Consider the equation $$ u'(t) = (Fu)(t) $$ where $F \colon L^2(0,T;\mathbb R^n) \to L^2(0,T;\mathbb R^n)$ is a causal (Volterra type) nonlinear operator. It means that the value of $(Fu)(t_0)$ depends on values $u(t)$ for $t \in (0,t_0)$.

I need results about solvability of this problem. The book by Gajewski et al. contains some results when the operator $F$ fulfills Lipschitz condition: $$ (*)\;\; \|Fu - Fv\|_{L^2(0,T;\mathbb R^n)} \leq L\|u - v\|_{L^2(0,T;\mathbb R^n)}. $$

But if $Fu$ contains, for instance, square $u^2$ then it fulfills only local Lipshitz condition, i.e. $(*)$ is fulfilled only for $u, v \in B(u_0, r)$ where $B$ is a ball.

Where can I find results for solvability of this equation with local Lipschitz condition?