Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^n$ (or more generally a compact manifold). I'm interested in well-posedness (existence most importantly) of equations of the form
$$u_t(t) - \Delta A(u(t)) + b(t)u(t)= f(t)\quad\text{on $\Omega$}$$ $$u(0) = u_0$$ where $A:\mathbb{R} \to \mathbb{R}$ passes through the origin, is invertible, Lipschitz and monotone, $b \in L^\infty((0,T)\times \Omega)$ could be negative, and $f \in L^2(0,T;L^2)$. The weak formulation I am interested in is: $$\text{Find $u \in H^1(0,T;L^2) \cap L^2(0,T;H^1)$ with $u(0) = u_0$ such that}$$ $$\int_0^T \int_\Omega u_t(t)\varphi(t) + \int_0^T \int_\Omega \nabla A(u(t)) \nabla \varphi(t) + b(t)u(t)\varphi(t) = \int_0^T \int_\Omega f(t)\varphi(t)$$ $$\text{for all $\varphi \in L^2(0,T;H^1)$}$$
The main issue with using a Faedo-Galerkin method is that I am unable to obtain bound in $L^2(0,T;L^2)$ on the derivatives $\dot u_n$, which is necessary since by compactness, it would give a strongly convergent sequence $u_n \to u$. This is necessary to identify the limit $A(u_n(t)) \rightharpoonup A(u)$ via monotonicity. Obtaining a bound on $u_n$ in $L^2(0,T;H^1) \cap L^\infty(0,T;L^2)$ is no problem. In other words, I end up with the equation $$u_t(t) -\Delta \eta(t) + b(t)u(t) = f(t)$$ but am unable to identify $\eta = A(u).$ Please note that I have simplified the problem I am looking at so I do apologise if the conditions I gave above are too easy to obtain the bound on $\dot u_m$. I am mainly interested in techniques to solve problems where such a bound is not possible rather than specifics of the example.
I am wondering if there is a weaker notion of a weak solution that I should be looking at in this case. Or maybe there is a different method for existence without using monotonicity? Can anyone please advise me? Surely this type of problem is well-studied in the literature. Can anyone suggest me something to read? Or should I be looking at a different notion of a weak solution??
