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?

  • 2
    $\begingroup$ $u^2$ does not map $L^2$ to $L^2$. But there is no need to use $L^2$ as a function space. Examine the usual existence proof for ODEs. It is based on converting the ODE to a Volterra equation! $\endgroup$ – Michael Renardy Feb 16 '15 at 4:02
  • $\begingroup$ @MichaelRenardy Thank you. I need $L^2$ space because in my problem Lipschitz condition is fulfilled in $L^2$. $\endgroup$ – jokersobak Feb 16 '15 at 8:56

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