Causal differential equation with local Lipschitz condition

Question

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?

Answer 1

Write the original equation as an integral equation $$ u(t) = u_{0}+\int_{0}^{t}F(u(s))ds. $$ Begin the standard Picard iteration, but don't worry about details of domain and range at this point: $$ u_{1}(t) = u_{0}+\int_{0}^{t}F(u_{0})ds,\\ u_{2}(t) = u_{0}+\int_{0}^{t}F(u_{1}(s))ds,\\ \cdots \\ u_{n+1}(t)=u_{0}+\int_{0}^{t}(F(u_{n}(s))ds. $$ Then, assume the iterates $u_{n}$ trace out paths in the region where $F$ is Lipschitz with constant $M$: $$ \|u_{n+1}(t)-u_{n}(t)\| \le \int_{0}^{t}\|F(u_{n}(s))-F(u_{n-1}(s))\|ds \\ \le M\int_{0}^{t}\|u_{n}(t_{n})-u_{n-1}(t_{n})\|dt_{n} \\ \le M^{2}\int_{0}^{t}\int_{0}^{t_{n}}\|u_{n-1}(t_{n-1})-u_{n-2}(t_{n-1})\|dt_{n-1}dt_{n} \\ \le M^{n}\int_{0}^{t}\int_{0}^{t_{n}}\cdots\int_{0}^{t_{2}}\|u_{1}(t_{1})-u_{0}\|dt_{1}dt_{2}\cdots dt_{n} \\ \le M^{n}\|F(u_{0})\|\int_{0}^{t}\int_{0}^{t_{n}}\cdots\int_{0}^{t_{2}}t_{1}dt_{1}dt_{2}\cdots dt_{n} \\ = M^{n}\frac{t^{n+1}}{(n+1)!}. $$ Therefore, $$ u_{n+1}(t)-u_{0} = \sum_{k=1}^{n+1}(u_{k}-u_{k-1}) $$ is uniformly bounded by $$ \|u_{n+1}(t)-u_{0}\| \le \sum_{k=1}^{n+1}\frac{t^{n+1}M^{n}}{(n+1)!} \le \frac{1}{M}(e^{tM}-1). $$ By choosing the interval $[0,T]$ small enough, you can guarantee that the iterates remain in the region where $F$ is locally Lipschitz with Lipschitz constant $M$, which justifies the above estimates and guarantees the uniform convergence of the iterates through the above telescoping series solution. So that's enough to get a local solution on $[0,T]$, where $T$ is chosen small enough that $\frac{1}{M}(e^{TM}-1) \le r$. You can then solve over $[T,T+T']$ for $v$ such that $v(T)=u(T)$, and continue, with some different Lipschitz constant $M'$ and radius $r'$ for the function $F$ in a neighborhood of $u(T)$. A connectedness and continuity argument should allow you to continue.