I want to derive the weak formulation of the following problem
\begin{align} \frac{\partial f(x,t)}{\partial t} &= \frac{1}{2}\int_{0}^{x} Q(x-y , y) f(x-y,t) f(y,t) dy - f(x,t) \int_{I} Q(x, y) f(y,t) dy, \forall \: x \in I\\ f(x,0)&= f_0(x), \: \forall \: x \in I, \end{align} where $I=[0,x_{max}]$ is a bounded interval.
I assume that $Q \in L^{\infty}(I \times I)$. Let $f \in U_{T}=U \times (0, T]$ for some fixed time $T>0$, a solution of the previous problem, where $U$ is the functional solution space. Let also $\phi \in V$ a test function, where $V$ is a test functional space to be defined satisfying regularity constraints.
I defined a mapping $\mathbf{f}:[0,T] \to U$ such that \begin{align*} \left[ \mathbf{f}(t)\right] (x):=f(x,t) \: \: (x \in I, \: t \in [0,T]) . \end{align*}
Then by multiplying the pde in the strong form by $\phi$ and integrating over the domain $I$, I showed that
\begin{align} \int_{I} \mathbf{f}_t \phi dx &= \frac{1}{2}\int_{I} \int_{I} \left[ \phi(x+y)-\phi(x)-\phi(y) \right] Q(x,y) [\mathbf{f}(t)](x) [\mathbf{f}(t)](y) dx dy. \end{align}
If I introduce a continuous trilinear form, $T$, on $U\times U \times V$ 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 U \times U \times V . \end{align}
Then using H\"older inequality, we obtain \begin{align} \mid T(\mathbf{f},\mathbf{f},\phi) \mid \leq C(I) \left( \int_{I} \int_{I} \left[ \phi(x+y)-\phi(x)-\phi(y) \right]^{2} dx dy \right)^{1/2} \left( \int_{I} \int_{I} \mathbf{f}^{4} dx dy \right)^{1/2} , \end{align} where $C(I)$ is a constant.
Therefore, the weak form holds for any $\phi \in L^{2}(I)$ and it is well defined for any $\mathbf{f}(t) \in L^{4}(I)$, with $\mathbf{f}_t \in (L^{4}(I))^{\star}$ (I mean the here the dual of $L^{4}(I)$.
Is it correct the last statement or we could use the Sobolev embedding of $H^1(I)$ into $L^4(I)$ to have better result?
Well, you have \begin{align} |T(f, f, \phi)| = &\ \mid \int dxdy\ [\phi(x+y)-\phi(x)-\phi(y)]Q(x, y)[f(x)f(y)] \mid \\ \leq&\ \left(\int dxdy\ [\phi(x+y)-\phi(x)-\phi(y)]^2\right)^{1/2}\left(\int dxdy\ |Q(x, y)|^2|f(x)|^2|f(y)|^2\right)^{1/2}\\ \leq&\ ||Q||_{L^\infty}\left(\int dxdy\ [\phi(x+y)-\phi(x)-\phi(y)]^2\right)^{1/2}\left( \int dx \ |f(x)|^2 \int dy\ |f(y)|^2\right)^{1/2}\\ =& \mid \ ||Q||_{L^\infty}\left(\int dxdy\ [\phi(x+y)-\phi(x)-\phi(y)]^2\right)^{1/2} \int dx \ |f(x)|^2 \mid . \end{align}