  Let $(R,\frak m)$ be a Noetherian local ring of prime characteristic $p>0$,
and $t$ an integer such that $H_{\frak m}^j(R)/0^F_{H^j_{\frak m}(R)}$ has
finite length for all $j<t$. The aim of this paper is to show that there exists
an uniform bound for Frobenius test exponents of ideals generated by filter
regular sequences of length at most $t$.
