considering here the non-degenerate case $F'(u)>cst>0$ you can basically compare with the heat equation, i-e the best case scenario $F(u)=u$. In general for the HE you only get $\partial\_t u\in L^2(0,T;H^{-1})$ as a **global** regularity. Again for the HE, the global regularity strongly depends on your initial data: for example $\partial\_tu\in L^2(0,T;L^2)$ essentially requires $u\_0\in H^2$. But then you cannot simply use the energy methods or natural $H^1$ fixed point, since $H^2$ is not the natural space. The general strategy both for linear and non-linear equations is: 1) obtain existence of weak solutions in the natural energy space, and 2) prove that your energy solutions actually enjoy more regularity. Step 1 is usually the easy one (this is your fixed point), but step 2 is essentially a parabolic regularity result hence much more involved. There's no such thing as a free meal...


You should look at Vázquez's book [the porous medium equation] chapter 3 (in particular what he calls "generalized porous medium with "good" $\Phi$", which would be your $F$ here), or the classical reference [Ladyzenskaya, Solonnikov, Ural'ceva - Linear and quasilinear equations of parabolic type] (hard to read).


