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In this work, we explore the eternal collapsing phenomenon of a stellarsystem (e.g., a star) within the framework of $f(R)$ gravity and investigatesome new aspects of the continued homogeneous gravitational collapse withperfect fluid distribution. The exact solutions of field equations have beenobtained in an independent way by the parameterization of the expansion scalar($\Theta$) governed by the interior spherically symmetric FLRW metric. Weimpose the Darmois junction condition required for the smooth matching of theinterior region to the Schwarzschild exterior metric across the boundaryhypersurface of the star. The junction conditions demand that the pressure isnon-vanishing at the boundary and is proportional to the non-linear terms of$f(R)$ gravity, and the mass function $m(t, r)$ is equal to Schwarzschild mass$M$. The eight massive stars, namely $Westerhout 49-2, BAT99-98, R136a1,R136a2, WR 24, Pismis 24-1$, $\lambda- Cephei$, and $\beta -Canis Majoris$ withtheir known astrophysical data (masses and radii) are used to estimate thenumerical values of the model parameters which allows us to study the solutionsnumerically and graphically. Here we have discussed two $f(R)$ gravity modelsdescribing the collapse phenomenon. The singularity analysis of models isdiscussed via the apparent horizon and we have shown that stars tend tocollapse for an infinite co-moving time in order to attain the singularity (aneternal collapsing phenomenon). We have also shown that our models satisfy theenergy conditions and stability requirements for stellar systems.

An Eternal gravitational collapse in $f(R)$ theory of gravity and their astrophysical implications