 Solution

Choose $R_0$ with $K\subset \{(y,w):|y|\le R_0,|w|\le R_0\}$ and let $A_0={\max }_{|w|\le R_0}|a(w)|$, which is finite by continuity. Every point transported from $K$ satisfiesHence anycontains the support of a function in the required phase space ball throughout $[0,T]$. Equivalently, the image of the compact set $[0,T]\times K$ under the characteristic flow map is compact. This uniform support control puts the solution in ${\mathcal{C}}_T$ and justifies the following differentiations under the integral.