= Solution
If $f$ and $\widetilde f$ are two weak solutions with the same initial value and the required local time bound, their difference $w$ satisfies $w=\tau w$. By <linearity> this implies $w=\tau^nw$ for every $n$. On $[0,T]$ put $A_T=\sup_{s\leq T}\|w(s)\|_1<\infty$. The <factorial bound for a Volterra iterate> now gives
$$
\|w(t)\|_1\leq A_T\frac{(2T)^n}{n!},\qquad0\leq t\leq T.
$$
For fixed $T$, the factor tends to zero; its successive-term ratio is $2T/(n+1)$. Therefore $w(t)=0$ as an $L^1$ element at every time on this interval. Since $T$ is arbitrary, \b[the locally time-bounded $L^1$ weak solution is unique globally]. This proof uses precisely the additional condition requested, rather than assuming arbitrary pointwise-in-time integrability alone supplies a finite uniform bound.
Back to article page
