A version on Gronwall's inequality with interval $[-\tau, t]$

Question

Let $\tau$ and $T$ be positive constants and $u:[-\tau,T] \rightarrow \mathbb R$ be a Riemann integrable function. Assume that for all $t \in [0, T],$ $$u(t) \leq a+b \int_{t-\tau}^{t}u(s)ds.$$ Is it true that then $u(t) \leq a e^{b \tau}$ for for all $t \in [0, T]$?