= Solution
At maximum expansion, $a_{\max}=A/k$. With $a_0=1$, the present <Friedmann equation> gives $kc^2=H_0^2(\Omega_{M,0}-1)$, so positive curvature requires $\Omega_{M,0}>1$. Taking $H_0>0$ for the presently expanding branch gives
$$
\boxed{a_{\max}=\frac{\Omega_{M,0}}{\Omega_{M,0}-1},\qquad
t_{\rm life}=\frac{\pi\Omega_{M,0}}{H_0(\Omega_{M,0}-1)^{3/2}},\qquad
q_0=\frac{\Omega_{M,0}}2.}
$$
The <lifetime of a closed dust universe> follows by setting $\theta=2\pi$. The <deceleration parameter> follows from the <Friedmann acceleration equation>, $\ddot a/a=-4\pi G\rho_M/3$. Thus this model always decelerates while expanding. Its total lifetime includes the elapsed age; it is not the remaining lifetime.
The <supernova discovery of accelerating expansion> does not support this dust-only recollapse model as a description of the observed expansion history. Already the 1998 and 1999 <Type Ia supernova> analyses favoured <accelerating expansion>, requiring ingredients beyond this zero-<cosmological constant>, positive-curvature dust model within the usual <FLRW metric> interpretation.<Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant by Adam Riess and others>[Abstract: evidence for present acceleration]{ref}<Measurements of Omega and Lambda from 42 High-Redshift Supernovae by Saul Perlmutter and others>[Abstract: positive cosmological constant preferred]{ref}
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