is said to be [*globally exponentially stable on the set $\Omega$*]{} if for all $\varphi\in\Omega$ there exist $c>0$ and $d>0$ such that, for all $(t,m)\in[\overline{\tau},+\infty)\times[0,1]$, $$|N^{\varphi}(t,m)-\overline{N}(t,m)|\leq ce^{-d(t-\overline{\tau})}.$$ We will say [*globally stable*]{} (respectively [*globally exponentially stable*]{}) if $\Omega=C([0,\overline{\tau}]\times[0,1])$.

From the result of local stability obtained in Section \[sectionlocalstability\], we are able to establish a first result about global stability for the trivial solution of Equation (\[mildformulation\]). This result is presented in the following proposition.

\[stabgloblip\] Suppose that the mapping $x\mapsto x\beta(m,x)$ is Lipschitz continuous on $[0,+\infty)$ for all $m\in[0,1]$, with a Lipschitz constant $L$, such that $$\label{condstab}
L(2\tilde{\zeta}+1)<I.$$ Then the trivial solution $N\equiv 0$ of Problem (\[mildformulation\]) is globally exponentially stable.

We first show, as in the proof of Proposition \[localstability\], that for all $\varphi\in
C([0,\overline{\tau}]\times[0,1])$ and $(t,m)\in[0,+\infty)\times[0,1]$, $$|N^{\varphi}(t,m)| \leq ||\varphi||.$$ Then, we have (using the same arguments as in the proof of Theorem \[localexpstability\]) that, $$\lim_{t\to\infty}N^{\varphi}(t,m)=0 \quad \textrm{ exponentially},$$ for all $\varphi\in C([0,\overline{\tau}]\times[0,1])$ and $m\in[0,1]$.

The inequality $$L(2\tilde{\zeta}+1)<I,$$ is satisfied if $I$ or $\nu$ are large or if $L$ is small. Recall that $I$ and $\nu$ represent the mortality rates in the resting and the proliferating phase and that $L$ is a bound of the quantity of reintroduced cells. 