Context is probably not important. What does it mean to say "$u_{a,b}$ depends $C^{m}$-ly on $a$ and $b$"? I know what it means to say a solution depends continuously on the data (change in solutions is less than or equal to change in data, informally).
Context: $u_{a,b} = u(at, x+bt)$ and $u(x,t)$ solves some PDE.