I have the following evolution PDE problem \begin{align} \frac{\partial f(x,t)}{\partial t} &= \frac{1}{2}\int_{y\leq x,x \in I} Q(x-y , y) f(x-y,t) f(y,t) dy - f(x,t) \int_{I} Q(x, y) f(y,t) dy \\ &= U(t,f(x,t))\\ f(x,0)&= f_0(x), \: \forall \: x \in I , \end{align}
Let us assume that $Q \in L^{\infty}(I \times I)$. Let $f \in F_{T}=F \times (0,T]$ for some fixed time $T>0$, a solution of the previous problem, where $F$ is the functional solution space and let $\phi \in V$ a test function, where $V$ is a test functional space to be defined satisfying regularity constraints.
I introduced a continuous trilinear form, $T$, on $F\times F \times V$ such that \begin{align} T(u,v,w)&= \frac{1}{2}\int_{I} \int_{I} \left[ w(x+y)-w(x)-w(y) \right] Q(x,y) u(x) v(y) dx dy , \: \forall \: (u,v,w) \in F \times F \times V , \end{align}
and the mapping $\mathbf{f}:[0,T] \to F$, defined by \begin{align*} \left[ \mathbf{f}(t)\right] (x):=f(x,t) \: \: (x \in I, \: t \in [0,T]) . \end{align*} I derived the weak formulation of this problem and I end up with
\begin{align} &\bigg\vert \text{Find} \: \mathbf{f} \in H^{1} \left( 0,T,L^{2}(I) \right) ,\: \text{such that:} \nonumber\\ &\bigg\vert i) \langle \mathbf{f}_t , \phi \rangle=T(\mathbf{f},\mathbf{f},\phi), \forall \phi \in L^{2}(I) \nonumber \\ & \bigg\vert ii) \mathbf{f}(0)=f_0 , \end{align}
where $H^{1} \left(0,T;L^{2}(I) \right)=\{ u; u \in L^{2}\left(0,T;L^{2}(I)\right), u_t \in L^{2}\left( 0,T;L^{2} (I)\right) \}$ ($u_t$: is the weak derivative of $u$ with respect to time).
Now I want to investigate the well-posedness of the PDE problem. Since I am working with a Hilbert space $(L^{2}(I))$, there is a nice result in page 136 of the book(Differential equations in abstract spaces, GE Ladas and V Lakshmikantham) which states that with some mild conditions on $U$, it is sufficient to show the well-posedness of the PDE problem (I stated before) if we show that $\exists \: M$ such that \begin{equation} Re \langle U(t,f) - U(t,g), f-g\rangle \geq M \mid \mid f-g \mid \mid^2,\: t_0 \le t \le t_0+\alpha,\: f,g \in \mathbf{X} , \end{equation}
where $\mathbf{X}$ is the solution space and in my case it is $L^2(I)$. I am still not able to get the desired inequality. Any help in this direction? Are there any other results that could be used to show the well-posedness?